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1 |
Abstract
I Introduction
I-A Models
I-A1 Correlated Gaussian Mixture Models
I-A2 Correlated Contextual Stochastic Block Models
I-B Prior Works
I-B 1 Graph Matching
Matching correlated random graphs
Database alignment
Attributed graph matching
I-B2 Community Recovery in Correlated Random Graphs
I-C Our Contributions
I-D Notation
II Correlated Gaussian Mixture Models
II-A Exact Matching on Correlated Gaussian Mixture Models
II-B Exact Community Recovery in Correlated Gaussian Mixture Models
III Correlated Contextual Stochastic Block Models
III-A Exact Matching
III-B Exact Community Recovery
IV Outline of the Proof
IV-A Proof Sketch of Theorem 1
IV-B Proof Sketch of Theorem 5
V Discussion and Open Problems
V-1 Closing the information-theoretic gap for exact matching
V-2 Closing the information-theoretic gap for exact community recovery
V-3 Generalizing to more communities
V-4 Multiple correlated graphs
V-5 Efficient algorithms
VI Proof of Theorem 1: Achievability of Exact Matching in Correlated Gaussian Mixture Models
VII Proof of Theorem 2: Impossibility of Exact Matching in Correlated Gaussian Mixture Models
VIII Proof of Theorem 3: Achievability of Exact Community Recovery in Correlated Gaussian Mixture Models
IX Proof of Theorem 5 : Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-A The k𝑘kitalic_k-core matching and the proof of Theorem 5
IX-B Lemmas for the analysis of k𝑘kitalic_k-core matching
IX-C Proofs of Theorems 12 and 13 and Lemmas 5 and 6
X Proof of Theorem 6 : Impossibility of Exact Matching in Correlated Contextual Stochastic Block Models
X-A Proof of Lemma 9
XI Proof of Theorem 7: Achievability of Exact Community Recovery in Correlated Contextual Stochastic Block Models
XII Proof of Theorem 8 : Impossiblity of Exact Community Recovery in Correlated Contextual Stochastic Block Models
XII-A Proof of Lemma 11
XIII Proof of Lemma 1
XIV Technical Tools
References
| 480 |
Table of Contents
Abstract
I Introduction
I-A Models
I-A1 Correlated Gaussian Mixture Models
I-A2 Correlated Contextual Stochastic Block Models
I-B Prior Works
I-B 1 Graph Matching
Matching correlated random graphs
Database alignment
Attributed graph matching
I-B2 Community Recovery in Correlated Random Graphs
I-C Our Contributions
I-D Notation
II Correlated Gaussian Mixture Models
II-A Exact Matching on Correlated Gaussian Mixture Models
II-B Exact Community Recovery in Correlated Gaussian Mixture Models
III Correlated Contextual Stochastic Block Models
III-A Exact Matching
III-B Exact Community Recovery
IV Outline of the Proof
IV-A Proof Sketch of Theorem 1
IV-B Proof Sketch of Theorem 5
V Discussion and Open Problems
V-1 Closing the information-theoretic gap for exact matching
V-2 Closing the information-theoretic gap for exact community recovery
V-3 Generalizing to more communities
V-4 Multiple correlated graphs
V-5 Efficient algorithms
VI Proof of Theorem 1: Achievability of Exact Matching in Correlated Gaussian Mixture Models
VII Proof of Theorem 2: Impossibility of Exact Matching in Correlated Gaussian Mixture Models
VIII Proof of Theorem 3: Achievability of Exact Community Recovery in Correlated Gaussian Mixture Models
IX Proof of Theorem 5 : Achievability of Exact Matching in Correlated Contextual Stochastic Block Models
IX-A The k𝑘kitalic_k-core matching and the proof of Theorem 5
IX-B Lemmas for the analysis of k𝑘kitalic_k-core matching
IX-C Proofs of Theorems 12 and 13 and Lemmas 5 and 6
X Proof of Theorem 6 : Impossibility of Exact Matching in Correlated Contextual Stochastic Block Models
X-A Proof of Lemma 9
XI Proof of Theorem 7: Achievability of Exact Community Recovery in Correlated Contextual Stochastic Block Models
XII Proof of Theorem 8 : Impossiblity of Exact Community Recovery in Correlated Contextual Stochastic Block Models
XII-A Proof of Lemma 11
XIII Proof of Lemma 1
XIV Technical Tools
References
| 484 |
2 |
We study community detection in multiple networks whose nodes and edges are jointly correlated. This setting arises naturally in applications such as social platforms, where a shared set of users may exhibit both correlated friendship patterns and correlated attributes across different platforms. Extending the classical Stochastic Block Model (SBM) and its contextual counterpart (CSBM), we introduce the correlated CSBM, which incorporates structural and attribute correlations across graphs. To build intuition, we first analyze correlated Gaussian Mixture Models, wherein only correlated node attributes are available without edges, and identify the conditions under which an estimator minimizing the distance between attributes achieves exact matching of nodes across the two databases. For correlated CSBMs, we develop a two-step procedure that first applies k𝑘kitalic_k-core matching to most nodes using edge information, then refines the matching for the remaining unmatched nodes by leveraging their attributes with a distance-based estimator. We identify the conditions under which the algorithm recovers the exact node correspondence, enabling us to merge the correlated edges and average the correlated attributes for enhanced community detection. Crucially, by aligning and combining graphs, we identify regimes in which community detection is impossible in a single graph but becomes feasible when side information from correlated graphs is incorporated. Our results illustrate how the interplay between graph matching and community recovery can boost performance, broadening the scope of multi-graph, attribute-based community detection.
| 275 |
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
Abstract
We study community detection in multiple networks whose nodes and edges are jointly correlated. This setting arises naturally in applications such as social platforms, where a shared set of users may exhibit both correlated friendship patterns and correlated attributes across different platforms. Extending the classical Stochastic Block Model (SBM) and its contextual counterpart (CSBM), we introduce the correlated CSBM, which incorporates structural and attribute correlations across graphs. To build intuition, we first analyze correlated Gaussian Mixture Models, wherein only correlated node attributes are available without edges, and identify the conditions under which an estimator minimizing the distance between attributes achieves exact matching of nodes across the two databases. For correlated CSBMs, we develop a two-step procedure that first applies k𝑘kitalic_k-core matching to most nodes using edge information, then refines the matching for the remaining unmatched nodes by leveraging their attributes with a distance-based estimator. We identify the conditions under which the algorithm recovers the exact node correspondence, enabling us to merge the correlated edges and average the correlated attributes for enhanced community detection. Crucially, by aligning and combining graphs, we identify regimes in which community detection is impossible in a single graph but becomes feasible when side information from correlated graphs is incorporated. Our results illustrate how the interplay between graph matching and community recovery can boost performance, broadening the scope of multi-graph, attribute-based community detection.
| 291 |
3 |
Identifying community labels of nodes from a given graph or database–often referred to as community recovery or community detection–is a fundamental problem in network analysis, with wide-ranging applications in machine learning, social network analysis, and biology. The principal insight behind many community detection approaches is that nodes within the same community are typically more strongly connected or share similar attributes compared to nodes in different communities. However, real-world networks frequently deviate from such idealized patterns due to noise or the presence of anomalous nodes whose behaviors do not conform to typical intra-community connectivity or attribute similarities, thereby complicating the community recovery process.
A variety of probabilistic models has been developed to provide rigorous frameworks for community detection. Among these, the Stochastic Block Model (SBM) introduced by Holland, Laskey, and Leinhardt [1] remains one of the most widely studied. In the classical SBM, n𝑛nitalic_n nodes are partitioned into r𝑟ritalic_r communities, and edges form with probability p∈[0,1]𝑝01p\in[0,1]italic_p ∈ [ 0 , 1 ] between nodes in the same community, versus q∈[0,p)𝑞0𝑝q\in[0,p)italic_q ∈ [ 0 , italic_p ) between nodes in different communities. Such models capture community structures effectively in various settings–for example, social networks where edges represent friendships or interactions. In SBMs with p=alognn𝑝𝑎𝑛𝑛p=\frac{a\log n}{n}italic_p = divide start_ARG italic_a roman_log italic_n end_ARG start_ARG italic_n end_ARG, q=blognn𝑞𝑏𝑛𝑛q=\frac{b\log n}{n}italic_q = divide start_ARG italic_b roman_log italic_n end_ARG start_ARG italic_n end_ARG for constants a,b>0𝑎𝑏0a,b>0italic_a , italic_b > 0 and r=Θ(1)𝑟Θ1r=\Theta(1)italic_r = roman_Θ ( 1 ), it has been established in [2, 3, 4] that exact community recovery is information-theoretically achievable if and only if a−b>r𝑎𝑏𝑟\sqrt{a}-\sqrt{b}>\sqrt{r}square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG > square-root start_ARG italic_r end_ARG. Although SBMs incorporate only network structure, node attributes can play an equally important role in determining community memberships.
Since the standard SBM focuses on graph connectivity alone, it neglects potentially informative node attributes. To remedy this, the Contextual Stochastic Block Model (CSBM) includes node attributes alongside structural connectivity. For instance, in [5], the authors consider a two-community CSBM in which each node has a Gaussian-distributed attribute vector of dimension d𝑑ditalic_d, with mean either 𝝁𝝁{\boldsymbol{\mu}}bold_italic_μ or −𝝁𝝁-{\boldsymbol{\mu}}- bold_italic_μ (depending on the node’s community) and covariance 𝑰dsubscript𝑰𝑑{\boldsymbol{I}}_{d}bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT. This augmented framework leverages both graph structure and node attributes, leading to improved community recovery. Indeed, it is shown in [5] that the Signal-to-Noise Ratio (SNR), derived from both edges and attributes, governs the feasibility of exact community recovery, thereby demonstrating that combining these two sources of information can outperform methods relying on only one.
While CSBMs incorporate attributes in a single-network context, many real-world settings naturally feature multiple, correlated networks. For example, users often participate in more than one social platform (e.g., Facebook and LinkedIn), giving rise to correlated friendship relationships. However, due to privacy and security concerns, user identities are anonymized across different platforms, making it nontrivial to match corresponding users. This task, known as graph matching, is critical for leveraging correlated graphs for downstream tasks. Indeed, finding the exact correspondence between all nodes (i.e., exact matching) enables the construction of a combined graph by merging edges from both platforms, thereby facilitating more accurate community detection. Past research [6, 7] has shown that once exact matching is attainable in correlated SBMs, community recovery becomes easier than if only one graph were available. Moreover, Gaudio et al. [8] established precise information-theoretic limits for exact community recovery under correlated SBMs.
Building on these insights, this work addresses scenarios in which both edges and node attributes are correlated across multiple networks. For example, a user on platforms like Facebook and LinkedIn may exhibit similar friendship connections as well as comparable personal attributes. We posit that these correlated sources of information can significantly enhance community recovery. Concretely, we propose extending Contextual Stochastic Block Models to the correlated setting, resulting in correlated CSBMs.
As a preliminary step, we first investigate correlated Gaussian Mixture Models, which capture correlated attributes alone (i.e., without edges). By generalizing the database alignment techniques from [9, 10], we identify the conditions under which an estimator minimizing the distance between node attributes can reliably recover the underlying permutation π∗:[n]→[n]:subscript𝜋→delimited-[]𝑛delimited-[]𝑛\pi_{*}:[n]\to[n]italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : [ italic_n ] → [ italic_n ], even when community memberships are initially unknown. For correlated CSBMs, we introduce a two-step algorithm for exact matching: in the first step, we apply the k𝑘kitalic_k-core matching approach [11], which leverages edge information to recover matching for n−o(n)𝑛𝑜𝑛n-o(n)italic_n - italic_o ( italic_n ) nodes; in the second step, we employ the minimum-distance estimator using the node attributes of the remaining unmatched nodes, thereby finalizing the alignment. We derive conditions where this two-step algorithm achieves exact matching.
Once the node alignment is established, merging the correlated edges yields a denser graph structure, while averaging correlated node attributes increases the effective SNR. Crucially, by aligning and combining graphs, we identify regimes in which community detection is impossible in a single graph but becomes feasible when side information from correlated graphs is incorporated. This strategy offers deeper insights into how the interplay between graph matching and community detection can improve overall performance. To the best of our knowledge, we are the first to investigate community recovery in correlated graphs that incorporate correlated node attributes, thereby broadening the scope of existing research on multi-graph and attribute-based community detection.
| 1,455 |
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
Identifying community labels of nodes from a given graph or database–often referred to as community recovery or community detection–is a fundamental problem in network analysis, with wide-ranging applications in machine learning, social network analysis, and biology. The principal insight behind many community detection approaches is that nodes within the same community are typically more strongly connected or share similar attributes compared to nodes in different communities. However, real-world networks frequently deviate from such idealized patterns due to noise or the presence of anomalous nodes whose behaviors do not conform to typical intra-community connectivity or attribute similarities, thereby complicating the community recovery process.
A variety of probabilistic models has been developed to provide rigorous frameworks for community detection. Among these, the Stochastic Block Model (SBM) introduced by Holland, Laskey, and Leinhardt [1] remains one of the most widely studied. In the classical SBM, n𝑛nitalic_n nodes are partitioned into r𝑟ritalic_r communities, and edges form with probability p∈[0,1]𝑝01p\in[0,1]italic_p ∈ [ 0 , 1 ] between nodes in the same community, versus q∈[0,p)𝑞0𝑝q\in[0,p)italic_q ∈ [ 0 , italic_p ) between nodes in different communities. Such models capture community structures effectively in various settings–for example, social networks where edges represent friendships or interactions. In SBMs with p=alognn𝑝𝑎𝑛𝑛p=\frac{a\log n}{n}italic_p = divide start_ARG italic_a roman_log italic_n end_ARG start_ARG italic_n end_ARG, q=blognn𝑞𝑏𝑛𝑛q=\frac{b\log n}{n}italic_q = divide start_ARG italic_b roman_log italic_n end_ARG start_ARG italic_n end_ARG for constants a,b>0𝑎𝑏0a,b>0italic_a , italic_b > 0 and r=Θ(1)𝑟Θ1r=\Theta(1)italic_r = roman_Θ ( 1 ), it has been established in [2, 3, 4] that exact community recovery is information-theoretically achievable if and only if a−b>r𝑎𝑏𝑟\sqrt{a}-\sqrt{b}>\sqrt{r}square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG > square-root start_ARG italic_r end_ARG. Although SBMs incorporate only network structure, node attributes can play an equally important role in determining community memberships.
Since the standard SBM focuses on graph connectivity alone, it neglects potentially informative node attributes. To remedy this, the Contextual Stochastic Block Model (CSBM) includes node attributes alongside structural connectivity. For instance, in [5], the authors consider a two-community CSBM in which each node has a Gaussian-distributed attribute vector of dimension d𝑑ditalic_d, with mean either 𝝁𝝁{\boldsymbol{\mu}}bold_italic_μ or −𝝁𝝁-{\boldsymbol{\mu}}- bold_italic_μ (depending on the node’s community) and covariance 𝑰dsubscript𝑰𝑑{\boldsymbol{I}}_{d}bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT. This augmented framework leverages both graph structure and node attributes, leading to improved community recovery. Indeed, it is shown in [5] that the Signal-to-Noise Ratio (SNR), derived from both edges and attributes, governs the feasibility of exact community recovery, thereby demonstrating that combining these two sources of information can outperform methods relying on only one.
While CSBMs incorporate attributes in a single-network context, many real-world settings naturally feature multiple, correlated networks. For example, users often participate in more than one social platform (e.g., Facebook and LinkedIn), giving rise to correlated friendship relationships. However, due to privacy and security concerns, user identities are anonymized across different platforms, making it nontrivial to match corresponding users. This task, known as graph matching, is critical for leveraging correlated graphs for downstream tasks. Indeed, finding the exact correspondence between all nodes (i.e., exact matching) enables the construction of a combined graph by merging edges from both platforms, thereby facilitating more accurate community detection. Past research [6, 7] has shown that once exact matching is attainable in correlated SBMs, community recovery becomes easier than if only one graph were available. Moreover, Gaudio et al. [8] established precise information-theoretic limits for exact community recovery under correlated SBMs.
Building on these insights, this work addresses scenarios in which both edges and node attributes are correlated across multiple networks. For example, a user on platforms like Facebook and LinkedIn may exhibit similar friendship connections as well as comparable personal attributes. We posit that these correlated sources of information can significantly enhance community recovery. Concretely, we propose extending Contextual Stochastic Block Models to the correlated setting, resulting in correlated CSBMs.
As a preliminary step, we first investigate correlated Gaussian Mixture Models, which capture correlated attributes alone (i.e., without edges). By generalizing the database alignment techniques from [9, 10], we identify the conditions under which an estimator minimizing the distance between node attributes can reliably recover the underlying permutation π∗:[n]→[n]:subscript𝜋→delimited-[]𝑛delimited-[]𝑛\pi_{*}:[n]\to[n]italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : [ italic_n ] → [ italic_n ], even when community memberships are initially unknown. For correlated CSBMs, we introduce a two-step algorithm for exact matching: in the first step, we apply the k𝑘kitalic_k-core matching approach [11], which leverages edge information to recover matching for n−o(n)𝑛𝑜𝑛n-o(n)italic_n - italic_o ( italic_n ) nodes; in the second step, we employ the minimum-distance estimator using the node attributes of the remaining unmatched nodes, thereby finalizing the alignment. We derive conditions where this two-step algorithm achieves exact matching.
Once the node alignment is established, merging the correlated edges yields a denser graph structure, while averaging correlated node attributes increases the effective SNR. Crucially, by aligning and combining graphs, we identify regimes in which community detection is impossible in a single graph but becomes feasible when side information from correlated graphs is incorporated. This strategy offers deeper insights into how the interplay between graph matching and community detection can improve overall performance. To the best of our knowledge, we are the first to investigate community recovery in correlated graphs that incorporate correlated node attributes, thereby broadening the scope of existing research on multi-graph and attribute-based community detection.
| 1,472 |
4 |
We introduce two new models to capture correlations: correlated Gaussian Mixture Models, which focus on node attributes alone, and correlated Contextual Stochastic Block Models, which integrate both node attributes and graph structure.
| 40 |
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-A Models
We introduce two new models to capture correlations: correlated Gaussian Mixture Models, which focus on node attributes alone, and correlated Contextual Stochastic Block Models, which integrate both node attributes and graph structure.
| 61 |
5 |
First, we assign d𝑑ditalic_d dimensional features (or attributes) to n𝑛nitalic_n nodes. Let V1:=[n]assignsubscript𝑉1delimited-[]𝑛V_{1}:=[n]italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT := [ italic_n ] denote the set of nodes in the first database, and for each node i∈V1𝑖subscript𝑉1i\in V_{1}italic_i ∈ italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, the node attribute is given by
, 1 = 𝒙i=𝝁σi+𝒛i,subscript𝒙𝑖𝝁subscript𝜎𝑖subscript𝒛𝑖{\boldsymbol{x}}_{i}=\boldsymbol{\mu}\sigma_{i}+{\boldsymbol{z}}_{i},bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ,. , 2 = . , 3 = (1)
where 𝝁∈ℝd𝝁superscriptℝ𝑑\boldsymbol{\mu}\in\mathbb{R}^{d}bold_italic_μ ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT, σi∈{−1,+1}subscript𝜎𝑖11\sigma_{i}\in\{-1,+1\}italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ { - 1 , + 1 } and 𝒛i∼𝒩(0,𝑰d)similar-tosubscript𝒛𝑖𝒩0subscript𝑰𝑑{\boldsymbol{z}}_{i}\sim\mathcal{N}(0,{\boldsymbol{I}}_{d})bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∼ caligraphic_N ( 0 , bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) for all i∈[n]𝑖delimited-[]𝑛i\in[n]italic_i ∈ [ italic_n ].
Let 𝝈:={σi}i=1nassign𝝈superscriptsubscriptsubscript𝜎𝑖𝑖1𝑛{\boldsymbol{\sigma}}:=\{\sigma_{i}\}_{i=1}^{n}bold_italic_σ := { italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT be the vector of community labels associated with the nodes in V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT.
Next, for the node set V2:=[n]assignsubscript𝑉2delimited-[]𝑛V_{2}:=[n]italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT := [ italic_n ], we assign the attribute
, 1 = 𝒚i=𝝁σi+ρ𝒛i+1−ρ2𝒘i,subscript𝒚𝑖𝝁subscript𝜎𝑖𝜌subscript𝒛𝑖1superscript𝜌2subscript𝒘𝑖{\boldsymbol{y}}_{i}=\boldsymbol{\mu}\sigma_{i}+\rho{\boldsymbol{z}}_{i}+\sqrt%
{1-\rho^{2}}{\boldsymbol{w}}_{i},bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_ρ bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + square-root start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ,. , 2 = . , 3 = (2)
where ρ∈[0,1]𝜌01\rho\in[0,1]italic_ρ ∈ [ 0 , 1 ] and 𝒘i∼𝒩(0,𝑰d)similar-tosubscript𝒘𝑖𝒩0subscript𝑰𝑑{\boldsymbol{w}}_{i}\sim\mathcal{N}(0,{\boldsymbol{I}}_{d})bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∼ caligraphic_N ( 0 , bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) for all i∈[n]𝑖delimited-[]𝑛i\in[n]italic_i ∈ [ italic_n ].
Equivalently, for each i∈[n]𝑖delimited-[]𝑛i\in[n]italic_i ∈ [ italic_n ], the pair (𝒙i,𝒚i)subscript𝒙𝑖subscript𝒚𝑖({\boldsymbol{x}}_{i},{\boldsymbol{y}}_{i})( bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) can be represented as
, 1 = (𝒙i,𝒚i)∼𝒩([𝝁σi𝝁σi],Σd),similar-tosubscript𝒙𝑖subscript𝒚𝑖𝒩delimited-[]matrix𝝁subscript𝜎𝑖𝝁subscript𝜎𝑖subscriptΣ𝑑({\boldsymbol{x}}_{i},{\boldsymbol{y}}_{i})\sim\mathcal{N}\left(\left[\begin{%
matrix}\boldsymbol{\mu}\sigma_{i}&\boldsymbol{\mu}\sigma_{i}\end{matrix}\right%
],\Sigma_{d}\right),( bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ∼ caligraphic_N ( [ start_ARG start_ROW start_CELL bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL start_CELL bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] , roman_Σ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) ,. , 2 = . , 3 = (3)
where
, 1 = Σd:=[𝑰ddiag(ρ)diag(ρ)𝑰d].assignsubscriptΣ𝑑delimited-[]matrixsubscript𝑰𝑑diag𝜌diag𝜌subscript𝑰𝑑\Sigma_{d}:=\left[\begin{matrix}{\boldsymbol{I}}_{d}&\text{diag}(\rho)\\
\text{diag}(\rho)&{\boldsymbol{I}}_{d}\\
\end{matrix}\right].roman_Σ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT := [ start_ARG start_ROW start_CELL bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_CELL start_CELL diag ( italic_ρ ) end_CELL end_ROW start_ROW start_CELL diag ( italic_ρ ) end_CELL start_CELL bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] .. , 2 = . , 3 = (4)
| 1,732 |
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-A Models
I-A1 Correlated Gaussian Mixture Models
First, we assign d𝑑ditalic_d dimensional features (or attributes) to n𝑛nitalic_n nodes. Let V1:=[n]assignsubscript𝑉1delimited-[]𝑛V_{1}:=[n]italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT := [ italic_n ] denote the set of nodes in the first database, and for each node i∈V1𝑖subscript𝑉1i\in V_{1}italic_i ∈ italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, the node attribute is given by
, 1 = 𝒙i=𝝁σi+𝒛i,subscript𝒙𝑖𝝁subscript𝜎𝑖subscript𝒛𝑖{\boldsymbol{x}}_{i}=\boldsymbol{\mu}\sigma_{i}+{\boldsymbol{z}}_{i},bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ,. , 2 = . , 3 = (1)
where 𝝁∈ℝd𝝁superscriptℝ𝑑\boldsymbol{\mu}\in\mathbb{R}^{d}bold_italic_μ ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT, σi∈{−1,+1}subscript𝜎𝑖11\sigma_{i}\in\{-1,+1\}italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ { - 1 , + 1 } and 𝒛i∼𝒩(0,𝑰d)similar-tosubscript𝒛𝑖𝒩0subscript𝑰𝑑{\boldsymbol{z}}_{i}\sim\mathcal{N}(0,{\boldsymbol{I}}_{d})bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∼ caligraphic_N ( 0 , bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) for all i∈[n]𝑖delimited-[]𝑛i\in[n]italic_i ∈ [ italic_n ].
Let 𝝈:={σi}i=1nassign𝝈superscriptsubscriptsubscript𝜎𝑖𝑖1𝑛{\boldsymbol{\sigma}}:=\{\sigma_{i}\}_{i=1}^{n}bold_italic_σ := { italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT be the vector of community labels associated with the nodes in V1subscript𝑉1V_{1}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT.
Next, for the node set V2:=[n]assignsubscript𝑉2delimited-[]𝑛V_{2}:=[n]italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT := [ italic_n ], we assign the attribute
, 1 = 𝒚i=𝝁σi+ρ𝒛i+1−ρ2𝒘i,subscript𝒚𝑖𝝁subscript𝜎𝑖𝜌subscript𝒛𝑖1superscript𝜌2subscript𝒘𝑖{\boldsymbol{y}}_{i}=\boldsymbol{\mu}\sigma_{i}+\rho{\boldsymbol{z}}_{i}+\sqrt%
{1-\rho^{2}}{\boldsymbol{w}}_{i},bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_ρ bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + square-root start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ,. , 2 = . , 3 = (2)
where ρ∈[0,1]𝜌01\rho\in[0,1]italic_ρ ∈ [ 0 , 1 ] and 𝒘i∼𝒩(0,𝑰d)similar-tosubscript𝒘𝑖𝒩0subscript𝑰𝑑{\boldsymbol{w}}_{i}\sim\mathcal{N}(0,{\boldsymbol{I}}_{d})bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∼ caligraphic_N ( 0 , bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) for all i∈[n]𝑖delimited-[]𝑛i\in[n]italic_i ∈ [ italic_n ].
Equivalently, for each i∈[n]𝑖delimited-[]𝑛i\in[n]italic_i ∈ [ italic_n ], the pair (𝒙i,𝒚i)subscript𝒙𝑖subscript𝒚𝑖({\boldsymbol{x}}_{i},{\boldsymbol{y}}_{i})( bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) can be represented as
, 1 = (𝒙i,𝒚i)∼𝒩([𝝁σi𝝁σi],Σd),similar-tosubscript𝒙𝑖subscript𝒚𝑖𝒩delimited-[]matrix𝝁subscript𝜎𝑖𝝁subscript𝜎𝑖subscriptΣ𝑑({\boldsymbol{x}}_{i},{\boldsymbol{y}}_{i})\sim\mathcal{N}\left(\left[\begin{%
matrix}\boldsymbol{\mu}\sigma_{i}&\boldsymbol{\mu}\sigma_{i}\end{matrix}\right%
],\Sigma_{d}\right),( bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ∼ caligraphic_N ( [ start_ARG start_ROW start_CELL bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL start_CELL bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] , roman_Σ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) ,. , 2 = . , 3 = (3)
where
, 1 = Σd:=[𝑰ddiag(ρ)diag(ρ)𝑰d].assignsubscriptΣ𝑑delimited-[]matrixsubscript𝑰𝑑diag𝜌diag𝜌subscript𝑰𝑑\Sigma_{d}:=\left[\begin{matrix}{\boldsymbol{I}}_{d}&\text{diag}(\rho)\\
\text{diag}(\rho)&{\boldsymbol{I}}_{d}\\
\end{matrix}\right].roman_Σ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT := [ start_ARG start_ROW start_CELL bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_CELL start_CELL diag ( italic_ρ ) end_CELL end_ROW start_ROW start_CELL diag ( italic_ρ ) end_CELL start_CELL bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] .. , 2 = . , 3 = (4)
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We view these assigned attributes as two “databases,” which can be represented by the matrices X:=[𝒙1,𝒙2,…,𝒙n]⊤∈ℝn×dassign𝑋superscriptsubscript𝒙1subscript𝒙2…subscript𝒙𝑛topsuperscriptℝ𝑛𝑑X:=[{\boldsymbol{x}}_{1},{\boldsymbol{x}}_{2},\ldots,{\boldsymbol{x}}_{n}]^{%
\top}\in\mathbb{R}^{n\times d}italic_X := [ bold_italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , bold_italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × italic_d end_POSTSUPERSCRIPT and Y′:=[𝒚1,𝒚2,…,𝒚n]⊤∈ℝn×dassignsuperscript𝑌′superscriptsubscript𝒚1subscript𝒚2…subscript𝒚𝑛topsuperscriptℝ𝑛𝑑Y^{\prime}:=[{\boldsymbol{y}}_{1},{\boldsymbol{y}}_{2},\ldots,{\boldsymbol{y}}%
_{n}]^{\top}\in\mathbb{R}^{n\times d}italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := [ bold_italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , bold_italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × italic_d end_POSTSUPERSCRIPT. Finally, for a permutation π∗:[n]→[n]:subscript𝜋→delimited-[]𝑛delimited-[]𝑛\pi_{*}:[n]\to[n]italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : [ italic_n ] → [ italic_n ], define Y=[𝒚π∗(1),𝒚π∗(2),…,𝒚π∗(n)]⊤∈ℝn×d𝑌superscriptsubscript𝒚subscript𝜋1subscript𝒚subscript𝜋2…subscript𝒚subscript𝜋𝑛topsuperscriptℝ𝑛𝑑Y=[{\boldsymbol{y}}_{\pi_{*}(1)},{\boldsymbol{y}}_{\pi_{*}(2)},\ldots,{%
\boldsymbol{y}}_{\pi_{*}(n)}]^{\top}\in\mathbb{R}^{n\times d}italic_Y = [ bold_italic_y start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT , … , bold_italic_y start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_n ) end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × italic_d end_POSTSUPERSCRIPT. We assume π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT is uniformly distributed over Snsubscript𝑆𝑛S_{n}italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, the set of all permutations on n𝑛nitalic_n elements. The community label vectors for X𝑋Xitalic_X and Y𝑌Yitalic_Y are given by 𝝈1:=𝝈assignsuperscript𝝈1𝝈{\boldsymbol{\sigma}}^{1}:={\boldsymbol{\sigma}}bold_italic_σ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT := bold_italic_σ and 𝝈2:=𝝈∘π∗−1assignsuperscript𝝈2𝝈superscriptsubscript𝜋1{\boldsymbol{\sigma}}^{2}:={\boldsymbol{\sigma}}\circ\pi_{*}^{-1}bold_italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT := bold_italic_σ ∘ italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT, respectively. We write the resulting pair of databases as
, 1 = (X,Y)∼CGMMs(n,𝝁,d,ρ).similar-to𝑋𝑌CGMMs𝑛𝝁𝑑𝜌(X,\,Y)\;\sim\;\text{CGMMs}\bigl{(}n,\boldsymbol{\mu},d,\rho\bigr{)}.( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) .. , 2 =
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-A Models
I-A1 Correlated Gaussian Mixture Models
We view these assigned attributes as two “databases,” which can be represented by the matrices X:=[𝒙1,𝒙2,…,𝒙n]⊤∈ℝn×dassign𝑋superscriptsubscript𝒙1subscript𝒙2…subscript𝒙𝑛topsuperscriptℝ𝑛𝑑X:=[{\boldsymbol{x}}_{1},{\boldsymbol{x}}_{2},\ldots,{\boldsymbol{x}}_{n}]^{%
\top}\in\mathbb{R}^{n\times d}italic_X := [ bold_italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , bold_italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × italic_d end_POSTSUPERSCRIPT and Y′:=[𝒚1,𝒚2,…,𝒚n]⊤∈ℝn×dassignsuperscript𝑌′superscriptsubscript𝒚1subscript𝒚2…subscript𝒚𝑛topsuperscriptℝ𝑛𝑑Y^{\prime}:=[{\boldsymbol{y}}_{1},{\boldsymbol{y}}_{2},\ldots,{\boldsymbol{y}}%
_{n}]^{\top}\in\mathbb{R}^{n\times d}italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := [ bold_italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , bold_italic_y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × italic_d end_POSTSUPERSCRIPT. Finally, for a permutation π∗:[n]→[n]:subscript𝜋→delimited-[]𝑛delimited-[]𝑛\pi_{*}:[n]\to[n]italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : [ italic_n ] → [ italic_n ], define Y=[𝒚π∗(1),𝒚π∗(2),…,𝒚π∗(n)]⊤∈ℝn×d𝑌superscriptsubscript𝒚subscript𝜋1subscript𝒚subscript𝜋2…subscript𝒚subscript𝜋𝑛topsuperscriptℝ𝑛𝑑Y=[{\boldsymbol{y}}_{\pi_{*}(1)},{\boldsymbol{y}}_{\pi_{*}(2)},\ldots,{%
\boldsymbol{y}}_{\pi_{*}(n)}]^{\top}\in\mathbb{R}^{n\times d}italic_Y = [ bold_italic_y start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( 1 ) end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( 2 ) end_POSTSUBSCRIPT , … , bold_italic_y start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_n ) end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_n × italic_d end_POSTSUPERSCRIPT. We assume π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT is uniformly distributed over Snsubscript𝑆𝑛S_{n}italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT, the set of all permutations on n𝑛nitalic_n elements. The community label vectors for X𝑋Xitalic_X and Y𝑌Yitalic_Y are given by 𝝈1:=𝝈assignsuperscript𝝈1𝝈{\boldsymbol{\sigma}}^{1}:={\boldsymbol{\sigma}}bold_italic_σ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT := bold_italic_σ and 𝝈2:=𝝈∘π∗−1assignsuperscript𝝈2𝝈superscriptsubscript𝜋1{\boldsymbol{\sigma}}^{2}:={\boldsymbol{\sigma}}\circ\pi_{*}^{-1}bold_italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT := bold_italic_σ ∘ italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT, respectively. We write the resulting pair of databases as
, 1 = (X,Y)∼CGMMs(n,𝝁,d,ρ).similar-to𝑋𝑌CGMMs𝑛𝝁𝑑𝜌(X,\,Y)\;\sim\;\text{CGMMs}\bigl{(}n,\boldsymbol{\mu},d,\rho\bigr{)}.( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) .. , 2 =
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Let V=[n]𝑉delimited-[]𝑛V=[n]italic_V = [ italic_n ] be the vertex set, and let 𝝈:={σi}i=1nassign𝝈superscriptsubscriptsubscript𝜎𝑖𝑖1𝑛{\boldsymbol{\sigma}}:=\{\sigma_{i}\}_{i=1}^{n}bold_italic_σ := { italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT be the community labels, where each σi∈{−1,+1}subscript𝜎𝑖11\sigma_{i}\in\{-1,+1\}italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ { - 1 , + 1 } is drawn independently and uniformly at random. We generate a “parent” graph G∼SBM(n,p,q)similar-to𝐺SBM𝑛𝑝𝑞G\sim\text{SBM}(n,p,q)italic_G ∼ SBM ( italic_n , italic_p , italic_q ) with p,q∈[0,1]𝑝𝑞01p,q\in[0,1]italic_p , italic_q ∈ [ 0 , 1 ], p>q𝑝𝑞p>qitalic_p > italic_q, and q=Θ(p)𝑞Θ𝑝q=\Theta(p)italic_q = roman_Θ ( italic_p ) in the following manner. Partition V𝑉Vitalic_V into V+:={i∈[n]:σi=+1}assignsuperscript𝑉conditional-set𝑖delimited-[]𝑛subscript𝜎𝑖1V^{+}:=\{i\in[n]:\sigma_{i}=+1\}italic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT := { italic_i ∈ [ italic_n ] : italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = + 1 } and V−:={i∈[n]:σi=−1}assignsuperscript𝑉conditional-set𝑖delimited-[]𝑛subscript𝜎𝑖1V^{-}:=\{i\in[n]:\sigma_{i}=-1\}italic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT := { italic_i ∈ [ italic_n ] : italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = - 1 }. If σuσv=+1subscript𝜎𝑢subscript𝜎𝑣1\sigma_{u}\sigma_{v}=+1italic_σ start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT = + 1, an edge (u,v)𝑢𝑣(u,v)( italic_u , italic_v ) is placed with probability p𝑝pitalic_p; if σuσv=−1subscript𝜎𝑢subscript𝜎𝑣1\sigma_{u}\sigma_{v}=-1italic_σ start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT = - 1, an edge is placed with probability q𝑞qitalic_q. The graph G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is then obtained by sampling every edge of G𝐺Gitalic_G independently with probability s𝑠sitalic_s. Similarly, G2′subscriptsuperscript𝐺′2G^{\prime}_{2}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is generated by the same sampling procedure, ensuring that G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2′subscriptsuperscript𝐺′2G^{\prime}_{2}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are subgraphs of G𝐺Gitalic_G with
ℙ{(u,v)∈ℰ(G2′)|(u,v)∈ℰ(G1)}=sℙconditional-set𝑢𝑣ℰsubscriptsuperscript𝐺′2𝑢𝑣ℰsubscript𝐺1𝑠\mathbb{P}\left\{(u,v)\in\mathcal{E}(G^{\prime}_{2})|(u,v)\in\mathcal{E}(G_{1}%
)\right\}=sblackboard_P { ( italic_u , italic_v ) ∈ caligraphic_E ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) | ( italic_u , italic_v ) ∈ caligraphic_E ( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) } = italic_s
for all distinct u,v∈[n]𝑢𝑣delimited-[]𝑛u,v\in[n]italic_u , italic_v ∈ [ italic_n ], where ℰ(G)ℰ𝐺\mathcal{E}(G)caligraphic_E ( italic_G ) denotes the edge set of graph G𝐺Gitalic_G.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-A Models
I-A2 Correlated Contextual Stochastic Block Models
Let V=[n]𝑉delimited-[]𝑛V=[n]italic_V = [ italic_n ] be the vertex set, and let 𝝈:={σi}i=1nassign𝝈superscriptsubscriptsubscript𝜎𝑖𝑖1𝑛{\boldsymbol{\sigma}}:=\{\sigma_{i}\}_{i=1}^{n}bold_italic_σ := { italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT be the community labels, where each σi∈{−1,+1}subscript𝜎𝑖11\sigma_{i}\in\{-1,+1\}italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ { - 1 , + 1 } is drawn independently and uniformly at random. We generate a “parent” graph G∼SBM(n,p,q)similar-to𝐺SBM𝑛𝑝𝑞G\sim\text{SBM}(n,p,q)italic_G ∼ SBM ( italic_n , italic_p , italic_q ) with p,q∈[0,1]𝑝𝑞01p,q\in[0,1]italic_p , italic_q ∈ [ 0 , 1 ], p>q𝑝𝑞p>qitalic_p > italic_q, and q=Θ(p)𝑞Θ𝑝q=\Theta(p)italic_q = roman_Θ ( italic_p ) in the following manner. Partition V𝑉Vitalic_V into V+:={i∈[n]:σi=+1}assignsuperscript𝑉conditional-set𝑖delimited-[]𝑛subscript𝜎𝑖1V^{+}:=\{i\in[n]:\sigma_{i}=+1\}italic_V start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT := { italic_i ∈ [ italic_n ] : italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = + 1 } and V−:={i∈[n]:σi=−1}assignsuperscript𝑉conditional-set𝑖delimited-[]𝑛subscript𝜎𝑖1V^{-}:=\{i\in[n]:\sigma_{i}=-1\}italic_V start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT := { italic_i ∈ [ italic_n ] : italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = - 1 }. If σuσv=+1subscript𝜎𝑢subscript𝜎𝑣1\sigma_{u}\sigma_{v}=+1italic_σ start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT = + 1, an edge (u,v)𝑢𝑣(u,v)( italic_u , italic_v ) is placed with probability p𝑝pitalic_p; if σuσv=−1subscript𝜎𝑢subscript𝜎𝑣1\sigma_{u}\sigma_{v}=-1italic_σ start_POSTSUBSCRIPT italic_u end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT = - 1, an edge is placed with probability q𝑞qitalic_q. The graph G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is then obtained by sampling every edge of G𝐺Gitalic_G independently with probability s𝑠sitalic_s. Similarly, G2′subscriptsuperscript𝐺′2G^{\prime}_{2}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is generated by the same sampling procedure, ensuring that G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2′subscriptsuperscript𝐺′2G^{\prime}_{2}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are subgraphs of G𝐺Gitalic_G with
ℙ{(u,v)∈ℰ(G2′)|(u,v)∈ℰ(G1)}=sℙconditional-set𝑢𝑣ℰsubscriptsuperscript𝐺′2𝑢𝑣ℰsubscript𝐺1𝑠\mathbb{P}\left\{(u,v)\in\mathcal{E}(G^{\prime}_{2})|(u,v)\in\mathcal{E}(G_{1}%
)\right\}=sblackboard_P { ( italic_u , italic_v ) ∈ caligraphic_E ( italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) | ( italic_u , italic_v ) ∈ caligraphic_E ( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) } = italic_s
for all distinct u,v∈[n]𝑢𝑣delimited-[]𝑛u,v\in[n]italic_u , italic_v ∈ [ italic_n ], where ℰ(G)ℰ𝐺\mathcal{E}(G)caligraphic_E ( italic_G ) denotes the edge set of graph G𝐺Gitalic_G.
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Each node in G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2′subscriptsuperscript𝐺′2G^{\prime}_{2}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is assigned correlated Gaussian attributes {𝒙i}subscript𝒙𝑖\{{\boldsymbol{x}}_{i}\}{ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } and {𝒚i}subscript𝒚𝑖\{{\boldsymbol{y}}_{i}\}{ bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } as defined in (3), where 𝝁𝝁\boldsymbol{\mu}bold_italic_μ is uniformly distributed over the set
{𝝁∈ℝd:‖𝝁‖2=R}conditional-set𝝁superscriptℝ𝑑superscriptnorm𝝁2𝑅\bigl{\{}\boldsymbol{\mu}\in\mathbb{R}^{d}:\|\boldsymbol{\mu}\|^{2}=R\bigr{\}}{ bold_italic_μ ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT : ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_R }
for some R>0𝑅0R>0italic_R > 0. Finally, G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is obtained by permuting the nodes of G2′subscriptsuperscript𝐺′2G^{\prime}_{2}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT using π∗:[n]→[n]:subscript𝜋→delimited-[]𝑛delimited-[]𝑛\pi_{*}:[n]\to[n]italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : [ italic_n ] → [ italic_n ]. The community labels for G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are given by 𝝈1:=𝝈assignsuperscript𝝈1𝝈{\boldsymbol{\sigma}}^{1}:={\boldsymbol{\sigma}}bold_italic_σ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT := bold_italic_σ and 𝝈2:=𝝈∘π∗−1assignsuperscript𝝈2𝝈superscriptsubscript𝜋1{\boldsymbol{\sigma}}^{2}:={\boldsymbol{\sigma}}\circ\pi_{*}^{-1}bold_italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT := bold_italic_σ ∘ italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT, respectively. Denoting the database (attribute) matrices by X,Y′𝑋superscript𝑌′X,Y^{\prime}italic_X , italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, and Y𝑌Yitalic_Y, and the adjacency matrices by A,B′𝐴superscript𝐵′A,B^{\prime}italic_A , italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, and B𝐵Bitalic_B, we denote the resulting graphs by
, 1 = (G1,G2)∼CCSBMs(n,p,q,s;R,d,ρ).similar-tosubscript𝐺1subscript𝐺2CCSBMs𝑛𝑝𝑞𝑠𝑅𝑑𝜌(G_{1},G_{2})\;\sim\;\text{CCSBMs}\bigl{(}n,p,q,s;\,R,d,\rho\bigr{)}.( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CCSBMs ( italic_n , italic_p , italic_q , italic_s ; italic_R , italic_d , italic_ρ ) .. , 2 =
| 966 |
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-A Models
I-A2 Correlated Contextual Stochastic Block Models
Each node in G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2′subscriptsuperscript𝐺′2G^{\prime}_{2}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is assigned correlated Gaussian attributes {𝒙i}subscript𝒙𝑖\{{\boldsymbol{x}}_{i}\}{ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } and {𝒚i}subscript𝒚𝑖\{{\boldsymbol{y}}_{i}\}{ bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } as defined in (3), where 𝝁𝝁\boldsymbol{\mu}bold_italic_μ is uniformly distributed over the set
{𝝁∈ℝd:‖𝝁‖2=R}conditional-set𝝁superscriptℝ𝑑superscriptnorm𝝁2𝑅\bigl{\{}\boldsymbol{\mu}\in\mathbb{R}^{d}:\|\boldsymbol{\mu}\|^{2}=R\bigr{\}}{ bold_italic_μ ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT : ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_R }
for some R>0𝑅0R>0italic_R > 0. Finally, G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is obtained by permuting the nodes of G2′subscriptsuperscript𝐺′2G^{\prime}_{2}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT using π∗:[n]→[n]:subscript𝜋→delimited-[]𝑛delimited-[]𝑛\pi_{*}:[n]\to[n]italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT : [ italic_n ] → [ italic_n ]. The community labels for G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are given by 𝝈1:=𝝈assignsuperscript𝝈1𝝈{\boldsymbol{\sigma}}^{1}:={\boldsymbol{\sigma}}bold_italic_σ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT := bold_italic_σ and 𝝈2:=𝝈∘π∗−1assignsuperscript𝝈2𝝈superscriptsubscript𝜋1{\boldsymbol{\sigma}}^{2}:={\boldsymbol{\sigma}}\circ\pi_{*}^{-1}bold_italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT := bold_italic_σ ∘ italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT, respectively. Denoting the database (attribute) matrices by X,Y′𝑋superscript𝑌′X,Y^{\prime}italic_X , italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, and Y𝑌Yitalic_Y, and the adjacency matrices by A,B′𝐴superscript𝐵′A,B^{\prime}italic_A , italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, and B𝐵Bitalic_B, we denote the resulting graphs by
, 1 = (G1,G2)∼CCSBMs(n,p,q,s;R,d,ρ).similar-tosubscript𝐺1subscript𝐺2CCSBMs𝑛𝑝𝑞𝑠𝑅𝑑𝜌(G_{1},G_{2})\;\sim\;\text{CCSBMs}\bigl{(}n,p,q,s;\,R,d,\rho\bigr{)}.( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CCSBMs ( italic_n , italic_p , italic_q , italic_s ; italic_R , italic_d , italic_ρ ) .. , 2 =
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In Table I, we present various graph models–including our newly introduced ones–classified according to whether they incorporate community structure, edges, node attributes, or correlated graphs. Table II provides a summary of information-theoretic limits for graph matching in correlated graphs and community recovery in graphs with community structure, highlighting the performance gains achievable in our proposed models by incorporating correlated edges and/or node attributes.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-B Prior Works
In Table I, we present various graph models–including our newly introduced ones–classified according to whether they incorporate community structure, edges, node attributes, or correlated graphs. Table II provides a summary of information-theoretic limits for graph matching in correlated graphs and community recovery in graphs with community structure, highlighting the performance gains achievable in our proposed models by incorporating correlated edges and/or node attributes.
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One of the most extensively studied settings for graph matching is the correlated Erdős–Rényi (ER) model, first proposed in [15]. In this model, the parent graph G𝐺Gitalic_G is drawn from 𝒢(n,p)𝒢𝑛𝑝\mathcal{G}(n,p)caligraphic_G ( italic_n , italic_p ) (an ER graph), and G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2′subscriptsuperscript𝐺′2G^{\prime}_{2}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are obtained by independently sampling every edge of G𝐺Gitalic_G with probability s𝑠sitalic_s twice. Cullina and Kiyavash [16, 17] provided the first information-theoretic limits for exact matching, showing that exact matching is possible if nps2≥logn+ω(1)𝑛𝑝superscript𝑠2𝑛𝜔1nps^{2}\geq\log n+\omega(1)italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ roman_log italic_n + italic_ω ( 1 ) under the condition p≤O(1/(logn)3)𝑝𝑂1superscript𝑛3p\leq O(1/(\log n)^{3})italic_p ≤ italic_O ( 1 / ( roman_log italic_n ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ). More recently, Wu et al. [12] showed that exact matching remains feasible whenever p=o(1)𝑝𝑜1p=o(1)italic_p = italic_o ( 1 ) and nps2≥(1+ϵ)logn𝑛𝑝superscript𝑠21italic-ϵ𝑛nps^{2}\geq(1+\epsilon)\log nitalic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + italic_ϵ ) roman_log italic_n for any fixed ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0. However, these proofs rely on checking all permutations, yielding time complexity on the order of Θ(n!)Θ𝑛\Theta(n!)roman_Θ ( italic_n ! ). Consequently, a significant effort has focused on more efficient algorithms. Quasi-polynomial time (nO(logn)superscript𝑛𝑂𝑛n^{O(\log n)}italic_n start_POSTSUPERSCRIPT italic_O ( roman_log italic_n ) end_POSTSUPERSCRIPT) approaches were proposed in [18, 19], while polynomial-time algorithms in [20, 21, 22] achieve exact matching under s=1−o(1)𝑠1𝑜1s=1-o(1)italic_s = 1 - italic_o ( 1 ). Recently, the first polynomial-time algorithms for constant correlation s≥α𝑠𝛼s\geq\alphaitalic_s ≥ italic_α (for a suitable constant α𝛼\alphaitalic_α) appeared in [23, 24], using subgraph counting or large-neighborhood statistics.
Graph matching under correlated Stochastic Block Models (SBMs), where the parent graph is an SBM, has also been investigated [25, 26]. Assuming known community labels in each graph, Onaran et al. [25] showed that exact matching is possible when s(1−1−s2)p+q2≥3logn𝑠11superscript𝑠2𝑝𝑞23𝑛s(1-\sqrt{1-s^{2}})\frac{p+q}{2}\geq 3\log nitalic_s ( 1 - square-root start_ARG 1 - italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG ≥ 3 roman_log italic_n for two communities. Cullina et al. [26] extended this to r𝑟ritalic_r communities, where p=alognn𝑝𝑎𝑛𝑛p=\frac{a\log n}{n}italic_p = divide start_ARG italic_a roman_log italic_n end_ARG start_ARG italic_n end_ARG and q=blognn𝑞𝑏𝑛𝑛q=\frac{b\log n}{n}italic_q = divide start_ARG italic_b roman_log italic_n end_ARG start_ARG italic_n end_ARG, demonstrating that exact matching holds if s2a+(r−1)br>2superscript𝑠2𝑎𝑟1𝑏𝑟2s^{2}\frac{a+(r-1)b}{r}>2italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_a + ( italic_r - 1 ) italic_b end_ARG start_ARG italic_r end_ARG > 2. Notably, in the special case p=q𝑝𝑞p=qitalic_p = italic_q, correlated SBMs reduce to correlated ER graphs; even under known labels, the bounds in [25, 26] differ from the information-theoretic limit in the correlated ER setting. Rácz and Sridhar [6] refined these results for r=2𝑟2r=2italic_r = 2, proving that exact matching is possible if s2a+b2>1superscript𝑠2𝑎𝑏21s^{2}\,\frac{a+b}{2}>1italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_a + italic_b end_ARG start_ARG 2 end_ARG > 1. Yang and Chung [7] generalized these findings to SBMs with r𝑟ritalic_r communities, showing that exact matching holds if ns2p+(r−1)qr≥(1+ϵ)logn𝑛superscript𝑠2𝑝𝑟1𝑞𝑟1italic-ϵ𝑛ns^{2}\,\frac{p+(r-1)q}{r}\geq(1+\epsilon)\log nitalic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + ( italic_r - 1 ) italic_q end_ARG start_ARG italic_r end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n, under mild assumptions. As before, achieving this bound requires a time complexity of Θ(n!)Θ𝑛\Theta(n!)roman_Θ ( italic_n ! ). Yang et al. [27] designed a polynomial-time algorithm under constant correlation when community labels are known, and Chai and Rácz [28] recently devised a polynomial-time method that obviates label information.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-B Prior Works
I-B1 Graph Matching
Matching correlated random graphs
One of the most extensively studied settings for graph matching is the correlated Erdős–Rényi (ER) model, first proposed in [15]. In this model, the parent graph G𝐺Gitalic_G is drawn from 𝒢(n,p)𝒢𝑛𝑝\mathcal{G}(n,p)caligraphic_G ( italic_n , italic_p ) (an ER graph), and G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2′subscriptsuperscript𝐺′2G^{\prime}_{2}italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are obtained by independently sampling every edge of G𝐺Gitalic_G with probability s𝑠sitalic_s twice. Cullina and Kiyavash [16, 17] provided the first information-theoretic limits for exact matching, showing that exact matching is possible if nps2≥logn+ω(1)𝑛𝑝superscript𝑠2𝑛𝜔1nps^{2}\geq\log n+\omega(1)italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ roman_log italic_n + italic_ω ( 1 ) under the condition p≤O(1/(logn)3)𝑝𝑂1superscript𝑛3p\leq O(1/(\log n)^{3})italic_p ≤ italic_O ( 1 / ( roman_log italic_n ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ). More recently, Wu et al. [12] showed that exact matching remains feasible whenever p=o(1)𝑝𝑜1p=o(1)italic_p = italic_o ( 1 ) and nps2≥(1+ϵ)logn𝑛𝑝superscript𝑠21italic-ϵ𝑛nps^{2}\geq(1+\epsilon)\log nitalic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + italic_ϵ ) roman_log italic_n for any fixed ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0. However, these proofs rely on checking all permutations, yielding time complexity on the order of Θ(n!)Θ𝑛\Theta(n!)roman_Θ ( italic_n ! ). Consequently, a significant effort has focused on more efficient algorithms. Quasi-polynomial time (nO(logn)superscript𝑛𝑂𝑛n^{O(\log n)}italic_n start_POSTSUPERSCRIPT italic_O ( roman_log italic_n ) end_POSTSUPERSCRIPT) approaches were proposed in [18, 19], while polynomial-time algorithms in [20, 21, 22] achieve exact matching under s=1−o(1)𝑠1𝑜1s=1-o(1)italic_s = 1 - italic_o ( 1 ). Recently, the first polynomial-time algorithms for constant correlation s≥α𝑠𝛼s\geq\alphaitalic_s ≥ italic_α (for a suitable constant α𝛼\alphaitalic_α) appeared in [23, 24], using subgraph counting or large-neighborhood statistics.
Graph matching under correlated Stochastic Block Models (SBMs), where the parent graph is an SBM, has also been investigated [25, 26]. Assuming known community labels in each graph, Onaran et al. [25] showed that exact matching is possible when s(1−1−s2)p+q2≥3logn𝑠11superscript𝑠2𝑝𝑞23𝑛s(1-\sqrt{1-s^{2}})\frac{p+q}{2}\geq 3\log nitalic_s ( 1 - square-root start_ARG 1 - italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG ≥ 3 roman_log italic_n for two communities. Cullina et al. [26] extended this to r𝑟ritalic_r communities, where p=alognn𝑝𝑎𝑛𝑛p=\frac{a\log n}{n}italic_p = divide start_ARG italic_a roman_log italic_n end_ARG start_ARG italic_n end_ARG and q=blognn𝑞𝑏𝑛𝑛q=\frac{b\log n}{n}italic_q = divide start_ARG italic_b roman_log italic_n end_ARG start_ARG italic_n end_ARG, demonstrating that exact matching holds if s2a+(r−1)br>2superscript𝑠2𝑎𝑟1𝑏𝑟2s^{2}\frac{a+(r-1)b}{r}>2italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_a + ( italic_r - 1 ) italic_b end_ARG start_ARG italic_r end_ARG > 2. Notably, in the special case p=q𝑝𝑞p=qitalic_p = italic_q, correlated SBMs reduce to correlated ER graphs; even under known labels, the bounds in [25, 26] differ from the information-theoretic limit in the correlated ER setting. Rácz and Sridhar [6] refined these results for r=2𝑟2r=2italic_r = 2, proving that exact matching is possible if s2a+b2>1superscript𝑠2𝑎𝑏21s^{2}\,\frac{a+b}{2}>1italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_a + italic_b end_ARG start_ARG 2 end_ARG > 1. Yang and Chung [7] generalized these findings to SBMs with r𝑟ritalic_r communities, showing that exact matching holds if ns2p+(r−1)qr≥(1+ϵ)logn𝑛superscript𝑠2𝑝𝑟1𝑞𝑟1italic-ϵ𝑛ns^{2}\,\frac{p+(r-1)q}{r}\geq(1+\epsilon)\log nitalic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + ( italic_r - 1 ) italic_q end_ARG start_ARG italic_r end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n, under mild assumptions. As before, achieving this bound requires a time complexity of Θ(n!)Θ𝑛\Theta(n!)roman_Θ ( italic_n ! ). Yang et al. [27] designed a polynomial-time algorithm under constant correlation when community labels are known, and Chai and Rácz [28] recently devised a polynomial-time method that obviates label information.
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Database alignment [9, 29, 30, 31] addresses the problem of finding a one-to-one correspondence between nodes in two “databases,” where each node is associated with correlated attributes. Similar to graph matching, various models have been proposed, among which the correlated Gaussian database model is popular. In this model, each pair of corresponding nodes (𝒙i,𝒚i)subscript𝒙𝑖subscript𝒚𝑖({\boldsymbol{x}}_{i},{\boldsymbol{y}}_{i})( bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) is drawn i.i.d. from 𝒩(𝝁,Σd)𝒩𝝁subscriptΣ𝑑\mathcal{N}(\boldsymbol{\mu},\Sigma_{d})caligraphic_N ( bold_italic_μ , roman_Σ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ), where 𝝁∈ℝ2d𝝁superscriptℝ2𝑑\boldsymbol{\mu}\in\mathbb{R}^{2d}bold_italic_μ ∈ blackboard_R start_POSTSUPERSCRIPT 2 italic_d end_POSTSUPERSCRIPT and
Σd=[𝑰ddiag(ρ)diag(ρ)𝑰d].subscriptΣ𝑑matrixsubscript𝑰𝑑diag𝜌diag𝜌subscript𝑰𝑑\Sigma_{d}=\begin{bmatrix}{\boldsymbol{I}}_{d}&\text{diag}(\rho)\\
\text{diag}(\rho)&{\boldsymbol{I}}_{d}\end{bmatrix}.roman_Σ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = [ start_ARG start_ROW start_CELL bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_CELL start_CELL diag ( italic_ρ ) end_CELL end_ROW start_ROW start_CELL diag ( italic_ρ ) end_CELL start_CELL bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] .
Dai et al. [9] showed that exact alignment is possible if d4log11−ρ2≥logn+ω(1)𝑑411superscript𝜌2𝑛𝜔1\tfrac{d}{4}\log\tfrac{1}{1-\rho^{2}}\geq\log n+\omega(1)divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ roman_log italic_n + italic_ω ( 1 ). Their method uses the maximum a posteriori (MAP) estimator, with a time complexity of O(n2d+n3)𝑂superscript𝑛2𝑑superscript𝑛3O(n^{2}d+n^{3})italic_O ( italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d + italic_n start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ).
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-B Prior Works
I-B1 Graph Matching
Database alignment
Database alignment [9, 29, 30, 31] addresses the problem of finding a one-to-one correspondence between nodes in two “databases,” where each node is associated with correlated attributes. Similar to graph matching, various models have been proposed, among which the correlated Gaussian database model is popular. In this model, each pair of corresponding nodes (𝒙i,𝒚i)subscript𝒙𝑖subscript𝒚𝑖({\boldsymbol{x}}_{i},{\boldsymbol{y}}_{i})( bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) is drawn i.i.d. from 𝒩(𝝁,Σd)𝒩𝝁subscriptΣ𝑑\mathcal{N}(\boldsymbol{\mu},\Sigma_{d})caligraphic_N ( bold_italic_μ , roman_Σ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ), where 𝝁∈ℝ2d𝝁superscriptℝ2𝑑\boldsymbol{\mu}\in\mathbb{R}^{2d}bold_italic_μ ∈ blackboard_R start_POSTSUPERSCRIPT 2 italic_d end_POSTSUPERSCRIPT and
Σd=[𝑰ddiag(ρ)diag(ρ)𝑰d].subscriptΣ𝑑matrixsubscript𝑰𝑑diag𝜌diag𝜌subscript𝑰𝑑\Sigma_{d}=\begin{bmatrix}{\boldsymbol{I}}_{d}&\text{diag}(\rho)\\
\text{diag}(\rho)&{\boldsymbol{I}}_{d}\end{bmatrix}.roman_Σ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = [ start_ARG start_ROW start_CELL bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_CELL start_CELL diag ( italic_ρ ) end_CELL end_ROW start_ROW start_CELL diag ( italic_ρ ) end_CELL start_CELL bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ] .
Dai et al. [9] showed that exact alignment is possible if d4log11−ρ2≥logn+ω(1)𝑑411superscript𝜌2𝑛𝜔1\tfrac{d}{4}\log\tfrac{1}{1-\rho^{2}}\geq\log n+\omega(1)divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ roman_log italic_n + italic_ω ( 1 ). Their method uses the maximum a posteriori (MAP) estimator, with a time complexity of O(n2d+n3)𝑂superscript𝑛2𝑑superscript𝑛3O(n^{2}d+n^{3})italic_O ( italic_n start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d + italic_n start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ).
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In many social networks, users (nodes) have both connections (edges) and personal attributes. The attributed graph alignment problem aims to match nodes across two correlated graphs while exploiting both edge structure and node features. In the correlated Gaussian-attributed ER model [13], the edges come from correlated ER graphs, and node attributes come from correlated Gaussian databases. It was shown that exact matching is possible if
, 1 = nps2+d4log11−ρ2≥(1+ϵ)logn,𝑛𝑝superscript𝑠2𝑑411superscript𝜌21italic-ϵ𝑛nps^{2}+\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1+\epsilon)\log n,italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n ,. , 2 = . , 3 = (5)
indicating that the effective SNR is an additive combination of edge- and attribute-based signals.
Zhang et al. [32] introduced an alternative attributed ER pair model with n𝑛nitalic_n user nodes and m𝑚mitalic_m attribute nodes, assuming that the m𝑚mitalic_m attribute nodes are pre-aligned. Edges between user nodes appear with probability p𝑝pitalic_p, while edges between users and attribute nodes appear with probability q𝑞qitalic_q. Similar to the correlated ER framework, edges are independently subsampled with probabilities spsubscript𝑠𝑝s_{p}italic_s start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT and sqsubscript𝑠𝑞s_{q}italic_s start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT for user-user and user-attribute edges, respectively, and a permutation π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT is applied to yield G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Exact matching is possible if npsp2+mqsq2≥logn+ω(1)𝑛𝑝superscriptsubscript𝑠𝑝2𝑚𝑞superscriptsubscript𝑠𝑞2𝑛𝜔1nps_{p}^{2}+mqs_{q}^{2}\geq\log n+\omega(1)italic_n italic_p italic_s start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_m italic_q italic_s start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ roman_log italic_n + italic_ω ( 1 ). Polynomial-time algorithms for recovering π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT in this setting have been explored in [33].
| 746 |
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-B Prior Works
I-B1 Graph Matching
Attributed graph matching
In many social networks, users (nodes) have both connections (edges) and personal attributes. The attributed graph alignment problem aims to match nodes across two correlated graphs while exploiting both edge structure and node features. In the correlated Gaussian-attributed ER model [13], the edges come from correlated ER graphs, and node attributes come from correlated Gaussian databases. It was shown that exact matching is possible if
, 1 = nps2+d4log11−ρ2≥(1+ϵ)logn,𝑛𝑝superscript𝑠2𝑑411superscript𝜌21italic-ϵ𝑛nps^{2}+\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1+\epsilon)\log n,italic_n italic_p italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n ,. , 2 = . , 3 = (5)
indicating that the effective SNR is an additive combination of edge- and attribute-based signals.
Zhang et al. [32] introduced an alternative attributed ER pair model with n𝑛nitalic_n user nodes and m𝑚mitalic_m attribute nodes, assuming that the m𝑚mitalic_m attribute nodes are pre-aligned. Edges between user nodes appear with probability p𝑝pitalic_p, while edges between users and attribute nodes appear with probability q𝑞qitalic_q. Similar to the correlated ER framework, edges are independently subsampled with probabilities spsubscript𝑠𝑝s_{p}italic_s start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT and sqsubscript𝑠𝑞s_{q}italic_s start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT for user-user and user-attribute edges, respectively, and a permutation π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT is applied to yield G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Exact matching is possible if npsp2+mqsq2≥logn+ω(1)𝑛𝑝superscriptsubscript𝑠𝑝2𝑚𝑞superscriptsubscript𝑠𝑞2𝑛𝜔1nps_{p}^{2}+mqs_{q}^{2}\geq\log n+\omega(1)italic_n italic_p italic_s start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_m italic_q italic_s start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ roman_log italic_n + italic_ω ( 1 ). Polynomial-time algorithms for recovering π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT in this setting have been explored in [33].
| 779 |
13 |
Rácz and Sridhar [6] first investigated exact community recovery in the presence of two or more correlated networks. Focusing on correlated SBMs with p=alognn𝑝𝑎𝑛𝑛p=\frac{a\log n}{n}italic_p = divide start_ARG italic_a roman_log italic_n end_ARG start_ARG italic_n end_ARG and q=blognn𝑞𝑏𝑛𝑛q=\frac{b\log n}{n}italic_q = divide start_ARG italic_b roman_log italic_n end_ARG start_ARG italic_n end_ARG (for a,b>0𝑎𝑏0a,b>0italic_a , italic_b > 0) and two communities, they established conditions under which exact matching is possible. Once the exact matching is achieved, they construct a union graph G1∨π∗G2∼SBM(n,p(1−(1−s)2),q(1−(1−s)2))similar-tosubscriptsubscript𝜋subscript𝐺1subscript𝐺2SBM𝑛𝑝1superscript1𝑠2𝑞1superscript1𝑠2G_{1}\vee_{\pi_{*}}G_{2}\sim\text{SBM}\bigl{(}n,p(1-(1-s)^{2}),q(1-(1-s)^{2})%
\bigr{)}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∨ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ SBM ( italic_n , italic_p ( 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) , italic_q ( 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ), which is denser than the individual graphs G1,G2∼SBM(n,ps,qs)similar-tosubscript𝐺1subscript𝐺2SBM𝑛𝑝𝑠𝑞𝑠G_{1},G_{2}\sim\text{SBM}\bigl{(}n,ps,qs\bigr{)}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ SBM ( italic_n , italic_p italic_s , italic_q italic_s ). By doing so, the threshold for exact community recovery becomes less stringent: while a single graph G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT or G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT requires a−b>2s𝑎𝑏2𝑠\sqrt{a}-\sqrt{b}>\tfrac{2}{s}square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG > divide start_ARG 2 end_ARG start_ARG italic_s end_ARG, the union graph only needs a−b>21−(1−s)2𝑎𝑏21superscript1𝑠2\sqrt{a}-\sqrt{b}>\tfrac{2}{\sqrt{1-(1-s)^{2}}}square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG > divide start_ARG 2 end_ARG start_ARG square-root start_ARG 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG, illustrating a regime in which exact recovery is infeasible with a single graph but feasible with two correlated ones.
Although [6] narrowed the gap between achievability and impossibility for two-community correlated SBMs, Gaudio et al. [8] completely characterized the information-theoretic limit for exact recovery by leveraging partial matching, even in cases where perfect matching is not possible. Subsequent work has extended these ideas to correlated SBMs with more communities or more than two correlated graphs. Yang and Chung [7] generalized the results of [6] to SBMs with r𝑟ritalic_r communities that may scale with n𝑛nitalic_n, while Rácz and Zhang [34] built on [8] to determine the exact information-theoretic threshold for community recovery in scenarios involving more than two correlated SBM graphs.
| 995 |
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-B Prior Works
I-B2 Community Recovery in Correlated Random Graphs
Rácz and Sridhar [6] first investigated exact community recovery in the presence of two or more correlated networks. Focusing on correlated SBMs with p=alognn𝑝𝑎𝑛𝑛p=\frac{a\log n}{n}italic_p = divide start_ARG italic_a roman_log italic_n end_ARG start_ARG italic_n end_ARG and q=blognn𝑞𝑏𝑛𝑛q=\frac{b\log n}{n}italic_q = divide start_ARG italic_b roman_log italic_n end_ARG start_ARG italic_n end_ARG (for a,b>0𝑎𝑏0a,b>0italic_a , italic_b > 0) and two communities, they established conditions under which exact matching is possible. Once the exact matching is achieved, they construct a union graph G1∨π∗G2∼SBM(n,p(1−(1−s)2),q(1−(1−s)2))similar-tosubscriptsubscript𝜋subscript𝐺1subscript𝐺2SBM𝑛𝑝1superscript1𝑠2𝑞1superscript1𝑠2G_{1}\vee_{\pi_{*}}G_{2}\sim\text{SBM}\bigl{(}n,p(1-(1-s)^{2}),q(1-(1-s)^{2})%
\bigr{)}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∨ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ SBM ( italic_n , italic_p ( 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) , italic_q ( 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ), which is denser than the individual graphs G1,G2∼SBM(n,ps,qs)similar-tosubscript𝐺1subscript𝐺2SBM𝑛𝑝𝑠𝑞𝑠G_{1},G_{2}\sim\text{SBM}\bigl{(}n,ps,qs\bigr{)}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ SBM ( italic_n , italic_p italic_s , italic_q italic_s ). By doing so, the threshold for exact community recovery becomes less stringent: while a single graph G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT or G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT requires a−b>2s𝑎𝑏2𝑠\sqrt{a}-\sqrt{b}>\tfrac{2}{s}square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG > divide start_ARG 2 end_ARG start_ARG italic_s end_ARG, the union graph only needs a−b>21−(1−s)2𝑎𝑏21superscript1𝑠2\sqrt{a}-\sqrt{b}>\tfrac{2}{\sqrt{1-(1-s)^{2}}}square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG > divide start_ARG 2 end_ARG start_ARG square-root start_ARG 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG, illustrating a regime in which exact recovery is infeasible with a single graph but feasible with two correlated ones.
Although [6] narrowed the gap between achievability and impossibility for two-community correlated SBMs, Gaudio et al. [8] completely characterized the information-theoretic limit for exact recovery by leveraging partial matching, even in cases where perfect matching is not possible. Subsequent work has extended these ideas to correlated SBMs with more communities or more than two correlated graphs. Yang and Chung [7] generalized the results of [6] to SBMs with r𝑟ritalic_r communities that may scale with n𝑛nitalic_n, while Rácz and Zhang [34] built on [8] to determine the exact information-theoretic threshold for community recovery in scenarios involving more than two correlated SBM graphs.
| 1,029 |
14 |
This paper introduces and analyzes two new models that jointly consider correlated graphs and correlated node attributes to better reflect hidden community structures. Specifically, we focus on correlated Gaussian Mixture Models (GMMs) and correlated Contextual Stochastic Block Models (CSBMs), as defined in Section I-A, with the ultimate goal of determining the conditions for exact community recovery. A key preliminary step is to establish exact matching–that is, to recover the one-to-one correspondence between the nodes of two correlated graphs–so that the second graph (or database) can serve as side information for community detection. We characterize the regimes under which exact matching is achievable or impossible in each proposed model.
In the correlated GMM setting, we adopt an estimator that minimizes the sum of squared distances between node attributes,
, 1 = π^:=argminπ∈Sn∑i=1n∥𝒙i−𝒚π(i)∥2,assign^𝜋subscriptargmin𝜋subscript𝑆𝑛superscriptsubscript𝑖1𝑛superscriptdelimited-∥∥subscript𝒙𝑖subscript𝒚𝜋𝑖2\hat{\pi}:=\operatorname*{arg\,min}_{\pi\in S_{n}}\sum_{i=1}^{n}\bigl{\lVert}{%
\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{\pi(i)}\bigr{\rVert}^{2},over^ start_ARG italic_π end_ARG := start_OPERATOR roman_arg roman_min end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,. , 2 = . , 3 = (6)
and establish threshold conditions for exact alignment. For the correlated CSBMs, we develop a two-step algorithm that first performs k𝑘kitalic_k-core matching using only edge information to match the majority of nodes, and then applies the distance-based attribute estimator (6) to align the remaining unmatched nodes. This two-step strategy, previously employed for correlated Gaussian-attributed Erdős–Rényi graphs [13], is shown here to be effective even when community labels are unknown. In particular, the k𝑘kitalic_k-core matching [11, 8, 35, 13], which selects the largest matching (the number of matched nodes) with a minimum degree of at least k𝑘kitalic_k in the intersection graph under a particular permutation, turns out to be successfully recovery n−o(n)𝑛𝑜𝑛n-o(n)italic_n - italic_o ( italic_n ) nodes with a proper choice of k𝑘kitalic_k. We also analyze the MAP estimator to characterize the regimes in which exact matching becomes information-theoretically impossible.
Having established the conditions for exact matching, we then investigate exact community recovery in these correlated models. When matching is successful, one can merge the two correlated graphs by taking their union, thereby creating a denser graph, and average their correlated Gaussian attributes to reduce variance. Consequently, the achievable range for exact community detection expands relative to the scenario of having only a single graph or database, as illustrated in Figures 1 and 3. In particular, for correlated GMMs (X,Y)∼CGMMs(n,𝝁,d,ρ)similar-to𝑋𝑌CGMMs𝑛𝝁𝑑𝜌(X,Y)\sim\text{CGMMs}\bigl{(}n,\boldsymbol{\mu},d,\rho\bigr{)}( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) with ‖𝝁‖2=Rsuperscriptnorm𝝁2𝑅\|\boldsymbol{\mu}\|^{2}=R∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_R, the effective signal-to-noise ratio for exact recovery increases from
, 1 = R2R+dnto(21+ρR)221+ρR+dn,superscript𝑅2𝑅𝑑𝑛tosuperscript21𝜌𝑅221𝜌𝑅𝑑𝑛\frac{R^{2}}{\,R+\tfrac{d}{n}\,}\quad\text{to}\quad\frac{\bigl{(}\tfrac{2}{1+%
\rho}\,R\bigr{)}^{2}}{\,\tfrac{2}{1+\rho}\,R+\tfrac{d}{n}\,},divide start_ARG italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_R + divide start_ARG italic_d end_ARG start_ARG italic_n end_ARG end_ARG to divide start_ARG ( divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R + divide start_ARG italic_d end_ARG start_ARG italic_n end_ARG end_ARG ,. , 2 =
while for the contextual SBMs (G1,G2)∼CCSBMs(n,p,q,s;R,d,ρ)similar-tosubscript𝐺1subscript𝐺2CCSBMs𝑛𝑝𝑞𝑠𝑅𝑑𝜌(G_{1},G_{2})\sim\text{CCSBMs}\bigl{(}n,p,q,s;R,d,\rho\bigr{)}( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CCSBMs ( italic_n , italic_p , italic_q , italic_s ; italic_R , italic_d , italic_ρ ), the corresponding SNR improves from
, 1 = s(a−b)2+c 2to(1−(1−s)2)(a−b)2+c′ 2,𝑠superscript𝑎𝑏2𝑐2to1superscript1𝑠2superscript𝑎𝑏2superscript𝑐′2\frac{s\bigl{(}\sqrt{a}-\sqrt{b}\bigr{)}^{2}+c}{\,2\,}\quad\text{to}\quad\frac%
{\bigl{(}1-(1-s)^{2}\bigr{)}\bigl{(}\sqrt{a}-\sqrt{b}\bigr{)}^{2}+c^{\prime}}{%
\,2\,},divide start_ARG italic_s ( square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c end_ARG start_ARG 2 end_ARG to divide start_ARG ( 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ( square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG ,. , 2 =
| 1,685 |
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-C Our Contributions
This paper introduces and analyzes two new models that jointly consider correlated graphs and correlated node attributes to better reflect hidden community structures. Specifically, we focus on correlated Gaussian Mixture Models (GMMs) and correlated Contextual Stochastic Block Models (CSBMs), as defined in Section I-A, with the ultimate goal of determining the conditions for exact community recovery. A key preliminary step is to establish exact matching–that is, to recover the one-to-one correspondence between the nodes of two correlated graphs–so that the second graph (or database) can serve as side information for community detection. We characterize the regimes under which exact matching is achievable or impossible in each proposed model.
In the correlated GMM setting, we adopt an estimator that minimizes the sum of squared distances between node attributes,
, 1 = π^:=argminπ∈Sn∑i=1n∥𝒙i−𝒚π(i)∥2,assign^𝜋subscriptargmin𝜋subscript𝑆𝑛superscriptsubscript𝑖1𝑛superscriptdelimited-∥∥subscript𝒙𝑖subscript𝒚𝜋𝑖2\hat{\pi}:=\operatorname*{arg\,min}_{\pi\in S_{n}}\sum_{i=1}^{n}\bigl{\lVert}{%
\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{\pi(i)}\bigr{\rVert}^{2},over^ start_ARG italic_π end_ARG := start_OPERATOR roman_arg roman_min end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,. , 2 = . , 3 = (6)
and establish threshold conditions for exact alignment. For the correlated CSBMs, we develop a two-step algorithm that first performs k𝑘kitalic_k-core matching using only edge information to match the majority of nodes, and then applies the distance-based attribute estimator (6) to align the remaining unmatched nodes. This two-step strategy, previously employed for correlated Gaussian-attributed Erdős–Rényi graphs [13], is shown here to be effective even when community labels are unknown. In particular, the k𝑘kitalic_k-core matching [11, 8, 35, 13], which selects the largest matching (the number of matched nodes) with a minimum degree of at least k𝑘kitalic_k in the intersection graph under a particular permutation, turns out to be successfully recovery n−o(n)𝑛𝑜𝑛n-o(n)italic_n - italic_o ( italic_n ) nodes with a proper choice of k𝑘kitalic_k. We also analyze the MAP estimator to characterize the regimes in which exact matching becomes information-theoretically impossible.
Having established the conditions for exact matching, we then investigate exact community recovery in these correlated models. When matching is successful, one can merge the two correlated graphs by taking their union, thereby creating a denser graph, and average their correlated Gaussian attributes to reduce variance. Consequently, the achievable range for exact community detection expands relative to the scenario of having only a single graph or database, as illustrated in Figures 1 and 3. In particular, for correlated GMMs (X,Y)∼CGMMs(n,𝝁,d,ρ)similar-to𝑋𝑌CGMMs𝑛𝝁𝑑𝜌(X,Y)\sim\text{CGMMs}\bigl{(}n,\boldsymbol{\mu},d,\rho\bigr{)}( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) with ‖𝝁‖2=Rsuperscriptnorm𝝁2𝑅\|\boldsymbol{\mu}\|^{2}=R∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_R, the effective signal-to-noise ratio for exact recovery increases from
, 1 = R2R+dnto(21+ρR)221+ρR+dn,superscript𝑅2𝑅𝑑𝑛tosuperscript21𝜌𝑅221𝜌𝑅𝑑𝑛\frac{R^{2}}{\,R+\tfrac{d}{n}\,}\quad\text{to}\quad\frac{\bigl{(}\tfrac{2}{1+%
\rho}\,R\bigr{)}^{2}}{\,\tfrac{2}{1+\rho}\,R+\tfrac{d}{n}\,},divide start_ARG italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_R + divide start_ARG italic_d end_ARG start_ARG italic_n end_ARG end_ARG to divide start_ARG ( divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R + divide start_ARG italic_d end_ARG start_ARG italic_n end_ARG end_ARG ,. , 2 =
while for the contextual SBMs (G1,G2)∼CCSBMs(n,p,q,s;R,d,ρ)similar-tosubscript𝐺1subscript𝐺2CCSBMs𝑛𝑝𝑞𝑠𝑅𝑑𝜌(G_{1},G_{2})\sim\text{CCSBMs}\bigl{(}n,p,q,s;R,d,\rho\bigr{)}( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CCSBMs ( italic_n , italic_p , italic_q , italic_s ; italic_R , italic_d , italic_ρ ), the corresponding SNR improves from
, 1 = s(a−b)2+c 2to(1−(1−s)2)(a−b)2+c′ 2,𝑠superscript𝑎𝑏2𝑐2to1superscript1𝑠2superscript𝑎𝑏2superscript𝑐′2\frac{s\bigl{(}\sqrt{a}-\sqrt{b}\bigr{)}^{2}+c}{\,2\,}\quad\text{to}\quad\frac%
{\bigl{(}1-(1-s)^{2}\bigr{)}\bigl{(}\sqrt{a}-\sqrt{b}\bigr{)}^{2}+c^{\prime}}{%
\,2\,},divide start_ARG italic_s ( square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c end_ARG start_ARG 2 end_ARG to divide start_ARG ( 1 - ( 1 - italic_s ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ( square-root start_ARG italic_a end_ARG - square-root start_ARG italic_b end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG ,. , 2 =
| 1,707 |
15 |
where clogn=R2R+d/n𝑐𝑛superscript𝑅2𝑅𝑑𝑛c\log n=\frac{R^{2}}{\,R+d/n\,}italic_c roman_log italic_n = divide start_ARG italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_R + italic_d / italic_n end_ARG and c′logn=(21+ρR)221+ρR+d/n,superscript𝑐′𝑛superscript21𝜌𝑅221𝜌𝑅𝑑𝑛c^{\prime}\log n=\frac{\bigl{(}\tfrac{2}{1+\rho}\,R\bigr{)}^{2}}{\,\tfrac{2}{1%
+\rho}\,R+d/n\,},italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT roman_log italic_n = divide start_ARG ( divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R + italic_d / italic_n end_ARG , compared to having only one contextual SBM (G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT or G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT).
| 339 |
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-C Our Contributions
where clogn=R2R+d/n𝑐𝑛superscript𝑅2𝑅𝑑𝑛c\log n=\frac{R^{2}}{\,R+d/n\,}italic_c roman_log italic_n = divide start_ARG italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_R + italic_d / italic_n end_ARG and c′logn=(21+ρR)221+ρR+d/n,superscript𝑐′𝑛superscript21𝜌𝑅221𝜌𝑅𝑑𝑛c^{\prime}\log n=\frac{\bigl{(}\tfrac{2}{1+\rho}\,R\bigr{)}^{2}}{\,\tfrac{2}{1%
+\rho}\,R+d/n\,},italic_c start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT roman_log italic_n = divide start_ARG ( divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG divide start_ARG 2 end_ARG start_ARG 1 + italic_ρ end_ARG italic_R + italic_d / italic_n end_ARG , compared to having only one contextual SBM (G1subscript𝐺1G_{1}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT or G2subscript𝐺2G_{2}italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT).
| 361 |
16 |
For a positive integer n𝑛nitalic_n, write [n]:={1,2,…,n}assigndelimited-[]𝑛12…𝑛[n]:=\{1,2,\ldots,n\}[ italic_n ] := { 1 , 2 , … , italic_n }. For a graph G𝐺Gitalic_G on vertex set [n]delimited-[]𝑛[n][ italic_n ], let degG(i)subscriptdegree𝐺𝑖\deg_{G}(i)roman_deg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_i ) be the degree of node i𝑖iitalic_i, and let G{M}𝐺𝑀G\{M\}italic_G { italic_M } be the subgraph induced by M⊆[n]𝑀delimited-[]𝑛M\subseteq[n]italic_M ⊆ [ italic_n ]. Define dmin(G)subscript𝑑𝐺d_{\min}(G)italic_d start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_G ) as the minimum degree of G𝐺Gitalic_G. Let ℰ:={{i,j}:i,j∈[n],i≠j}assignℰconditional-set𝑖𝑗formulae-sequence𝑖𝑗delimited-[]𝑛𝑖𝑗\mathcal{E}:=\{\{i,j\}:i,j\in[n],i\neq j\}caligraphic_E := { { italic_i , italic_j } : italic_i , italic_j ∈ [ italic_n ] , italic_i ≠ italic_j } be the set of all unordered vertex pairs. For a community label vector 𝝈𝝈{\boldsymbol{\sigma}}bold_italic_σ, define
, 1 = ℰ+(𝝈):={{i,j}∈ℰ:σiσj=+1}andℰ−(𝝈):={{i,j}∈ℰ:σiσj=−1}.formulae-sequenceassignsuperscriptℰ𝝈conditional-set𝑖𝑗ℰsubscript𝜎𝑖subscript𝜎𝑗1andassignsuperscriptℰ𝝈conditional-set𝑖𝑗ℰsubscript𝜎𝑖subscript𝜎𝑗1\mathcal{E}^{+}({\boldsymbol{\sigma}}):=\{\{i,j\}\in\mathcal{E}:\sigma_{i}%
\sigma_{j}=+1\}\quad\text{and}\quad\mathcal{E}^{-}({\boldsymbol{\sigma}}):=\{%
\{i,j\}\in\mathcal{E}:\sigma_{i}\sigma_{j}=-1\}.caligraphic_E start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( bold_italic_σ ) := { { italic_i , italic_j } ∈ caligraphic_E : italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = + 1 } and caligraphic_E start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( bold_italic_σ ) := { { italic_i , italic_j } ∈ caligraphic_E : italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = - 1 } .. , 2 =
Then ℰ+(𝝈)superscriptℰ𝝈\mathcal{E}^{+}({\boldsymbol{\sigma}})caligraphic_E start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( bold_italic_σ ) and ℰ−(𝝈)superscriptℰ𝝈\mathcal{E}^{-}({\boldsymbol{\sigma}})caligraphic_E start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( bold_italic_σ ) partition ℰℰ\mathcal{E}caligraphic_E into intra- and inter-community node pairs. Let A,B′,B𝐴superscript𝐵′𝐵A,B^{\prime},Bitalic_A , italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B be the adjacency matrices of G1,G2′,G2subscript𝐺1subscriptsuperscript𝐺′2subscript𝐺2G_{1},G^{\prime}_{2},G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, respectively, and let X,Y′,Y𝑋superscript𝑌′𝑌X,Y^{\prime},Yitalic_X , italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_Y be the corresponding databases of node attributes. Denote by ∨\vee∨ and ∧\wedge∧ the entrywise max and min, respectively. For a permutation π𝜋\piitalic_π, define
, 1 = (A∨πB)i,j=max{Ai,j,Bπ(i),π(j)}and(A∧πB)i,j=min{Ai,j,Bπ(i),π(j)}.formulae-sequencesubscriptsubscript𝜋𝐴𝐵𝑖𝑗subscript𝐴𝑖𝑗subscript𝐵𝜋𝑖𝜋𝑗andsubscriptsubscript𝜋𝐴𝐵𝑖𝑗subscript𝐴𝑖𝑗subscript𝐵𝜋𝑖𝜋𝑗(A\vee_{\pi}B)_{i,j}=\max\{A_{i,j},\,B_{\pi(i),\pi(j)}\}\quad\text{and}\quad(A%
\wedge_{\pi}B)_{i,j}=\min\{A_{i,j},\,B_{\pi(i),\pi(j)}\}.( italic_A ∨ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT italic_B ) start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = roman_max { italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT italic_π ( italic_i ) , italic_π ( italic_j ) end_POSTSUBSCRIPT } and ( italic_A ∧ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT italic_B ) start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = roman_min { italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT italic_π ( italic_i ) , italic_π ( italic_j ) end_POSTSUBSCRIPT } .. , 2 =
| 1,550 |
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-D Notation
For a positive integer n𝑛nitalic_n, write [n]:={1,2,…,n}assigndelimited-[]𝑛12…𝑛[n]:=\{1,2,\ldots,n\}[ italic_n ] := { 1 , 2 , … , italic_n }. For a graph G𝐺Gitalic_G on vertex set [n]delimited-[]𝑛[n][ italic_n ], let degG(i)subscriptdegree𝐺𝑖\deg_{G}(i)roman_deg start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_i ) be the degree of node i𝑖iitalic_i, and let G{M}𝐺𝑀G\{M\}italic_G { italic_M } be the subgraph induced by M⊆[n]𝑀delimited-[]𝑛M\subseteq[n]italic_M ⊆ [ italic_n ]. Define dmin(G)subscript𝑑𝐺d_{\min}(G)italic_d start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ( italic_G ) as the minimum degree of G𝐺Gitalic_G. Let ℰ:={{i,j}:i,j∈[n],i≠j}assignℰconditional-set𝑖𝑗formulae-sequence𝑖𝑗delimited-[]𝑛𝑖𝑗\mathcal{E}:=\{\{i,j\}:i,j\in[n],i\neq j\}caligraphic_E := { { italic_i , italic_j } : italic_i , italic_j ∈ [ italic_n ] , italic_i ≠ italic_j } be the set of all unordered vertex pairs. For a community label vector 𝝈𝝈{\boldsymbol{\sigma}}bold_italic_σ, define
, 1 = ℰ+(𝝈):={{i,j}∈ℰ:σiσj=+1}andℰ−(𝝈):={{i,j}∈ℰ:σiσj=−1}.formulae-sequenceassignsuperscriptℰ𝝈conditional-set𝑖𝑗ℰsubscript𝜎𝑖subscript𝜎𝑗1andassignsuperscriptℰ𝝈conditional-set𝑖𝑗ℰsubscript𝜎𝑖subscript𝜎𝑗1\mathcal{E}^{+}({\boldsymbol{\sigma}}):=\{\{i,j\}\in\mathcal{E}:\sigma_{i}%
\sigma_{j}=+1\}\quad\text{and}\quad\mathcal{E}^{-}({\boldsymbol{\sigma}}):=\{%
\{i,j\}\in\mathcal{E}:\sigma_{i}\sigma_{j}=-1\}.caligraphic_E start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( bold_italic_σ ) := { { italic_i , italic_j } ∈ caligraphic_E : italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = + 1 } and caligraphic_E start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( bold_italic_σ ) := { { italic_i , italic_j } ∈ caligraphic_E : italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = - 1 } .. , 2 =
Then ℰ+(𝝈)superscriptℰ𝝈\mathcal{E}^{+}({\boldsymbol{\sigma}})caligraphic_E start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ( bold_italic_σ ) and ℰ−(𝝈)superscriptℰ𝝈\mathcal{E}^{-}({\boldsymbol{\sigma}})caligraphic_E start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( bold_italic_σ ) partition ℰℰ\mathcal{E}caligraphic_E into intra- and inter-community node pairs. Let A,B′,B𝐴superscript𝐵′𝐵A,B^{\prime},Bitalic_A , italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_B be the adjacency matrices of G1,G2′,G2subscript𝐺1subscriptsuperscript𝐺′2subscript𝐺2G_{1},G^{\prime}_{2},G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, respectively, and let X,Y′,Y𝑋superscript𝑌′𝑌X,Y^{\prime},Yitalic_X , italic_Y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_Y be the corresponding databases of node attributes. Denote by ∨\vee∨ and ∧\wedge∧ the entrywise max and min, respectively. For a permutation π𝜋\piitalic_π, define
, 1 = (A∨πB)i,j=max{Ai,j,Bπ(i),π(j)}and(A∧πB)i,j=min{Ai,j,Bπ(i),π(j)}.formulae-sequencesubscriptsubscript𝜋𝐴𝐵𝑖𝑗subscript𝐴𝑖𝑗subscript𝐵𝜋𝑖𝜋𝑗andsubscriptsubscript𝜋𝐴𝐵𝑖𝑗subscript𝐴𝑖𝑗subscript𝐵𝜋𝑖𝜋𝑗(A\vee_{\pi}B)_{i,j}=\max\{A_{i,j},\,B_{\pi(i),\pi(j)}\}\quad\text{and}\quad(A%
\wedge_{\pi}B)_{i,j}=\min\{A_{i,j},\,B_{\pi(i),\pi(j)}\}.( italic_A ∨ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT italic_B ) start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = roman_max { italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT italic_π ( italic_i ) , italic_π ( italic_j ) end_POSTSUBSCRIPT } and ( italic_A ∧ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT italic_B ) start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = roman_min { italic_A start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT , italic_B start_POSTSUBSCRIPT italic_π ( italic_i ) , italic_π ( italic_j ) end_POSTSUBSCRIPT } .. , 2 =
| 1,572 |
17 |
For v=[v1,…,vk]⊤∈ℝk𝑣superscriptsubscript𝑣1…subscript𝑣𝑘topsuperscriptℝ𝑘v=[v_{1},\ldots,v_{k}]^{\top}\in\mathbb{R}^{k}italic_v = [ italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT and i∈[k]𝑖delimited-[]𝑘i\in[k]italic_i ∈ [ italic_k ], let v−isubscript𝑣𝑖v_{-i}italic_v start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT be the vector obtained by removing visubscript𝑣𝑖v_{i}italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. For an event E𝐸Eitalic_E, let 𝟙(E)1𝐸\mathds{1}(E)blackboard_1 ( italic_E ) be its indicator. Write Φ(⋅)Φ⋅\Phi(\cdot)roman_Φ ( ⋅ ) for the tail distribution of a standard Gaussian. For two functions f,g:[n]→[m]:𝑓𝑔→delimited-[]𝑛delimited-[]𝑚f,g:[n]\to[m]italic_f , italic_g : [ italic_n ] → [ italic_m ], define their overlap as 𝐨𝐯(f,g):=1n∑i=1n𝟙(f(i)=g(i))assign𝐨𝐯𝑓𝑔1𝑛superscriptsubscript𝑖1𝑛1𝑓𝑖𝑔𝑖\mathbf{ov}(f,g):=\tfrac{1}{n}\sum_{i=1}^{n}\mathds{1}\bigl{(}f(i)=g(i)\bigr{)}bold_ov ( italic_f , italic_g ) := divide start_ARG 1 end_ARG start_ARG italic_n end_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT blackboard_1 ( italic_f ( italic_i ) = italic_g ( italic_i ) ). Lastly, asymptotic notation O(⋅),o(⋅),Ω(⋅),ω(⋅),Θ(⋅)𝑂⋅𝑜⋅Ω⋅𝜔⋅Θ⋅O(\cdot),o(\cdot),\Omega(\cdot),\omega(\cdot),\Theta(\cdot)italic_O ( ⋅ ) , italic_o ( ⋅ ) , roman_Ω ( ⋅ ) , italic_ω ( ⋅ ) , roman_Θ ( ⋅ ) is used with n→∞→𝑛n\to\inftyitalic_n → ∞.
| 713 |
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
I Introduction
I-D Notation
For v=[v1,…,vk]⊤∈ℝk𝑣superscriptsubscript𝑣1…subscript𝑣𝑘topsuperscriptℝ𝑘v=[v_{1},\ldots,v_{k}]^{\top}\in\mathbb{R}^{k}italic_v = [ italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_v start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT ⊤ end_POSTSUPERSCRIPT ∈ blackboard_R start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT and i∈[k]𝑖delimited-[]𝑘i\in[k]italic_i ∈ [ italic_k ], let v−isubscript𝑣𝑖v_{-i}italic_v start_POSTSUBSCRIPT - italic_i end_POSTSUBSCRIPT be the vector obtained by removing visubscript𝑣𝑖v_{i}italic_v start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. For an event E𝐸Eitalic_E, let 𝟙(E)1𝐸\mathds{1}(E)blackboard_1 ( italic_E ) be its indicator. Write Φ(⋅)Φ⋅\Phi(\cdot)roman_Φ ( ⋅ ) for the tail distribution of a standard Gaussian. For two functions f,g:[n]→[m]:𝑓𝑔→delimited-[]𝑛delimited-[]𝑚f,g:[n]\to[m]italic_f , italic_g : [ italic_n ] → [ italic_m ], define their overlap as 𝐨𝐯(f,g):=1n∑i=1n𝟙(f(i)=g(i))assign𝐨𝐯𝑓𝑔1𝑛superscriptsubscript𝑖1𝑛1𝑓𝑖𝑔𝑖\mathbf{ov}(f,g):=\tfrac{1}{n}\sum_{i=1}^{n}\mathds{1}\bigl{(}f(i)=g(i)\bigr{)}bold_ov ( italic_f , italic_g ) := divide start_ARG 1 end_ARG start_ARG italic_n end_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT blackboard_1 ( italic_f ( italic_i ) = italic_g ( italic_i ) ). Lastly, asymptotic notation O(⋅),o(⋅),Ω(⋅),ω(⋅),Θ(⋅)𝑂⋅𝑜⋅Ω⋅𝜔⋅Θ⋅O(\cdot),o(\cdot),\Omega(\cdot),\omega(\cdot),\Theta(\cdot)italic_O ( ⋅ ) , italic_o ( ⋅ ) , roman_Ω ( ⋅ ) , italic_ω ( ⋅ ) , roman_Θ ( ⋅ ) is used with n→∞→𝑛n\to\inftyitalic_n → ∞.
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In this section, we investigate the correlated Gaussian Mixture Models (GMMs) introduced in Section I-A1, with a primary goal of determining conditions for exact community recovery when two correlated databases are provided. Our approach consists of two steps: (i) establishing exact matching between the two databases, and (ii) merging the matched databases to identify regimes in which exact community recovery is significantly more tractable compared to using only a single database.
| 90 |
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
In this section, we investigate the correlated Gaussian Mixture Models (GMMs) introduced in Section I-A1, with a primary goal of determining conditions for exact community recovery when two correlated databases are provided. Our approach consists of two steps: (i) establishing exact matching between the two databases, and (ii) merging the matched databases to identify regimes in which exact community recovery is significantly more tractable compared to using only a single database.
| 112 |
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We begin by examining the requirements for exact matching in correlated GMMs. Theorem 1 below provides sufficient conditions under which an estimator (6) achieves perfect alignment with high probability.
| 37 |
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
II-A Exact Matching on Correlated Gaussian Mixture Models
We begin by examining the requirements for exact matching in correlated GMMs. Theorem 1 below provides sufficient conditions under which an estimator (6) achieves perfect alignment with high probability.
| 71 |
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Let (X,Y)∼CGMMs(n,𝛍,d,ρ)similar-to𝑋𝑌CGMMs𝑛𝛍𝑑𝜌(X,Y)\sim\textnormal{CGMMs}(n,\boldsymbol{\mu},d,\rho)( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) as defined in Section I-A1. Suppose that either
, 1 = d4log11−ρ2≥logn+ω(1)and‖𝝁‖2≥2logn+ω(1),formulae-sequence𝑑411superscript𝜌2𝑛𝜔1andsuperscriptnorm𝝁22𝑛𝜔1\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq\log n+\omega(1)\quad\text{and}\quad\|%
\boldsymbol{\mu}\|^{2}\geq 2\log n+\omega(1),divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ roman_log italic_n + italic_ω ( 1 ) and ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ) ,. , 2 = . , 3 = (7)
or
, 1 = d4log11−ρ2≥(1+ϵ)lognandd=ω(logn)formulae-sequence𝑑411superscript𝜌21italic-ϵ𝑛and𝑑𝜔𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1+\epsilon)\log n\quad\text{and}\quad d%
=\omega(\log n)divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n and italic_d = italic_ω ( roman_log italic_n ). , 2 = . , 3 = (8)
holds for an arbitrarily small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0. Then there exists an estimator π^(X,Y)^𝜋𝑋𝑌\hat{\pi}(X,Y)over^ start_ARG italic_π end_ARG ( italic_X , italic_Y ) such that π^(X,Y)=π∗^𝜋𝑋𝑌subscript𝜋\hat{\pi}(X,Y)=\pi_{*}over^ start_ARG italic_π end_ARG ( italic_X , italic_Y ) = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT with high probability.
In correlated GMMs, the MAP estimator can be written as
| 724 |
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
II-A Exact Matching on Correlated Gaussian Mixture Models
Theorem 1 (Achievability for Exact Matching).
Let (X,Y)∼CGMMs(n,𝛍,d,ρ)similar-to𝑋𝑌CGMMs𝑛𝛍𝑑𝜌(X,Y)\sim\textnormal{CGMMs}(n,\boldsymbol{\mu},d,\rho)( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) as defined in Section I-A1. Suppose that either
, 1 = d4log11−ρ2≥logn+ω(1)and‖𝝁‖2≥2logn+ω(1),formulae-sequence𝑑411superscript𝜌2𝑛𝜔1andsuperscriptnorm𝝁22𝑛𝜔1\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq\log n+\omega(1)\quad\text{and}\quad\|%
\boldsymbol{\mu}\|^{2}\geq 2\log n+\omega(1),divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ roman_log italic_n + italic_ω ( 1 ) and ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ) ,. , 2 = . , 3 = (7)
or
, 1 = d4log11−ρ2≥(1+ϵ)lognandd=ω(logn)formulae-sequence𝑑411superscript𝜌21italic-ϵ𝑛and𝑑𝜔𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1+\epsilon)\log n\quad\text{and}\quad d%
=\omega(\log n)divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n and italic_d = italic_ω ( roman_log italic_n ). , 2 = . , 3 = (8)
holds for an arbitrarily small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0. Then there exists an estimator π^(X,Y)^𝜋𝑋𝑌\hat{\pi}(X,Y)over^ start_ARG italic_π end_ARG ( italic_X , italic_Y ) such that π^(X,Y)=π∗^𝜋𝑋𝑌subscript𝜋\hat{\pi}(X,Y)=\pi_{*}over^ start_ARG italic_π end_ARG ( italic_X , italic_Y ) = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT with high probability.
In correlated GMMs, the MAP estimator can be written as
| 770 |
21 |
, 1 = π^^𝜋\displaystyle\hat{\pi}over^ start_ARG italic_π end_ARG. , 2 = =argmaxπ∈Snℙ(π∗=π|X,Y)absentsubscriptargmax𝜋subscript𝑆𝑛ℙsubscript𝜋conditional𝜋𝑋𝑌\displaystyle=\operatorname*{arg\,max}_{\pi\in S_{n}}\mathbb{P}(\pi_{*}=\pi|X,Y)= start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_P ( italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT = italic_π | italic_X , italic_Y ). , 3 = . , 4 = (9). , 1 = . , 2 = =(a)argmaxπ∈Snℙ(X,Y|π∗=π)superscript𝑎absentsubscriptargmax𝜋subscript𝑆𝑛ℙ𝑋conditional𝑌subscript𝜋𝜋\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\operatorname*{arg\,max}_{\pi\in
S%
_{n}}\mathbb{P}(X,Y|\pi_{*}=\pi)start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG ( italic_a ) end_ARG end_RELOP start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_P ( italic_X , italic_Y | italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT = italic_π ). , 3 = . , 4 = (9). , 1 = . , 2 = =argmaxπ∈Sn∑i=1n(−‖𝒙i−𝝁σi‖2+2ρ⟨𝒙i−𝝁σi,𝒚π(i)−𝝁σπ(i)⟩−‖𝒚π(i)−𝝁σπ(i)‖2)absentsubscriptargmax𝜋subscript𝑆𝑛subscriptsuperscript𝑛𝑖1superscriptnormsubscript𝒙𝑖𝝁subscript𝜎𝑖22𝜌subscript𝒙𝑖𝝁subscript𝜎𝑖subscript𝒚𝜋𝑖𝝁subscript𝜎𝜋𝑖superscriptnormsubscript𝒚𝜋𝑖𝝁subscript𝜎𝜋𝑖2\displaystyle=\operatorname*{arg\,max}_{\pi\in S_{n}}\sum^{n}_{i=1}\left(-\|{%
\boldsymbol{x}}_{i}-\boldsymbol{\mu}\sigma_{i}\|^{2}+2\rho\langle{\boldsymbol{%
x}}_{i}-\boldsymbol{\mu}\sigma_{i},{\boldsymbol{y}}_{\pi(i)}-\boldsymbol{\mu}%
\sigma_{\pi(i)}\rangle-\|{\boldsymbol{y}}_{\pi(i)}-\boldsymbol{\mu}\sigma_{\pi%
(i)}\|^{2}\right)= start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT ( - ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 italic_ρ ⟨ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT - bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ⟩ - ∥ bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT - bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ). , 3 = . , 4 = (9). , 1 = . , 2 = =argmaxπ∈Sn∑i=1n(−ρ‖𝒙i−𝒚π(i)‖2−(1−ρ)(‖𝒙i‖2+‖𝒚π(i)‖2))+f(𝝈,𝝈π)absentsubscriptargmax𝜋subscript𝑆𝑛subscriptsuperscript𝑛𝑖1𝜌superscriptnormsubscript𝒙𝑖subscript𝒚𝜋𝑖21𝜌superscriptnormsubscript𝒙𝑖2superscriptnormsubscript𝒚𝜋𝑖2𝑓𝝈subscript𝝈𝜋\displaystyle=\operatorname*{arg\,max}_{\pi\in S_{n}}\sum^{n}_{i=1}\left(-\rho%
\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{\pi(i)}\|^{2}-(1-\rho)(\|{\boldsymbol%
{x}}_{i}\|^{2}+\|{\boldsymbol{y}}_{\pi(i)}\|^{2})\right)+f({\boldsymbol{\sigma%
}},{\boldsymbol{\sigma}}_{\pi})= start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT ( - italic_ρ ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( 1 - italic_ρ ) ( ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ∥ bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ) + italic_f ( bold_italic_σ , bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT ). , 3 = . , 4 = (9). , 1 = . , 2 = =argminπ∈Sn∑i=1n‖𝒙i−𝒚π(i)‖2−1ρf(𝝈,𝝈π),absentsubscriptargmin𝜋subscript𝑆𝑛subscriptsuperscript𝑛𝑖1superscriptnormsubscript𝒙𝑖subscript𝒚𝜋𝑖21𝜌𝑓𝝈subscript𝝈𝜋\displaystyle=\operatorname*{arg\,min}_{\pi\in S_{n}}\sum^{n}_{i=1}\|{%
\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{\pi(i)}\|^{2}-\frac{1}{\rho}f({%
| 1,822 |
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
II-A Exact Matching on Correlated Gaussian Mixture Models
Theorem 1 (Achievability for Exact Matching).
, 1 = π^^𝜋\displaystyle\hat{\pi}over^ start_ARG italic_π end_ARG. , 2 = =argmaxπ∈Snℙ(π∗=π|X,Y)absentsubscriptargmax𝜋subscript𝑆𝑛ℙsubscript𝜋conditional𝜋𝑋𝑌\displaystyle=\operatorname*{arg\,max}_{\pi\in S_{n}}\mathbb{P}(\pi_{*}=\pi|X,Y)= start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_P ( italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT = italic_π | italic_X , italic_Y ). , 3 = . , 4 = (9). , 1 = . , 2 = =(a)argmaxπ∈Snℙ(X,Y|π∗=π)superscript𝑎absentsubscriptargmax𝜋subscript𝑆𝑛ℙ𝑋conditional𝑌subscript𝜋𝜋\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\operatorname*{arg\,max}_{\pi\in
S%
_{n}}\mathbb{P}(X,Y|\pi_{*}=\pi)start_RELOP SUPERSCRIPTOP start_ARG = end_ARG start_ARG ( italic_a ) end_ARG end_RELOP start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT blackboard_P ( italic_X , italic_Y | italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT = italic_π ). , 3 = . , 4 = (9). , 1 = . , 2 = =argmaxπ∈Sn∑i=1n(−‖𝒙i−𝝁σi‖2+2ρ⟨𝒙i−𝝁σi,𝒚π(i)−𝝁σπ(i)⟩−‖𝒚π(i)−𝝁σπ(i)‖2)absentsubscriptargmax𝜋subscript𝑆𝑛subscriptsuperscript𝑛𝑖1superscriptnormsubscript𝒙𝑖𝝁subscript𝜎𝑖22𝜌subscript𝒙𝑖𝝁subscript𝜎𝑖subscript𝒚𝜋𝑖𝝁subscript𝜎𝜋𝑖superscriptnormsubscript𝒚𝜋𝑖𝝁subscript𝜎𝜋𝑖2\displaystyle=\operatorname*{arg\,max}_{\pi\in S_{n}}\sum^{n}_{i=1}\left(-\|{%
\boldsymbol{x}}_{i}-\boldsymbol{\mu}\sigma_{i}\|^{2}+2\rho\langle{\boldsymbol{%
x}}_{i}-\boldsymbol{\mu}\sigma_{i},{\boldsymbol{y}}_{\pi(i)}-\boldsymbol{\mu}%
\sigma_{\pi(i)}\rangle-\|{\boldsymbol{y}}_{\pi(i)}-\boldsymbol{\mu}\sigma_{\pi%
(i)}\|^{2}\right)= start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT ( - ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 italic_ρ ⟨ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT - bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ⟩ - ∥ bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT - bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ). , 3 = . , 4 = (9). , 1 = . , 2 = =argmaxπ∈Sn∑i=1n(−ρ‖𝒙i−𝒚π(i)‖2−(1−ρ)(‖𝒙i‖2+‖𝒚π(i)‖2))+f(𝝈,𝝈π)absentsubscriptargmax𝜋subscript𝑆𝑛subscriptsuperscript𝑛𝑖1𝜌superscriptnormsubscript𝒙𝑖subscript𝒚𝜋𝑖21𝜌superscriptnormsubscript𝒙𝑖2superscriptnormsubscript𝒚𝜋𝑖2𝑓𝝈subscript𝝈𝜋\displaystyle=\operatorname*{arg\,max}_{\pi\in S_{n}}\sum^{n}_{i=1}\left(-\rho%
\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{\pi(i)}\|^{2}-(1-\rho)(\|{\boldsymbol%
{x}}_{i}\|^{2}+\|{\boldsymbol{y}}_{\pi(i)}\|^{2})\right)+f({\boldsymbol{\sigma%
}},{\boldsymbol{\sigma}}_{\pi})= start_OPERATOR roman_arg roman_max end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT ( - italic_ρ ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( 1 - italic_ρ ) ( ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ∥ bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ) + italic_f ( bold_italic_σ , bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT ). , 3 = . , 4 = (9). , 1 = . , 2 = =argminπ∈Sn∑i=1n‖𝒙i−𝒚π(i)‖2−1ρf(𝝈,𝝈π),absentsubscriptargmin𝜋subscript𝑆𝑛subscriptsuperscript𝑛𝑖1superscriptnormsubscript𝒙𝑖subscript𝒚𝜋𝑖21𝜌𝑓𝝈subscript𝝈𝜋\displaystyle=\operatorname*{arg\,min}_{\pi\in S_{n}}\sum^{n}_{i=1}\|{%
\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{\pi(i)}\|^{2}-\frac{1}{\rho}f({%
| 1,868 |
22 |
\boldsymbol{\sigma}},{\boldsymbol{\sigma}}_{\pi}),= start_OPERATOR roman_arg roman_min end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG italic_ρ end_ARG italic_f ( bold_italic_σ , bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT ) ,. , 3 = . , 4 = (9)
where
, 1 = f(𝝈,𝝈π)=∑i=1n(2⟨𝒙i,𝝁σi⟩+2⟨𝒚π(i),𝝁σπ(i)⟩−2ρ⟨𝒙i,𝝁σπ(i)⟩−2ρ⟨𝒚π(i),𝝁σi⟩+2ρ‖𝝁‖2σiσπ(i)).𝑓𝝈subscript𝝈𝜋superscriptsubscript𝑖1𝑛2subscript𝒙𝑖𝝁subscript𝜎𝑖2subscript𝒚𝜋𝑖𝝁subscript𝜎𝜋𝑖2𝜌subscript𝒙𝑖𝝁subscript𝜎𝜋𝑖2𝜌subscript𝒚𝜋𝑖𝝁subscript𝜎𝑖2𝜌superscriptnorm𝝁2subscript𝜎𝑖subscript𝜎𝜋𝑖f\bigl{(}{\boldsymbol{\sigma}},{\boldsymbol{\sigma}}_{\pi}\bigr{)}=\sum_{i=1}^%
{n}\Bigl{(}2\langle{\boldsymbol{x}}_{i},\boldsymbol{\mu}\sigma_{i}\rangle+2%
\langle{\boldsymbol{y}}_{\pi(i)},\boldsymbol{\mu}\sigma_{\pi(i)}\rangle-2\rho%
\langle{\boldsymbol{x}}_{i},\boldsymbol{\mu}\sigma_{\pi(i)}\rangle-2\rho%
\langle{\boldsymbol{y}}_{\pi(i)},\boldsymbol{\mu}\sigma_{i}\rangle+2\rho\|%
\boldsymbol{\mu}\|^{2}\sigma_{i}\sigma_{\pi(i)}\Bigr{)}.italic_f ( bold_italic_σ , bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( 2 ⟨ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ + 2 ⟨ bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT , bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ⟩ - 2 italic_ρ ⟨ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ⟩ - 2 italic_ρ ⟨ bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT , bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ + 2 italic_ρ ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ) .. , 2 =
Step (a)𝑎(a)( italic_a ) uses the fact that π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT is uniformly distributed over Snsubscript𝑆𝑛S_{n}italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT. Note also that
∑i=1n‖𝒙i‖2+‖𝒚π(i)‖2superscriptsubscript𝑖1𝑛superscriptnormsubscript𝒙𝑖2superscriptnormsubscript𝒚𝜋𝑖2\sum_{i=1}^{n}\|{\boldsymbol{x}}_{i}\|^{2}+\|{\boldsymbol{y}}_{\pi(i)}\|^{2}∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ∥ bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT
is invariant under permutation π𝜋\piitalic_π, ensuring this part does not affect the alignment decision. If the community labels 𝝈𝝈{\boldsymbol{\sigma}}bold_italic_σ and 𝝈πsubscript𝝈𝜋{\boldsymbol{\sigma}}_{\pi}bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT are known, then f(𝝈,𝝈π)𝑓𝝈subscript𝝈𝜋f\bigl{(}{\boldsymbol{\sigma}},{\boldsymbol{\sigma}}_{\pi}\bigr{)}italic_f ( bold_italic_σ , bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT ) is also fixed, so the MAP estimator reduces exactly to the simpler distance-based estimator in (6). Even without known labels, the proof of Theorem 1 shows that (6) attains tight bounds under the conditions ‖𝝁‖2≥2logn+ω(1)superscriptnorm𝝁22𝑛𝜔1\|\boldsymbol{\mu}\|^{2}\geq 2\log n+\omega(1)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ) or d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ); see Section VI for details.
| 1,594 |
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
II-A Exact Matching on Correlated Gaussian Mixture Models
Theorem 1 (Achievability for Exact Matching).
\boldsymbol{\sigma}},{\boldsymbol{\sigma}}_{\pi}),= start_OPERATOR roman_arg roman_min end_OPERATOR start_POSTSUBSCRIPT italic_π ∈ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG italic_ρ end_ARG italic_f ( bold_italic_σ , bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT ) ,. , 3 = . , 4 = (9)
where
, 1 = f(𝝈,𝝈π)=∑i=1n(2⟨𝒙i,𝝁σi⟩+2⟨𝒚π(i),𝝁σπ(i)⟩−2ρ⟨𝒙i,𝝁σπ(i)⟩−2ρ⟨𝒚π(i),𝝁σi⟩+2ρ‖𝝁‖2σiσπ(i)).𝑓𝝈subscript𝝈𝜋superscriptsubscript𝑖1𝑛2subscript𝒙𝑖𝝁subscript𝜎𝑖2subscript𝒚𝜋𝑖𝝁subscript𝜎𝜋𝑖2𝜌subscript𝒙𝑖𝝁subscript𝜎𝜋𝑖2𝜌subscript𝒚𝜋𝑖𝝁subscript𝜎𝑖2𝜌superscriptnorm𝝁2subscript𝜎𝑖subscript𝜎𝜋𝑖f\bigl{(}{\boldsymbol{\sigma}},{\boldsymbol{\sigma}}_{\pi}\bigr{)}=\sum_{i=1}^%
{n}\Bigl{(}2\langle{\boldsymbol{x}}_{i},\boldsymbol{\mu}\sigma_{i}\rangle+2%
\langle{\boldsymbol{y}}_{\pi(i)},\boldsymbol{\mu}\sigma_{\pi(i)}\rangle-2\rho%
\langle{\boldsymbol{x}}_{i},\boldsymbol{\mu}\sigma_{\pi(i)}\rangle-2\rho%
\langle{\boldsymbol{y}}_{\pi(i)},\boldsymbol{\mu}\sigma_{i}\rangle+2\rho\|%
\boldsymbol{\mu}\|^{2}\sigma_{i}\sigma_{\pi(i)}\Bigr{)}.italic_f ( bold_italic_σ , bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( 2 ⟨ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ + 2 ⟨ bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT , bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ⟩ - 2 italic_ρ ⟨ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ⟩ - 2 italic_ρ ⟨ bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT , bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ + 2 italic_ρ ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ) .. , 2 =
Step (a)𝑎(a)( italic_a ) uses the fact that π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT is uniformly distributed over Snsubscript𝑆𝑛S_{n}italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT. Note also that
∑i=1n‖𝒙i‖2+‖𝒚π(i)‖2superscriptsubscript𝑖1𝑛superscriptnormsubscript𝒙𝑖2superscriptnormsubscript𝒚𝜋𝑖2\sum_{i=1}^{n}\|{\boldsymbol{x}}_{i}\|^{2}+\|{\boldsymbol{y}}_{\pi(i)}\|^{2}∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ∥ bold_italic_y start_POSTSUBSCRIPT italic_π ( italic_i ) end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT
is invariant under permutation π𝜋\piitalic_π, ensuring this part does not affect the alignment decision. If the community labels 𝝈𝝈{\boldsymbol{\sigma}}bold_italic_σ and 𝝈πsubscript𝝈𝜋{\boldsymbol{\sigma}}_{\pi}bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT are known, then f(𝝈,𝝈π)𝑓𝝈subscript𝝈𝜋f\bigl{(}{\boldsymbol{\sigma}},{\boldsymbol{\sigma}}_{\pi}\bigr{)}italic_f ( bold_italic_σ , bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT ) is also fixed, so the MAP estimator reduces exactly to the simpler distance-based estimator in (6). Even without known labels, the proof of Theorem 1 shows that (6) attains tight bounds under the conditions ‖𝝁‖2≥2logn+ω(1)superscriptnorm𝝁22𝑛𝜔1\|\boldsymbol{\mu}\|^{2}\geq 2\log n+\omega(1)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ) or d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ); see Section VI for details.
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23 |
Assume d=o(nlogn)𝑑𝑜𝑛𝑛d=o(n\log n)italic_d = italic_o ( italic_n roman_log italic_n ). If ‖𝛍‖2≥(2+ϵ)lognsuperscriptnorm𝛍22italic-ϵ𝑛\|\boldsymbol{\mu}\|^{2}\geq(2+\epsilon)\log n∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 2 + italic_ϵ ) roman_log italic_n for some ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, then by [14], exact community recovery is already possible in each individual database X𝑋Xitalic_X or Y𝑌Yitalic_Y. In this situation, we only need to align nodes within the same community. Consequently, the distance-based estimator (6), which coincides with the MAP rule under known labels, matches nodes optimally. This explains the sufficiency of condition (7) for exact alignment when communities can first be recovered.
On the other hand, if ‖𝛍‖2=O(logn)superscriptnorm𝛍2𝑂𝑛\|\boldsymbol{\mu}\|^{2}=O(\log n)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_O ( roman_log italic_n ) is not large enough to recover community labels, exact matching can still be achieved by taking the condition (8). If d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ), ‖𝐱i−𝐲i‖2superscriptnormsubscript𝐱𝑖subscript𝐲𝑖2\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{i}\|^{2}∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, which originally follows the scaled chi-squared distribution, can be approximated as a normal distribution as follows:
, 1 = ‖𝒙i−𝒚i‖2≈(d)2𝒩((1−ρ)d,2(1−ρ)2d).superscript𝑑superscriptnormsubscript𝒙𝑖subscript𝒚𝑖22𝒩1𝜌𝑑2superscript1𝜌2𝑑\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{i}\|^{2}\stackrel{{\scriptstyle(d)}}{%
{\approx}}2\mathcal{N}((1-\rho)d,2(1-\rho)^{2}d).∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_RELOP SUPERSCRIPTOP start_ARG ≈ end_ARG start_ARG ( italic_d ) end_ARG end_RELOP 2 caligraphic_N ( ( 1 - italic_ρ ) italic_d , 2 ( 1 - italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d ) .. , 2 = . , 3 = (10)
Similarly, for different nodes within the same community, we obtain
, 1 = ‖𝒙i−𝒚j‖2≈(d)2𝒩(d,2d) for i≠j and σi=σj,superscript𝑑superscriptnormsubscript𝒙𝑖subscript𝒚𝑗22𝒩𝑑2𝑑 for 𝑖𝑗 and subscript𝜎𝑖subscript𝜎𝑗\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{j}\|^{2}\stackrel{{\scriptstyle(d)}}{%
{\approx}}2\mathcal{N}(d,2d)\;\text{ for }i\neq j\text{ and }\sigma_{i}=\sigma%
_{j},∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_RELOP SUPERSCRIPTOP start_ARG ≈ end_ARG start_ARG ( italic_d ) end_ARG end_RELOP 2 caligraphic_N ( italic_d , 2 italic_d ) for italic_i ≠ italic_j and italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ,. , 2 = . , 3 = (11)
and for nodes in different communities,
, 1 = ‖𝒙i−𝒚k‖2≈(d)2𝒩(2‖𝝁‖2+d,8‖𝝁‖2+2d) for i≠k and σi≠σk.superscript𝑑superscriptnormsubscript𝒙𝑖subscript𝒚𝑘22𝒩2superscriptnorm𝝁2𝑑8superscriptnorm𝝁22𝑑 for 𝑖𝑘 and subscript𝜎𝑖subscript𝜎𝑘\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{k}\|^{2}\stackrel{{\scriptstyle(d)}}{%
{\approx}}2\mathcal{N}(2\|\boldsymbol{\mu}\|^{2}+d,8\|\boldsymbol{\mu}\|^{2}+2%
d)\;\text{ for }i\neq k\text{ and }\sigma_{i}\neq\sigma_{k}.∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_RELOP SUPERSCRIPTOP start_ARG ≈ end_ARG start_ARG ( italic_d ) end_ARG end_RELOP 2 caligraphic_N ( 2 ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_d , 8 ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 italic_d ) for italic_i ≠ italic_k and italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≠ italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT .. , 2 = . , 3 = (12)
If d≫‖𝛍‖2much-greater-than𝑑superscriptnorm𝛍2d\gg\|\boldsymbol{\mu}\|^{2}italic_d ≫ ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, then we can consider ‖𝐱i−𝐲k‖2≈(d)2𝒩(d,2d)superscript𝑑superscriptnormsubscript𝐱𝑖subscript𝐲𝑘22𝒩𝑑2𝑑\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{k}\|^{2}\stackrel{{\scriptstyle(d)}}{%
{\approx}}2\mathcal{N}(d,2d)∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_RELOP SUPERSCRIPTOP start_ARG ≈ end_ARG start_ARG ( italic_d ) end_ARG end_RELOP 2 caligraphic_N ( italic_d , 2 italic_d ). Thus, through the approximations, for a fixed i𝑖iitalic_i, we can obtain that
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
II-A Exact Matching on Correlated Gaussian Mixture Models
Remark 1 (Interpretation of Conditions for Exact Matching).
Assume d=o(nlogn)𝑑𝑜𝑛𝑛d=o(n\log n)italic_d = italic_o ( italic_n roman_log italic_n ). If ‖𝛍‖2≥(2+ϵ)lognsuperscriptnorm𝛍22italic-ϵ𝑛\|\boldsymbol{\mu}\|^{2}\geq(2+\epsilon)\log n∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 2 + italic_ϵ ) roman_log italic_n for some ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, then by [14], exact community recovery is already possible in each individual database X𝑋Xitalic_X or Y𝑌Yitalic_Y. In this situation, we only need to align nodes within the same community. Consequently, the distance-based estimator (6), which coincides with the MAP rule under known labels, matches nodes optimally. This explains the sufficiency of condition (7) for exact alignment when communities can first be recovered.
On the other hand, if ‖𝛍‖2=O(logn)superscriptnorm𝛍2𝑂𝑛\|\boldsymbol{\mu}\|^{2}=O(\log n)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_O ( roman_log italic_n ) is not large enough to recover community labels, exact matching can still be achieved by taking the condition (8). If d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ), ‖𝐱i−𝐲i‖2superscriptnormsubscript𝐱𝑖subscript𝐲𝑖2\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{i}\|^{2}∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, which originally follows the scaled chi-squared distribution, can be approximated as a normal distribution as follows:
, 1 = ‖𝒙i−𝒚i‖2≈(d)2𝒩((1−ρ)d,2(1−ρ)2d).superscript𝑑superscriptnormsubscript𝒙𝑖subscript𝒚𝑖22𝒩1𝜌𝑑2superscript1𝜌2𝑑\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{i}\|^{2}\stackrel{{\scriptstyle(d)}}{%
{\approx}}2\mathcal{N}((1-\rho)d,2(1-\rho)^{2}d).∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_RELOP SUPERSCRIPTOP start_ARG ≈ end_ARG start_ARG ( italic_d ) end_ARG end_RELOP 2 caligraphic_N ( ( 1 - italic_ρ ) italic_d , 2 ( 1 - italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d ) .. , 2 = . , 3 = (10)
Similarly, for different nodes within the same community, we obtain
, 1 = ‖𝒙i−𝒚j‖2≈(d)2𝒩(d,2d) for i≠j and σi=σj,superscript𝑑superscriptnormsubscript𝒙𝑖subscript𝒚𝑗22𝒩𝑑2𝑑 for 𝑖𝑗 and subscript𝜎𝑖subscript𝜎𝑗\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{j}\|^{2}\stackrel{{\scriptstyle(d)}}{%
{\approx}}2\mathcal{N}(d,2d)\;\text{ for }i\neq j\text{ and }\sigma_{i}=\sigma%
_{j},∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_RELOP SUPERSCRIPTOP start_ARG ≈ end_ARG start_ARG ( italic_d ) end_ARG end_RELOP 2 caligraphic_N ( italic_d , 2 italic_d ) for italic_i ≠ italic_j and italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ,. , 2 = . , 3 = (11)
and for nodes in different communities,
, 1 = ‖𝒙i−𝒚k‖2≈(d)2𝒩(2‖𝝁‖2+d,8‖𝝁‖2+2d) for i≠k and σi≠σk.superscript𝑑superscriptnormsubscript𝒙𝑖subscript𝒚𝑘22𝒩2superscriptnorm𝝁2𝑑8superscriptnorm𝝁22𝑑 for 𝑖𝑘 and subscript𝜎𝑖subscript𝜎𝑘\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{k}\|^{2}\stackrel{{\scriptstyle(d)}}{%
{\approx}}2\mathcal{N}(2\|\boldsymbol{\mu}\|^{2}+d,8\|\boldsymbol{\mu}\|^{2}+2%
d)\;\text{ for }i\neq k\text{ and }\sigma_{i}\neq\sigma_{k}.∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_RELOP SUPERSCRIPTOP start_ARG ≈ end_ARG start_ARG ( italic_d ) end_ARG end_RELOP 2 caligraphic_N ( 2 ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_d , 8 ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 italic_d ) for italic_i ≠ italic_k and italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≠ italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT .. , 2 = . , 3 = (12)
If d≫‖𝛍‖2much-greater-than𝑑superscriptnorm𝛍2d\gg\|\boldsymbol{\mu}\|^{2}italic_d ≫ ∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, then we can consider ‖𝐱i−𝐲k‖2≈(d)2𝒩(d,2d)superscript𝑑superscriptnormsubscript𝐱𝑖subscript𝐲𝑘22𝒩𝑑2𝑑\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{k}\|^{2}\stackrel{{\scriptstyle(d)}}{%
{\approx}}2\mathcal{N}(d,2d)∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_RELOP SUPERSCRIPTOP start_ARG ≈ end_ARG start_ARG ( italic_d ) end_ARG end_RELOP 2 caligraphic_N ( italic_d , 2 italic_d ). Thus, through the approximations, for a fixed i𝑖iitalic_i, we can obtain that
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24 |
, 1 = . , 2 = ℙ(‖𝒙i−𝒚i‖2≥‖𝒙i−𝒚j‖2,∀j≠i)ℙformulae-sequencesuperscriptnormsubscript𝒙𝑖subscript𝒚𝑖2superscriptnormsubscript𝒙𝑖subscript𝒚𝑗2for-all𝑗𝑖\displaystyle\mathbb{P}\left(\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{i}\|^{2}%
\geq\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{j}\|^{2},\;\forall j\neq i\right)blackboard_P ( ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , ∀ italic_j ≠ italic_i ). , 3 = . , 4 = (13). , 1 = . , 2 = ≤(a)nℙ(𝒩((1−ρ)d,2(1−ρ)2d)≥𝒩(d,2d))superscript𝑎absent𝑛ℙ𝒩1𝜌𝑑2superscript1𝜌2𝑑𝒩𝑑2𝑑\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}n\mathbb{P}\left(\mathcal{N}(%
(1-\rho)d,2(1-\rho)^{2}d)\geq\mathcal{N}(d,2d)\right)start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_a ) end_ARG end_RELOP italic_n blackboard_P ( caligraphic_N ( ( 1 - italic_ρ ) italic_d , 2 ( 1 - italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d ) ≥ caligraphic_N ( italic_d , 2 italic_d ) ). , 3 = . , 4 = (13). , 1 = . , 2 = ≤nℙ(𝒩(0,2d(1+(1−ρ)2))≥ρd)absent𝑛ℙ𝒩02𝑑1superscript1𝜌2𝜌𝑑\displaystyle\leq n\mathbb{P}\left(\mathcal{N}(0,2d(1+(1-\rho)^{2}))\geq\rho d\right)≤ italic_n blackboard_P ( caligraphic_N ( 0 , 2 italic_d ( 1 + ( 1 - italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ) ≥ italic_ρ italic_d ). , 3 = . , 4 = (13). , 1 = . , 2 = ≤(b)exp(logn−14dρ2⋅11+(1−ρ)2).superscript𝑏absent𝑛⋅14𝑑superscript𝜌211superscript1𝜌2\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}\exp\left(\log n-\frac{1}{4}d%
\rho^{2}\cdot\frac{1}{1+(1-\rho)^{2}}\right).start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_b ) end_ARG end_RELOP roman_exp ( roman_log italic_n - divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_d italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⋅ divide start_ARG 1 end_ARG start_ARG 1 + ( 1 - italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) .. , 3 = . , 4 = (13)
The inequality (a)𝑎(a)( italic_a ) holds by approximating (12) as (11) and taking a union bound over j≠i𝑗𝑖j\neq iitalic_j ≠ italic_i, and the inequality (b)𝑏(b)( italic_b ) holds by the tail bound of normal distribution (Lemma 19).
Therefore, if 14dρ2≥(1+(1−ρ)2)logn+ω(1)14𝑑superscript𝜌21superscript1𝜌2𝑛𝜔1\frac{1}{4}d\rho^{2}\geq(1+(1-\rho)^{2})\log n+\omega(1)divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_d italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + ( 1 - italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) roman_log italic_n + italic_ω ( 1 ), exact matching becomes achievable even with a greedy algorithm, which attempts to match each point to its nearest neighbor.
If we consider the case where ρ=o(1)𝜌𝑜1\rho=o(1)italic_ρ = italic_o ( 1 ), the above condition becomes 14dρ2≥2logn+ω(1)14𝑑superscript𝜌22𝑛𝜔1\frac{1}{4}d\rho^{2}\geq 2\log n+\omega(1)divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_d italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ), while the bound in (8) is approximately d4log11−ρ2≈14dρ2≥(1+ϵ)logn𝑑411superscript𝜌214𝑑superscript𝜌21italic-ϵ𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\approx\frac{1}{4}d\rho^{2}\geq(1+\epsilon)\log
ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≈ divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_d italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + italic_ϵ ) roman_log italic_n. The tighter bound in (8) is obtained by carefully analyzing the distance-based estimator (6).
We now state the matching impossibility result for correlated GMMs.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
II-A Exact Matching on Correlated Gaussian Mixture Models
Remark 1 (Interpretation of Conditions for Exact Matching).
, 1 = . , 2 = ℙ(‖𝒙i−𝒚i‖2≥‖𝒙i−𝒚j‖2,∀j≠i)ℙformulae-sequencesuperscriptnormsubscript𝒙𝑖subscript𝒚𝑖2superscriptnormsubscript𝒙𝑖subscript𝒚𝑗2for-all𝑗𝑖\displaystyle\mathbb{P}\left(\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{i}\|^{2}%
\geq\|{\boldsymbol{x}}_{i}-{\boldsymbol{y}}_{j}\|^{2},\;\forall j\neq i\right)blackboard_P ( ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ∥ bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - bold_italic_y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , ∀ italic_j ≠ italic_i ). , 3 = . , 4 = (13). , 1 = . , 2 = ≤(a)nℙ(𝒩((1−ρ)d,2(1−ρ)2d)≥𝒩(d,2d))superscript𝑎absent𝑛ℙ𝒩1𝜌𝑑2superscript1𝜌2𝑑𝒩𝑑2𝑑\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}n\mathbb{P}\left(\mathcal{N}(%
(1-\rho)d,2(1-\rho)^{2}d)\geq\mathcal{N}(d,2d)\right)start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_a ) end_ARG end_RELOP italic_n blackboard_P ( caligraphic_N ( ( 1 - italic_ρ ) italic_d , 2 ( 1 - italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d ) ≥ caligraphic_N ( italic_d , 2 italic_d ) ). , 3 = . , 4 = (13). , 1 = . , 2 = ≤nℙ(𝒩(0,2d(1+(1−ρ)2))≥ρd)absent𝑛ℙ𝒩02𝑑1superscript1𝜌2𝜌𝑑\displaystyle\leq n\mathbb{P}\left(\mathcal{N}(0,2d(1+(1-\rho)^{2}))\geq\rho d\right)≤ italic_n blackboard_P ( caligraphic_N ( 0 , 2 italic_d ( 1 + ( 1 - italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ) ≥ italic_ρ italic_d ). , 3 = . , 4 = (13). , 1 = . , 2 = ≤(b)exp(logn−14dρ2⋅11+(1−ρ)2).superscript𝑏absent𝑛⋅14𝑑superscript𝜌211superscript1𝜌2\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}\exp\left(\log n-\frac{1}{4}d%
\rho^{2}\cdot\frac{1}{1+(1-\rho)^{2}}\right).start_RELOP SUPERSCRIPTOP start_ARG ≤ end_ARG start_ARG ( italic_b ) end_ARG end_RELOP roman_exp ( roman_log italic_n - divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_d italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⋅ divide start_ARG 1 end_ARG start_ARG 1 + ( 1 - italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) .. , 3 = . , 4 = (13)
The inequality (a)𝑎(a)( italic_a ) holds by approximating (12) as (11) and taking a union bound over j≠i𝑗𝑖j\neq iitalic_j ≠ italic_i, and the inequality (b)𝑏(b)( italic_b ) holds by the tail bound of normal distribution (Lemma 19).
Therefore, if 14dρ2≥(1+(1−ρ)2)logn+ω(1)14𝑑superscript𝜌21superscript1𝜌2𝑛𝜔1\frac{1}{4}d\rho^{2}\geq(1+(1-\rho)^{2})\log n+\omega(1)divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_d italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + ( 1 - italic_ρ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) roman_log italic_n + italic_ω ( 1 ), exact matching becomes achievable even with a greedy algorithm, which attempts to match each point to its nearest neighbor.
If we consider the case where ρ=o(1)𝜌𝑜1\rho=o(1)italic_ρ = italic_o ( 1 ), the above condition becomes 14dρ2≥2logn+ω(1)14𝑑superscript𝜌22𝑛𝜔1\frac{1}{4}d\rho^{2}\geq 2\log n+\omega(1)divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_d italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ), while the bound in (8) is approximately d4log11−ρ2≈14dρ2≥(1+ϵ)logn𝑑411superscript𝜌214𝑑superscript𝜌21italic-ϵ𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\approx\frac{1}{4}d\rho^{2}\geq(1+\epsilon)\log
ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≈ divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_d italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + italic_ϵ ) roman_log italic_n. The tighter bound in (8) is obtained by carefully analyzing the distance-based estimator (6).
We now state the matching impossibility result for correlated GMMs.
| 1,662 |
25 |
Let (X,Y)∼CGMMs(n,𝛍,d,ρ)similar-to𝑋𝑌CGMMs𝑛𝛍𝑑𝜌(X,Y)\sim\textnormal{CGMMs}(n,\boldsymbol{\mu},d,\rho)( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) be as in Section I-A1. Suppose either
, 1 = d4log11−ρ2≤(1−ϵ)lognand1≪d=O(logn),formulae-sequence𝑑411superscript𝜌21italic-ϵ𝑛andmuch-less-than1𝑑𝑂𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\leq(1-\epsilon)\log n\quad\text{and}\quad 1%
\ll d=O(\log n),divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≤ ( 1 - italic_ϵ ) roman_log italic_n and 1 ≪ italic_d = italic_O ( roman_log italic_n ) ,. , 2 = . , 3 = (14)
for any small ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, or
, 1 = d4log11−ρ2≤logn−logd+Cand1ρ2−1≤d40formulae-sequence𝑑411superscript𝜌2𝑛𝑑𝐶and1superscript𝜌21𝑑40\frac{d}{4}\log\frac{1}{1-\rho^{2}}\leq\log n-\log d+C\quad\text{and}\quad%
\frac{1}{\rho^{2}}-1\leq\frac{d}{40}divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≤ roman_log italic_n - roman_log italic_d + italic_C and divide start_ARG 1 end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG - 1 ≤ divide start_ARG italic_d end_ARG start_ARG 40 end_ARG. , 2 = . , 3 = (15)
for some positive constant C>0𝐶0C>0italic_C > 0. Under either set of conditions, there is no estimator that can achieve exact matching with high probability.
| 628 |
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
II-A Exact Matching on Correlated Gaussian Mixture Models
Theorem 2 (Impossibility for Exact Matching).
Let (X,Y)∼CGMMs(n,𝛍,d,ρ)similar-to𝑋𝑌CGMMs𝑛𝛍𝑑𝜌(X,Y)\sim\textnormal{CGMMs}(n,\boldsymbol{\mu},d,\rho)( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) be as in Section I-A1. Suppose either
, 1 = d4log11−ρ2≤(1−ϵ)lognand1≪d=O(logn),formulae-sequence𝑑411superscript𝜌21italic-ϵ𝑛andmuch-less-than1𝑑𝑂𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\leq(1-\epsilon)\log n\quad\text{and}\quad 1%
\ll d=O(\log n),divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≤ ( 1 - italic_ϵ ) roman_log italic_n and 1 ≪ italic_d = italic_O ( roman_log italic_n ) ,. , 2 = . , 3 = (14)
for any small ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, or
, 1 = d4log11−ρ2≤logn−logd+Cand1ρ2−1≤d40formulae-sequence𝑑411superscript𝜌2𝑛𝑑𝐶and1superscript𝜌21𝑑40\frac{d}{4}\log\frac{1}{1-\rho^{2}}\leq\log n-\log d+C\quad\text{and}\quad%
\frac{1}{\rho^{2}}-1\leq\frac{d}{40}divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≤ roman_log italic_n - roman_log italic_d + italic_C and divide start_ARG 1 end_ARG start_ARG italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG - 1 ≤ divide start_ARG italic_d end_ARG start_ARG 40 end_ARG. , 2 = . , 3 = (15)
for some positive constant C>0𝐶0C>0italic_C > 0. Under either set of conditions, there is no estimator that can achieve exact matching with high probability.
| 674 |
26 |
Comparing Theorems 1 and 2 shows that the limiting condition for exact matching is roughly
, 1 = d4log11−ρ2≥(1+ϵ)logn.𝑑411superscript𝜌21italic-ϵ𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1+\epsilon)\log n.divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n .. , 2 = . , 3 = (16)
However, to attain achievability, we also require ‖𝛍‖2≥2logn+ω(1)superscriptnorm𝛍22𝑛𝜔1\|\boldsymbol{\mu}\|^{2}\geq 2\log n+\omega(1)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ) or d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ). This gap may arise because the distance-based estimator (6) ignores the term f(𝛔,𝛔π)𝑓𝛔subscript𝛔𝜋f({\boldsymbol{\sigma}},{\boldsymbol{\sigma}}_{\pi})italic_f ( bold_italic_σ , bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT ) in (9). This simplification enables us to recover the matching without relying on the knowledge of the community labels 𝛔𝛔{\boldsymbol{\sigma}}bold_italic_σ and 𝛔πsubscript𝛔𝜋{\boldsymbol{\sigma}}_{\pi}bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT, but possibly at the cost of stricter requirements than the full MAP approach.
To prove the converse in Theorem 2, we show that the stated conditions imply failure of exact matching even if the community labels 𝝈1superscript𝝈1{\boldsymbol{\sigma}}^{1}bold_italic_σ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT and 𝝈2superscript𝝈2{\boldsymbol{\sigma}}^{2}bold_italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT are known, and a partial matching for all but |V′|superscript𝑉′|V^{\prime}|| italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | nodes is revealed (where V′:={i∈[n]:σi=+1}assignsuperscript𝑉′conditional-set𝑖delimited-[]𝑛subscript𝜎𝑖1V^{\prime}:=\{\,i\in[n]:\sigma_{i}=+1\}italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := { italic_i ∈ [ italic_n ] : italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = + 1 }). This reduces the task to a database alignment problem with |V′|superscript𝑉′|V^{\prime}|| italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | nodes in correlated Gaussian databases, for which previous impossibility arguments in [9, 36] directly apply. A complete proof is given in Section VII.
| 815 |
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
II-A Exact Matching on Correlated Gaussian Mixture Models
Remark 2 (Gaps in Achievability and Converse Results).
Comparing Theorems 1 and 2 shows that the limiting condition for exact matching is roughly
, 1 = d4log11−ρ2≥(1+ϵ)logn.𝑑411superscript𝜌21italic-ϵ𝑛\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1+\epsilon)\log n.divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n .. , 2 = . , 3 = (16)
However, to attain achievability, we also require ‖𝛍‖2≥2logn+ω(1)superscriptnorm𝛍22𝑛𝜔1\|\boldsymbol{\mu}\|^{2}\geq 2\log n+\omega(1)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ) or d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ). This gap may arise because the distance-based estimator (6) ignores the term f(𝛔,𝛔π)𝑓𝛔subscript𝛔𝜋f({\boldsymbol{\sigma}},{\boldsymbol{\sigma}}_{\pi})italic_f ( bold_italic_σ , bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT ) in (9). This simplification enables us to recover the matching without relying on the knowledge of the community labels 𝛔𝛔{\boldsymbol{\sigma}}bold_italic_σ and 𝛔πsubscript𝛔𝜋{\boldsymbol{\sigma}}_{\pi}bold_italic_σ start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT, but possibly at the cost of stricter requirements than the full MAP approach.
To prove the converse in Theorem 2, we show that the stated conditions imply failure of exact matching even if the community labels 𝝈1superscript𝝈1{\boldsymbol{\sigma}}^{1}bold_italic_σ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT and 𝝈2superscript𝝈2{\boldsymbol{\sigma}}^{2}bold_italic_σ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT are known, and a partial matching for all but |V′|superscript𝑉′|V^{\prime}|| italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | nodes is revealed (where V′:={i∈[n]:σi=+1}assignsuperscript𝑉′conditional-set𝑖delimited-[]𝑛subscript𝜎𝑖1V^{\prime}:=\{\,i\in[n]:\sigma_{i}=+1\}italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := { italic_i ∈ [ italic_n ] : italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = + 1 }). This reduces the task to a database alignment problem with |V′|superscript𝑉′|V^{\prime}|| italic_V start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | nodes in correlated Gaussian databases, for which previous impossibility arguments in [9, 36] directly apply. A complete proof is given in Section VII.
| 864 |
27 |
In [6], it was demonstrated that for correlated SBMs, once exact matching is achieved, combining correlated edges to form a denser union graph can extend the regime where exact community recovery is possible. Similarly, we now identify the conditions under which exact community recovery becomes feasible in correlated Gaussian Mixture Models (GMMs) when two correlated node-attribute databases are available.
| 75 |
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
II-B Exact Community Recovery in Correlated Gaussian Mixture Models
In [6], it was demonstrated that for correlated SBMs, once exact matching is achieved, combining correlated edges to form a denser union graph can extend the regime where exact community recovery is possible. Similarly, we now identify the conditions under which exact community recovery becomes feasible in correlated Gaussian Mixture Models (GMMs) when two correlated node-attribute databases are available.
| 110 |
28 |
Let (X,Y)∼CGMMs(n,𝛍,d,ρ)similar-to𝑋𝑌CGMMs𝑛𝛍𝑑𝜌(X,Y)\sim\textnormal{CGMMs}(n,\boldsymbol{\mu},d,\rho)( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) be as defined in Section I-A1. Suppose either (7) or (8) holds. If
, 1 = ‖𝝁‖2≥(1+ϵ)1+ρ2(1+1+2dnlogn)lognsuperscriptnorm𝝁21italic-ϵ1𝜌2112𝑑𝑛𝑛𝑛\|\boldsymbol{\mu}\|^{2}\geq(1+\epsilon)\frac{1+\rho}{2}\biggl{(}1+\sqrt{1+%
\frac{2d}{n\log n}}\biggr{)}\log n∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + italic_ϵ ) divide start_ARG 1 + italic_ρ end_ARG start_ARG 2 end_ARG ( 1 + square-root start_ARG 1 + divide start_ARG 2 italic_d end_ARG start_ARG italic_n roman_log italic_n end_ARG end_ARG ) roman_log italic_n. , 2 = . , 3 = (17)
for an arbitrarily small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, then there exists an estimator 𝛔^(X,Y)^𝛔𝑋𝑌\hat{{\boldsymbol{\sigma}}}(X,Y)over^ start_ARG bold_italic_σ end_ARG ( italic_X , italic_Y ) such that 𝛔^(X,Y)=𝛔^𝛔𝑋𝑌𝛔\hat{{\boldsymbol{\sigma}}}(X,Y)={\boldsymbol{\sigma}}over^ start_ARG bold_italic_σ end_ARG ( italic_X , italic_Y ) = bold_italic_σ with high probability.
If exact matching is possible–or if π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT is given–then the two correlated node attributes can be merged by taking their average. Specifically, if node i∈[n]𝑖delimited-[]𝑛i\in[n]italic_i ∈ [ italic_n ] has attributes 𝒙isubscript𝒙𝑖{\boldsymbol{x}}_{i}bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and 𝒚π∗(i)subscript𝒚subscript𝜋𝑖{\boldsymbol{y}}_{\pi_{*}(i)}bold_italic_y start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) end_POSTSUBSCRIPT, we form
, 1 = 𝒙i+𝒚π∗(i)2=𝝁σi+(1+ρ)𝒛i+1−ρ2𝒘i2,subscript𝒙𝑖subscript𝒚subscript𝜋𝑖2𝝁subscript𝜎𝑖1𝜌subscript𝒛𝑖1superscript𝜌2subscript𝒘𝑖2\frac{{\boldsymbol{x}}_{i}+{\boldsymbol{y}}_{\pi_{*}(i)}}{2}=\boldsymbol{\mu}%
\sigma_{i}+\frac{(1+\rho){\boldsymbol{z}}_{i}+\sqrt{1-\rho^{2}}{\boldsymbol{w}%
}_{i}}{2},divide start_ARG bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + bold_italic_y start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG = bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + divide start_ARG ( 1 + italic_ρ ) bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + square-root start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ,. , 2 = . , 3 = (18)
where 𝒙isubscript𝒙𝑖{\boldsymbol{x}}_{i}bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and 𝒚isubscript𝒚𝑖{\boldsymbol{y}}_{i}bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are defined in (1) and (2), respectively. Since 𝒛isubscript𝒛𝑖{\boldsymbol{z}}_{i}bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and 𝒘isubscript𝒘𝑖{\boldsymbol{w}}_{i}bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are independent d𝑑ditalic_d-dimensional Gaussian vectors,
(1+ρ)𝒛i+1−ρ2𝒘i21𝜌subscript𝒛𝑖1superscript𝜌2subscript𝒘𝑖2\frac{(1+\rho){\boldsymbol{z}}_{i}+\sqrt{1-\rho^{2}}{\boldsymbol{w}}_{i}}{2}divide start_ARG ( 1 + italic_ρ ) bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + square-root start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG
is a d𝑑ditalic_d-dimensional Gaussian with mean 00 and covariance 1+ρ2𝑰d1𝜌2subscript𝑰𝑑\frac{1+\rho}{2}\,{\boldsymbol{I}}_{d}divide start_ARG 1 + italic_ρ end_ARG start_ARG 2 end_ARG bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT. Therefore, the averaged attribute has the same mean as the individual node attributes but a smaller variance. By applying known results for Gaussian Mixture Models [14, 5] to this averaged feature, we establish Theorem 3. A detailed proof is provided in Section VIII.
| 1,523 |
Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
II-B Exact Community Recovery in Correlated Gaussian Mixture Models
Theorem 3 (Achievability for Exact Community Recovery).
Let (X,Y)∼CGMMs(n,𝛍,d,ρ)similar-to𝑋𝑌CGMMs𝑛𝛍𝑑𝜌(X,Y)\sim\textnormal{CGMMs}(n,\boldsymbol{\mu},d,\rho)( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) be as defined in Section I-A1. Suppose either (7) or (8) holds. If
, 1 = ‖𝝁‖2≥(1+ϵ)1+ρ2(1+1+2dnlogn)lognsuperscriptnorm𝝁21italic-ϵ1𝜌2112𝑑𝑛𝑛𝑛\|\boldsymbol{\mu}\|^{2}\geq(1+\epsilon)\frac{1+\rho}{2}\biggl{(}1+\sqrt{1+%
\frac{2d}{n\log n}}\biggr{)}\log n∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + italic_ϵ ) divide start_ARG 1 + italic_ρ end_ARG start_ARG 2 end_ARG ( 1 + square-root start_ARG 1 + divide start_ARG 2 italic_d end_ARG start_ARG italic_n roman_log italic_n end_ARG end_ARG ) roman_log italic_n. , 2 = . , 3 = (17)
for an arbitrarily small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, then there exists an estimator 𝛔^(X,Y)^𝛔𝑋𝑌\hat{{\boldsymbol{\sigma}}}(X,Y)over^ start_ARG bold_italic_σ end_ARG ( italic_X , italic_Y ) such that 𝛔^(X,Y)=𝛔^𝛔𝑋𝑌𝛔\hat{{\boldsymbol{\sigma}}}(X,Y)={\boldsymbol{\sigma}}over^ start_ARG bold_italic_σ end_ARG ( italic_X , italic_Y ) = bold_italic_σ with high probability.
If exact matching is possible–or if π∗subscript𝜋\pi_{*}italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT is given–then the two correlated node attributes can be merged by taking their average. Specifically, if node i∈[n]𝑖delimited-[]𝑛i\in[n]italic_i ∈ [ italic_n ] has attributes 𝒙isubscript𝒙𝑖{\boldsymbol{x}}_{i}bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and 𝒚π∗(i)subscript𝒚subscript𝜋𝑖{\boldsymbol{y}}_{\pi_{*}(i)}bold_italic_y start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) end_POSTSUBSCRIPT, we form
, 1 = 𝒙i+𝒚π∗(i)2=𝝁σi+(1+ρ)𝒛i+1−ρ2𝒘i2,subscript𝒙𝑖subscript𝒚subscript𝜋𝑖2𝝁subscript𝜎𝑖1𝜌subscript𝒛𝑖1superscript𝜌2subscript𝒘𝑖2\frac{{\boldsymbol{x}}_{i}+{\boldsymbol{y}}_{\pi_{*}(i)}}{2}=\boldsymbol{\mu}%
\sigma_{i}+\frac{(1+\rho){\boldsymbol{z}}_{i}+\sqrt{1-\rho^{2}}{\boldsymbol{w}%
}_{i}}{2},divide start_ARG bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + bold_italic_y start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT ( italic_i ) end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG = bold_italic_μ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + divide start_ARG ( 1 + italic_ρ ) bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + square-root start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ,. , 2 = . , 3 = (18)
where 𝒙isubscript𝒙𝑖{\boldsymbol{x}}_{i}bold_italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and 𝒚isubscript𝒚𝑖{\boldsymbol{y}}_{i}bold_italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are defined in (1) and (2), respectively. Since 𝒛isubscript𝒛𝑖{\boldsymbol{z}}_{i}bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and 𝒘isubscript𝒘𝑖{\boldsymbol{w}}_{i}bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are independent d𝑑ditalic_d-dimensional Gaussian vectors,
(1+ρ)𝒛i+1−ρ2𝒘i21𝜌subscript𝒛𝑖1superscript𝜌2subscript𝒘𝑖2\frac{(1+\rho){\boldsymbol{z}}_{i}+\sqrt{1-\rho^{2}}{\boldsymbol{w}}_{i}}{2}divide start_ARG ( 1 + italic_ρ ) bold_italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + square-root start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG bold_italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG
is a d𝑑ditalic_d-dimensional Gaussian with mean 00 and covariance 1+ρ2𝑰d1𝜌2subscript𝑰𝑑\frac{1+\rho}{2}\,{\boldsymbol{I}}_{d}divide start_ARG 1 + italic_ρ end_ARG start_ARG 2 end_ARG bold_italic_I start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT. Therefore, the averaged attribute has the same mean as the individual node attributes but a smaller variance. By applying known results for Gaussian Mixture Models [14, 5] to this averaged feature, we establish Theorem 3. A detailed proof is provided in Section VIII.
| 1,571 |
29 |
When only X𝑋Xitalic_X is available, [14, 5] showed that exact community recovery in a Gaussian Mixture Model requires
, 1 = ∥𝝁∥2≥(1+ϵ)(1+1+2dnlogn)logn.superscriptdelimited-∥∥𝝁21italic-ϵ112𝑑𝑛𝑛𝑛\lVert\boldsymbol{\mu}\rVert^{2}\geq(1+\epsilon)\left(1+\sqrt{1+\frac{2d}{n%
\log n}}\right)\log n.∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + italic_ϵ ) ( 1 + square-root start_ARG 1 + divide start_ARG 2 italic_d end_ARG start_ARG italic_n roman_log italic_n end_ARG end_ARG ) roman_log italic_n .. , 2 = . , 3 = (19)
Comparing (17) with (19) reveals that ‖𝛍‖2superscriptnorm𝛍2\|\boldsymbol{\mu}\|^{2}∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT can be smaller by a factor of 1+ρ21𝜌2\tfrac{1+\rho}{2}divide start_ARG 1 + italic_ρ end_ARG start_ARG 2 end_ARG when the correlated database Y𝑌Yitalic_Y is available and we can form the combined feature (18) via exact matching. From (8), such a gain is particularly relevant in the regime d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ). If d=O(logn)𝑑𝑂𝑛d=O(\log n)italic_d = italic_O ( roman_log italic_n ), then to satisfy (7) we need ‖𝛍‖2≥2logn+ω(1)superscriptnorm𝛍22𝑛𝜔1\|\boldsymbol{\mu}\|^{2}\geq 2\log n+\omega(1)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ) for exact matching, which can exceed the threshold (17) for community recovery. Hence, in low-dimensional regimes, the advantage of incorporating the second database Y𝑌Yitalic_Y is constrained by the stricter requirement on ‖𝛍‖2superscriptnorm𝛍2\|\boldsymbol{\mu}\|^{2}∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT for alignment.
We now provide the converse result, stating when exact community recovery is not possible in correlated GMMs.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
II-B Exact Community Recovery in Correlated Gaussian Mixture Models
Remark 3 (Comparison with Standard GMM Results).
When only X𝑋Xitalic_X is available, [14, 5] showed that exact community recovery in a Gaussian Mixture Model requires
, 1 = ∥𝝁∥2≥(1+ϵ)(1+1+2dnlogn)logn.superscriptdelimited-∥∥𝝁21italic-ϵ112𝑑𝑛𝑛𝑛\lVert\boldsymbol{\mu}\rVert^{2}\geq(1+\epsilon)\left(1+\sqrt{1+\frac{2d}{n%
\log n}}\right)\log n.∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 + italic_ϵ ) ( 1 + square-root start_ARG 1 + divide start_ARG 2 italic_d end_ARG start_ARG italic_n roman_log italic_n end_ARG end_ARG ) roman_log italic_n .. , 2 = . , 3 = (19)
Comparing (17) with (19) reveals that ‖𝛍‖2superscriptnorm𝛍2\|\boldsymbol{\mu}\|^{2}∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT can be smaller by a factor of 1+ρ21𝜌2\tfrac{1+\rho}{2}divide start_ARG 1 + italic_ρ end_ARG start_ARG 2 end_ARG when the correlated database Y𝑌Yitalic_Y is available and we can form the combined feature (18) via exact matching. From (8), such a gain is particularly relevant in the regime d=ω(logn)𝑑𝜔𝑛d=\omega(\log n)italic_d = italic_ω ( roman_log italic_n ). If d=O(logn)𝑑𝑂𝑛d=O(\log n)italic_d = italic_O ( roman_log italic_n ), then to satisfy (7) we need ‖𝛍‖2≥2logn+ω(1)superscriptnorm𝛍22𝑛𝜔1\|\boldsymbol{\mu}\|^{2}\geq 2\log n+\omega(1)∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 2 roman_log italic_n + italic_ω ( 1 ) for exact matching, which can exceed the threshold (17) for community recovery. Hence, in low-dimensional regimes, the advantage of incorporating the second database Y𝑌Yitalic_Y is constrained by the stricter requirement on ‖𝛍‖2superscriptnorm𝛍2\|\boldsymbol{\mu}\|^{2}∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT for alignment.
We now provide the converse result, stating when exact community recovery is not possible in correlated GMMs.
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Let (X,Y)∼CGMMs(n,𝛍,d,ρ)similar-to𝑋𝑌CGMMs𝑛𝛍𝑑𝜌(X,Y)\sim\textnormal{CGMMs}(n,\boldsymbol{\mu},d,\rho)( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) as in Section I-A1. Suppose
, 1 = ∥𝝁∥2≤(1−ϵ)1+ρ2(1+1+2dnlogn)lognsuperscriptdelimited-∥∥𝝁21italic-ϵ1𝜌2112𝑑𝑛𝑛𝑛\lVert\boldsymbol{\mu}\rVert^{2}\leq(1-\epsilon)\frac{1+\rho}{2}\left(1+\sqrt{%
1+\frac{2d}{n\log n}}\right)\log n∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ ( 1 - italic_ϵ ) divide start_ARG 1 + italic_ρ end_ARG start_ARG 2 end_ARG ( 1 + square-root start_ARG 1 + divide start_ARG 2 italic_d end_ARG start_ARG italic_n roman_log italic_n end_ARG end_ARG ) roman_log italic_n. , 2 = . , 3 = (20)
for an arbitrarily small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0. Then, there is no estimator that achieves exact community recovery with high probability.
The proof of Theorem 4 follows the same reasoning as in Theorem 8 for the special case p,q=0𝑝𝑞0p,q=0italic_p , italic_q = 0, thus it can be viewed as a corollary of that result.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
II-B Exact Community Recovery in Correlated Gaussian Mixture Models
Theorem 4 (Impossibility for Exact Community Recovery).
Let (X,Y)∼CGMMs(n,𝛍,d,ρ)similar-to𝑋𝑌CGMMs𝑛𝛍𝑑𝜌(X,Y)\sim\textnormal{CGMMs}(n,\boldsymbol{\mu},d,\rho)( italic_X , italic_Y ) ∼ CGMMs ( italic_n , bold_italic_μ , italic_d , italic_ρ ) as in Section I-A1. Suppose
, 1 = ∥𝝁∥2≤(1−ϵ)1+ρ2(1+1+2dnlogn)lognsuperscriptdelimited-∥∥𝝁21italic-ϵ1𝜌2112𝑑𝑛𝑛𝑛\lVert\boldsymbol{\mu}\rVert^{2}\leq(1-\epsilon)\frac{1+\rho}{2}\left(1+\sqrt{%
1+\frac{2d}{n\log n}}\right)\log n∥ bold_italic_μ ∥ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ ( 1 - italic_ϵ ) divide start_ARG 1 + italic_ρ end_ARG start_ARG 2 end_ARG ( 1 + square-root start_ARG 1 + divide start_ARG 2 italic_d end_ARG start_ARG italic_n roman_log italic_n end_ARG end_ARG ) roman_log italic_n. , 2 = . , 3 = (20)
for an arbitrarily small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0. Then, there is no estimator that achieves exact community recovery with high probability.
The proof of Theorem 4 follows the same reasoning as in Theorem 8 for the special case p,q=0𝑝𝑞0p,q=0italic_p , italic_q = 0, thus it can be viewed as a corollary of that result.
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Theorem 3 assumes that either (7) or (8) is satisfied to ensure exact matching. This raises a natural question: if matching is not feasible (i.e., d4log11−ρ2<logn𝑑411superscript𝜌2𝑛\tfrac{d}{4}\log\tfrac{1}{1-\rho^{2}}<\log ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG < roman_log italic_n), must exact community recovery also remain impossible, even if (17) holds? Addressing this question when exact matching is precluded is an interesting open problem, discussed further in Section V.
Figure 1 illustrates the threshold conditions for both exact matching and exact community recovery in correlated Gaussian Mixture Models. It visually highlights how the addition of correlated node attributes expands the parameter regime where community detection becomes feasible.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
II Correlated Gaussian Mixture Models
II-B Exact Community Recovery in Correlated Gaussian Mixture Models
Remark 4 (Information-Theoretic Gaps in Exact Community Recovery).
Theorem 3 assumes that either (7) or (8) is satisfied to ensure exact matching. This raises a natural question: if matching is not feasible (i.e., d4log11−ρ2<logn𝑑411superscript𝜌2𝑛\tfrac{d}{4}\log\tfrac{1}{1-\rho^{2}}<\log ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG < roman_log italic_n), must exact community recovery also remain impossible, even if (17) holds? Addressing this question when exact matching is precluded is an interesting open problem, discussed further in Section V.
Figure 1 illustrates the threshold conditions for both exact matching and exact community recovery in correlated Gaussian Mixture Models. It visually highlights how the addition of correlated node attributes expands the parameter regime where community detection becomes feasible.
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32 |
In this section, we consider the correlated Contextual Stochastic Block Models (CSBMs) introduced in Section I-A2. Similar to Section II, we first establish the conditions for exact matching and then derive the conditions for exact community recovery, assuming we can perfectly match the nodes between the two correlated graphs.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
III Correlated Contextual Stochastic Block Models
In this section, we consider the correlated Contextual Stochastic Block Models (CSBMs) introduced in Section I-A2. Similar to Section II, we first establish the conditions for exact matching and then derive the conditions for exact community recovery, assuming we can perfectly match the nodes between the two correlated graphs.
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The following theorem provides sufficient conditions for exact matching in correlated CSBMs.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
III Correlated Contextual Stochastic Block Models
III-A Exact Matching
The following theorem provides sufficient conditions for exact matching in correlated CSBMs.
| 44 |
34 |
Let (G1,G2)∼CCSBMs(n,p,q,s;R,d,ρ)similar-tosubscript𝐺1subscript𝐺2CCSBMs𝑛𝑝𝑞𝑠𝑅𝑑𝜌(G_{1},G_{2})\sim\textnormal{CCSBMs}(n,p,q,s;R,d,\rho)( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CCSBMs ( italic_n , italic_p , italic_q , italic_s ; italic_R , italic_d , italic_ρ ) be as defined in Section I-A2, and suppose
, 1 = p≤O(1e(loglogn)3),𝑝𝑂1superscript𝑒superscript𝑛3p\leq O\left(\frac{1}{e^{(\log\log n)^{3}}}\right),italic_p ≤ italic_O ( divide start_ARG 1 end_ARG start_ARG italic_e start_POSTSUPERSCRIPT ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT end_ARG ) ,. , 2 = . , 3 = (21)
as well as
, 1 = R≥2logn+ω(1) or d=ω(logn)𝑅2𝑛𝜔1 or 𝑑𝜔𝑛R\geq 2\log n+\omega(1)\text{ or }d=\omega(\log n)italic_R ≥ 2 roman_log italic_n + italic_ω ( 1 ) or italic_d = italic_ω ( roman_log italic_n ). , 2 = . , 3 = (22)
holds. If
, 1 = ns2p+q2+d4log11−ρ2≥(1+ϵ)logn𝑛superscript𝑠2𝑝𝑞2𝑑411superscript𝜌21italic-ϵ𝑛ns^{2}\frac{p+q}{2}+\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1+\epsilon)\log nitalic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG + divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n. , 2 = . , 3 = (23)
for an arbitrarily small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, then there exists an estimator π^(G1,G2)^𝜋subscript𝐺1subscript𝐺2\hat{\pi}(G_{1},G_{2})over^ start_ARG italic_π end_ARG ( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) such that π^(G1,G2)=π∗^𝜋subscript𝐺1subscript𝐺2subscript𝜋\hat{\pi}(G_{1},G_{2})=\pi_{*}over^ start_ARG italic_π end_ARG ( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT with high probability.
In [6, 35, 7], it was shown that for correlated SBMs, exact matching is possible if the average degree of the intersection graph ns2p+q2𝑛superscript𝑠2𝑝𝑞2ns^{2}\tfrac{p+q}{2}italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG exceeds logn𝑛\log nroman_log italic_n. Hence, we focus on the regime ns2p+q2=O(logn)𝑛superscript𝑠2𝑝𝑞2𝑂𝑛ns^{2}\tfrac{p+q}{2}=O(\log n)italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG = italic_O ( roman_log italic_n ), where we employ a two-step approach. First, we use k𝑘kitalic_k-core matching to align a large fraction of the nodes via edge information alone; the k𝑘kitalic_k-core approach has been extensively studied in [11, 8, 35] and is adapted here to handle more general regimes of p𝑝pitalic_p and q𝑞qitalic_q. By choosing an appropriate k𝑘kitalic_k, we obtain a partial matching that is both large (in terms of the number of matched nodes) and accurate (no mismatches). The remaining unmatched node pairs follow the correlated Gaussian Mixture Model framework, so we apply Theorem 1 to match those residual pairs via the node-attribute estimator.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
III Correlated Contextual Stochastic Block Models
III-A Exact Matching
Theorem 5 (Achievability for Exact Matching).
Let (G1,G2)∼CCSBMs(n,p,q,s;R,d,ρ)similar-tosubscript𝐺1subscript𝐺2CCSBMs𝑛𝑝𝑞𝑠𝑅𝑑𝜌(G_{1},G_{2})\sim\textnormal{CCSBMs}(n,p,q,s;R,d,\rho)( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ∼ CCSBMs ( italic_n , italic_p , italic_q , italic_s ; italic_R , italic_d , italic_ρ ) be as defined in Section I-A2, and suppose
, 1 = p≤O(1e(loglogn)3),𝑝𝑂1superscript𝑒superscript𝑛3p\leq O\left(\frac{1}{e^{(\log\log n)^{3}}}\right),italic_p ≤ italic_O ( divide start_ARG 1 end_ARG start_ARG italic_e start_POSTSUPERSCRIPT ( roman_log roman_log italic_n ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT end_ARG ) ,. , 2 = . , 3 = (21)
as well as
, 1 = R≥2logn+ω(1) or d=ω(logn)𝑅2𝑛𝜔1 or 𝑑𝜔𝑛R\geq 2\log n+\omega(1)\text{ or }d=\omega(\log n)italic_R ≥ 2 roman_log italic_n + italic_ω ( 1 ) or italic_d = italic_ω ( roman_log italic_n ). , 2 = . , 3 = (22)
holds. If
, 1 = ns2p+q2+d4log11−ρ2≥(1+ϵ)logn𝑛superscript𝑠2𝑝𝑞2𝑑411superscript𝜌21italic-ϵ𝑛ns^{2}\frac{p+q}{2}+\frac{d}{4}\log\frac{1}{1-\rho^{2}}\geq(1+\epsilon)\log nitalic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG + divide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n. , 2 = . , 3 = (23)
for an arbitrarily small constant ϵ>0italic-ϵ0\epsilon>0italic_ϵ > 0, then there exists an estimator π^(G1,G2)^𝜋subscript𝐺1subscript𝐺2\hat{\pi}(G_{1},G_{2})over^ start_ARG italic_π end_ARG ( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) such that π^(G1,G2)=π∗^𝜋subscript𝐺1subscript𝐺2subscript𝜋\hat{\pi}(G_{1},G_{2})=\pi_{*}over^ start_ARG italic_π end_ARG ( italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT with high probability.
In [6, 35, 7], it was shown that for correlated SBMs, exact matching is possible if the average degree of the intersection graph ns2p+q2𝑛superscript𝑠2𝑝𝑞2ns^{2}\tfrac{p+q}{2}italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG exceeds logn𝑛\log nroman_log italic_n. Hence, we focus on the regime ns2p+q2=O(logn)𝑛superscript𝑠2𝑝𝑞2𝑂𝑛ns^{2}\tfrac{p+q}{2}=O(\log n)italic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG = italic_O ( roman_log italic_n ), where we employ a two-step approach. First, we use k𝑘kitalic_k-core matching to align a large fraction of the nodes via edge information alone; the k𝑘kitalic_k-core approach has been extensively studied in [11, 8, 35] and is adapted here to handle more general regimes of p𝑝pitalic_p and q𝑞qitalic_q. By choosing an appropriate k𝑘kitalic_k, we obtain a partial matching that is both large (in terms of the number of matched nodes) and accurate (no mismatches). The remaining unmatched node pairs follow the correlated Gaussian Mixture Model framework, so we apply Theorem 1 to match those residual pairs via the node-attribute estimator.
| 1,272 |
35 |
Condition (21) ensures that the k𝑘kitalic_k-core algorithm yields the correct matching for nodes within the k𝑘kitalic_k-core of G1∧π∗G2subscriptsubscript𝜋subscript𝐺1subscript𝐺2G_{1}\wedge_{\pi_{*}}G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Condition (22) is required for matching the remaining nodes using node attributes. The combined inequality (23) highlights the benefit of leveraging both edge and node-attribute correlations. As a baseline, if d=0𝑑0d=0italic_d = 0 (no node attributes), the model reduces to correlated SBMs, and (23) becomes ns2p+q2≥(1+ϵ)logn𝑛superscript𝑠2𝑝𝑞21italic-ϵ𝑛ns^{2}\tfrac{p+q}{2}\geq(1+\epsilon)\,\log nitalic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n, matching the exact-matching threshold in [6, 35, 7]. Conversely, if p=q=0𝑝𝑞0p=q=0italic_p = italic_q = 0 (no edges), the model is just correlated GMMs, and (23) becomes d4log(11−ρ2)≥(1+ϵ)logn𝑑411superscript𝜌21italic-ϵ𝑛\frac{d}{4}\log\!\bigl{(}\tfrac{1}{1-\rho^{2}}\bigr{)}\geq(1+\epsilon)\log ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log ( divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ≥ ( 1 + italic_ϵ ) roman_log italic_n, consistent with Theorem 1.
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Exact Matching in Correlated Networks with Node Attributes for Improved Community Recovery
III Correlated Contextual Stochastic Block Models
III-A Exact Matching
Remark 5 (Interpretation of Conditions for Exact Matching).
Condition (21) ensures that the k𝑘kitalic_k-core algorithm yields the correct matching for nodes within the k𝑘kitalic_k-core of G1∧π∗G2subscriptsubscript𝜋subscript𝐺1subscript𝐺2G_{1}\wedge_{\pi_{*}}G_{2}italic_G start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∧ start_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_G start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. Condition (22) is required for matching the remaining nodes using node attributes. The combined inequality (23) highlights the benefit of leveraging both edge and node-attribute correlations. As a baseline, if d=0𝑑0d=0italic_d = 0 (no node attributes), the model reduces to correlated SBMs, and (23) becomes ns2p+q2≥(1+ϵ)logn𝑛superscript𝑠2𝑝𝑞21italic-ϵ𝑛ns^{2}\tfrac{p+q}{2}\geq(1+\epsilon)\,\log nitalic_n italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT divide start_ARG italic_p + italic_q end_ARG start_ARG 2 end_ARG ≥ ( 1 + italic_ϵ ) roman_log italic_n, matching the exact-matching threshold in [6, 35, 7]. Conversely, if p=q=0𝑝𝑞0p=q=0italic_p = italic_q = 0 (no edges), the model is just correlated GMMs, and (23) becomes d4log(11−ρ2)≥(1+ϵ)logn𝑑411superscript𝜌21italic-ϵ𝑛\frac{d}{4}\log\!\bigl{(}\tfrac{1}{1-\rho^{2}}\bigr{)}\geq(1+\epsilon)\log ndivide start_ARG italic_d end_ARG start_ARG 4 end_ARG roman_log ( divide start_ARG 1 end_ARG start_ARG 1 - italic_ρ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ≥ ( 1 + italic_ϵ ) roman_log italic_n, consistent with Theorem 1.
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