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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 ... | 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 ... | 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 classica... | 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 correla... | 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 detectio... | 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 ma... | 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... | 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... | 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_POSTS... | 1,763 |
6 | 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 := [ bol... | 1,180 | 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…... | 1,211 |
7 | 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_... | 1,219 | 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{... | 1,252 |
8 | 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 ... | 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_POSTSUPERSC... | 999 |
9 | 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 i... | 80 | 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. Ta... | 102 |
10 | 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_POSTSUBSCR... | 1,594 | 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,... | 1,627 |
11 | 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... | 711 | 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... | 742 |
12 | 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 correlate... | 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... | 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}i... | 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 w... | 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... | 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 o... | 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}... | 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_AR... | 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 ( i... | 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... | 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_POSTSUPE... | 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 sta... | 735 |
18 | 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, ... | 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 c... | 112 |
19 | 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 wh... | 71 |
20 | 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-sequence𝑑411supersc... | 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,\bol... | 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_... | 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... | 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_ital... | 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 ... | 1,640 |
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>0i... | 1,883 | 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 ro... | 1,930 |
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)black... | 1,615 | 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-sequencesuperscript... | 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-ϵ... | 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,\bol... | 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 - itali... | 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 matchin... | 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 G... | 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... | 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)lognsuperscriptnorm𝝁2... | 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\text... | 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... | 665 | 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 r... | 711 |
30 | 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\... | 440 | 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\text... | 488 |
31 | 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... | 217 | 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... | 267 |
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... | 64 | 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 exa... | 88 |
33 | The following theorem provides sufficient conditions for exact matching in correlated CSBMs. | 15 | 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... | 1,231 | 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\textno... | 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_POSTSUB... | 522 | 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 ... | 564 |
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