Correlated Stochastic Block Models: Exact Graph Matching with Applications to Recovering Communities

1 Introduction . Learning community structure in networks is a ubiquitous inference task in several domains , including biology [ 13 , 41 ] , sociology [ 26 ] , and machine learning [ 57 , 37 , 60 ] . Recent decades have therefore seen an explosion of work on the topic , leading to determining the fundamental information-theoretic limits for learning communities in probabilistic generative models [ 1 , 2 , 3 , 45 , 44 , 42 , 46 ] , as well as algorithms that work well in practice [ 55 , 33 , 22 ] . Typically , such algorithms only leverage the structure of the network ( i.e. , the configuration of node-node links ) . Increasingly , one often has access to side information that can greatly improve the performance of inference algorithms . There is a vast literature on designing algorithms that incorporate various types of side information to aid in recovering communities in networks . The works [ 21 , 38 , 47 , 32 , 8 , 63 , 11 , 56 , 39 ] leverage node-level information that is correlated with community memberships ; here the sharp limits for community detection were conjectured by Deshpande et al . [ 21 ] and recently proven by Lu and Sen [ 38 ] . Another line of work [ 30 , 5 , 51 , 52 , 36 , 4 , 7 , 43 , 39 ] recovers communities from a multilayer network , where the different layers are conditionally independent given the same community structure . Recently Ma and Nandy [ 39 ] synthesized these two strands of literature . In contrast to prior work , we explore scenarios where the side information comes in the form of multiple correlated networks , which is natural in several domains including social networks [ 49 , 53 , 35 ] , computational biology [ 59 ] , and machine learning [ 15 , 14 ] . In the context of social networks , for instance , many datasets are anonymized to protect the identity of users . Nevertheless , one may be able to infer additional information about users from additional networks by noting that the interaction patterns of the same set of users are likely to be similar across networks [ 49 , 53 , 35 ] . In computational biology , an important goal is to study the functional properties of protein groups 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . through a protein-protein interaction ( PPI ) network . Using the insight that functionally similar protein groups will have similar interaction structures , one can compare PPIs across species to infer protein functions [ 59 ] . In all of these examples , an important task , commonly known as graph matching , is to synthesize the information from multiple correlated networks in a sensible manner . To the best of our knowledge , we are the first to consider the use of multiple correlated networks for recovering communities . Specifically , we quantify , in an information-theoretic sense , how much information we can gain from correlated networks in order to infer community structure . To this end , we focus on correlated graphs G1 and G2 drawn marginally according to the stochastic block model ( SBM ) , which is widely recognized as the canonical probabilistic generative model for networks with community structure [ 31 , 1 ] . The reason for studying this probabilistic model is twofold . For one , it serves as a prototypical model for networks with community structure found in practice , hence the algorithms we develop will serve as a starting point for applications . Moreover , the SBM has well-defined ground-truth communities , so we can concretely study the correctness of algorithms in terms of whether the communities they output align with the ground truth . 2 Models and Questions . The stochastic block model ( SBM ) . The SBM is perhaps the simplest and most well-known probabilistic generative model for networks with community structure . It was initially proposed by Holland , Laskey , and Leinhardt [ 31 ] and subsequently used as a theoretical testbed for evaluating clustering algorithms on average-case networks ( see , e.g. , [ 24 , 12 , 9 ] ) . A striking fact about the SBM is that it exhibits sharp information-theoretic phase transitions for various inference tasks , leading to a precise understanding of when community information can be extracted from network data . Such phase transitions were first conjectured by Decelle et al . [ 20 ] and were subsequently proven rigorously by several authors [ 44 , 42 , 46 , 2 , 45 , 3 , 10 , 1 ] . In summary , the SBM is a well-motivated and mathematically rich setting for studying inference tasks . In this work we focus on the simplest setting , a SBM with two symmetric communities . For a positive integer n and p , q ∈ [ 0 , 1 ] , we construct G ∼ SBM ( n , p , q ) as follows . The graph G has n vertices , labeled by the elements of [ n ] : = { 1 , . . . , n } . Each vertex i ∈ [ n ] has a community label σi ∈ { +1 , −1 } ; these are drawn i.i.d . uniformly at random across all i ∈ [ n ] . Let σ : = { σi } ni=1 be the vector of community labels , with the two communities given by the sets V+ : = { i ∈ [ n ] : σi = +1 } and V− : = { i ∈ [ n ] : σi = −1 } . Then , given the community labels σ , the edges of G are drawn independently across vertex pairs as follows . For distinct i , j ∈ [ n ] , if σiσj = 1 , then the edge ( i , j ) is in G with probability p ; else ( i , j ) is in G with probability q . Community recovery . Generally speaking , a community recovery algorithm takes as input G ( without knowledge of the community labels σ ) and outputs a community labeling σ̂ . The overlap between the estimated labeling and the ground truth is given by ov ( σ̂ , σ ) : = 1 n ∣∣∣∣∣ n∑ i=1 σ̂iσi ∣∣∣∣∣ . In the formula for the overlap , we take an absolute value since the labelings σ and −σ specify the same community partition ( and it is only possible to recover σ up to its sign ) . Moreover , notice that ov ( σ̂ , σ ) is always between 0 and 1 , where a larger value corresponds to a better match between the estimated communities and the ground truth . Indeed , the algorithm succeeds in exactly recovering the communities ( i.e. , σ̂ = σ or σ̂ = −σ ) if and only if ov ( σ̂ , σ ) = 1 . In the logarithmic degree regime—that is , when p = α log ( n ) /n and q = β log ( n ) /n for some fixed constants α , β ≥ 0—it is well-known that there is a sharp information-theoretic threshold for exactly recovering communities in the SBM [ 2 , 45 , 3 , 1 ] . Specifically , if∣∣∣√α−√β∣∣∣ > √2 , ( 2.1 ) then exact recovery is possible : there is a polynomial-time algorithm which outputs an estimator σ̂ satisfying limn→∞ P ( ov ( σ̂ , σ ) = 1 ) = 1 . On the other hand , if∣∣∣√α−√β∣∣∣ < √2 , ( 2.2 ) then exact recovery is impossible : for any estimator σ̃ , we have that limn→∞ P ( ov ( σ̃ , σ ) = 1 ) = 0 . Correlated SBMs . The goal of our work is to understand how side information in the form of multiple correlated SBMs affects the threshold given by ( 2.1 ) and ( 2.2 ) . To construct a pair of correlated SBMs , we define an additional parameter s ∈ [ 0 , 1 ] which controls the level of correlation between the two graphs . Formally , we construct ( G1 , G2 ) ∼ CSBM ( n , p , q , s ) as follows . First , generate a parent graph G ∼ SBM ( n , p , q ) , and let σ denote the community labels . Next , given G , we construct G1 and G′2 by independent subsampling : each edge of G is included in G1 with probability s , independently of everything else , and non-edges of G remain non-edges in G1 ; we obtain G′2 independently in the same fashion . Note that G1 and G ′ 2 inherit the vertex labels from the parent graph G , and the community labels are given by σ in both graphs . Finally , we let π∗ be a uniformly random permutation of [ n ] , independently of everything else , and generate G2 by relabeling the vertices of G′2 according to π∗ ( e.g. , vertex i in G ′ 2 is relabeled to π∗ ( i ) in G2 ) . This last step in the construction of G2 reflects the observation that in applications , node labels are often obscured . This construction is visualized in Figure 1 . This model of correlated SBMs was first studied by Onaran , Erkip , and Garg [ 50 ] . This process of generating correlated graphs ( i.e. , by first generating a parent graph , independently subsampling it , and randomly permuting the labels ) is a natural and common approach for inducing correlation in the formation of edges , and has been employed to study correlated graphs from the Erdős-Rényi model ( see , e.g. , [ 53 ] , as well as further references in Section 4 ) , the Chung-Lu model [ 62 ] , and the preferential attachment model [ 35 ] . An important property of the construction is that marginally G1 and G2 are both SBMs . Specifically , since the subsampling probability is s , we have that G1 ∼ SBM ( n , ps , qs ) . In the logarithmic degree regime , where p = α log ( n ) /n and q = β log ( n ) /n , ( 2.1 ) implies that the communities can be exactly recovered from G1 alone if ∣∣∣√α−√β∣∣∣ > √2 s . ( 2.3 ) A central question of our work is how one can utilize the side information in G2 to go beyond the single-graph threshold ( 2.3 ) . This is formalized as follows . Objective 1 ( Exact community recovery ) . Given ( G1 , G2 ) ∼ CSBM ( n , α lognn , β logn n , s ) , deter- mine conditions on α , β , and s so that there exists an estimator σ̂ = σ̂ ( G1 , G2 ) satisfying lim n→∞ P ( ov ( σ̂ , σ ) = 1 ) = 1 . A key observation is that if the latent correspondence π∗ is known , then one can readily improve the achievability region in ( 2.3 ) . Indeed , if π∗ is known , then one can reconstruct G′2 from G2 . We can then construct a new graph H∗ by “ overlaying '' G1 and G′2 ( i.e. , taking their union ) . Formally , ( i , j ) is an edge in H∗ if and only if ( i , j ) is an edge in G1 or G′2 . An equivalent interpretation is that ( i , j ) is an edge in the parent graph G and it is included in either G1 or G′2 in the subsampling process . The probability that the edge is not included in either G1 or G′2 is ( 1− s ) 2 , so it follows that H∗ ∼ SBM ( n , α ( 1− ( 1− s ) 2 ) log ( n ) /n , β ( 1− ( 1− s ) 2 ) log ( n ) /n ) . By ( 2.1 ) it thus follows that exact community recovery is possible if∣∣∣√α−√β∣∣∣ > √ 2 1− ( 1− s ) 2 . ( 2.4 ) Since 1 − ( 1 − s ) 2 > s for s ∈ ( 0 , 1 ) , ( 2.4 ) is a strict improvement over ( 2.3 ) . Remarkably , this implies that if π∗ is known and if√ 2 s > ∣∣∣√α−√β∣∣∣ > √ 2 1− ( 1− s ) 2 , then it is information-theoretically impossible to exactly recover σ from G1 ( or G2 ) alone , but one can recover σ exactly by combining information from G1 and G2 . To make this rigorous , we study when it is possible to exactly recover π∗ from G1 and G2 . This task is known as graph matching . Objective 2 ( Exact graph matching ) . Given ( G1 , G2 ) ∼ CSBM ( n , α lognn , β logn n , s ) , determine conditions on α , β , and s so that there exists an estimator π̂ = π̂ ( G1 , G2 ) satisfying lim n→∞ P ( π̂ = π∗ ) = 1 . While we have motivated graph matching as an intermediate step in recovering communities , it is an important problem in its own right , with applications to data privacy in social networks [ 49 , 53 ] , protein-protein interaction networks [ 59 ] , and machine learning [ 15 , 14 ] , among others . In particular , it is well known that graph matching algorithms can be used to de-anonymize social networks [ 49 ] , showing that anonymity is not the same as privacy . Studying the fundamental limits of when graph matching is possible can serve to highlight the precise conditions when anonymity can indeed guarantee privacy , and when additional safeguards are necessary . Although Objective 2 has not been studied previously , there is strong evidence of a phase transition for exact recovery of π∗ in the logarithmic degree regime . In the special case of correlated Erdős-Rényi graphs—that is , when α = β—Cullina and Kiyavash [ 16 , 17 ] showed that the maximum likelihood estimate exactly recovers π∗ with probability tending to 1 if s2α > 1 . When α ̸= β , and assuming that the community labels are known in both graphs , Onaran , Garg , and Erkip [ 50 ] showed that exact recovery of π∗ is possible if s ( 1− √ 1− s2 ) ( α+ β ) /2 > 3 . Cullina et al . [ 19 ] , also assuming that community labels are known in both graphs , stated ( without proof ) that exact recovery is possible if s2 ( α+ β ) /2 > 2 . Since these works assume knowledge of community labels , it is unclear if these conditions allow to recover π∗ based on knowledge of only G1 and G2 . Nevertheless , they suggest that exact graph matching may be possible in the logarithmic degree regime . Turning to impossibility results , in correlated Erdős-Rényi graphs , if s2α < 1 , then there is no estimator which exactly recovers π∗ with probability bounded away from zero [ 16 , 17 , 61 ] . For correlated SBMs , Cullina et al . [ 19 ] showed that one can not exactly recover π∗ when s2 ( α+β ) /2 < 1 . In particular , for correlated Erdős-Rényi graphs the information-theoretic threshold s2α = 1 is the connectivity threshold for the intersection graph of G1 and G′2 . ( Given two graphs H1 and H2 , the edge ( i , j ) is in the intersection graph of H1 and H2 if and only if it is an edge in both H1 and H2 . ) For correlated SBMs the connectivity threshold for the intersection graph is s2 ( α+ β 2 ) = 1 . ( 2.5 ) This suggests that ( 2.5 ) may be the information-theoretic threshold for exact recovery of π∗ for correlated SBMs . Our main result , Theorem 3.1 , shows that this is indeed the case .

Summary: This paper provides necessary and sufficient conditions for exact recovery problems in a two-community correlated stochastic block model (SBMs). In this model, correlated SBMs are generated on the same set of vertices in a natural way and then the vertices of  one of the SBMs is randomly permuted. The paper gives necessary and sufficient conditions for 1) recovering the (unknown) permutation to recover the correspondence between the two SBMs and 2) recovering the underlying community memberships.  The extension to multiple correlated SBMs in also considered.