Information Recovery in Shuffled Graphs via Graph Matching

Abstract

While many multiple graph inference methodologies operate under the implicit assumption that an explicit vertex correspondence is known across the vertex sets of the graphs, in practice these correspondences may only be partially or errorfully known. Herein, we provide an information theoretic foundation for understanding the practical impact that errorfully observed vertex correspondences can have on subsequent inference, and the capacity of graph matching methods to recover the lost vertex alignment and inferential performance. Working in the correlated stochastic blockmodel setting, we establish a duality between the loss of mutual information due to an errorfully observed vertex correspondence and the ability of graph matching algorithms to recover the true correspondence across graphs. In the process, we establish a phase transition for graph matchability in terms of the correlation across graphs, and we conjecture the analogous phase transition for the relative information loss due to shuffling vertex labels. We demonstrate the practical effect that graph shufflingand matching-can have on subsequent inference, with examples from two sample graph hypothesis testing and joint spectral graph clustering.

Matchability of heterogeneous networks pairs

“…In this setting, the growth condition of Theorem 6, p = ω( log n/n), can hold in the dense setting (p = Θ(1)) as well as when p/ = Ω(1). In the dense setting of p = Θ(1), δ GMP -matchability transitions at 2 = Θ( log n n ) [38,40], which our Theorem recovers (asymptotically).…”

“…In recent work addressing the question of δ GMP -matchability, results have been established for the R = J n , Q 1 = Q 2 = pJ n setting (see, for example, [46,40,4,18]), in the correlated stochastic blockmodel setting (see, for example, [45,38]), in the correlated heterogeneous Erdős-Renyi model (see, for example, [39,41]), and in the general R and general Q 1 = Q 2 = Q setting (see, for example, [49,42]). In the non-identically distributed model setting, the work in [14,15,16] considers R = J n , Q 1 = pJ n , and Q 2 = qJ n .…”

“…For each P in Π(n), define The proof of Theorem 6 relies on a now standard application of McDiarmid's inequality and is similar to the proofs of analogous matchability results in [40,39,38]; details of the proof can be found in Appendix A.1.…”

“…However, recent approaches that leverage efficient representation/learning methodologies (see, for example, [5,59,26]) have shown excellent empirical performance matching networks with up to millions of nodes.In addition to algorithmic advancements in graph matching, there has been a flurry of activity studying the closely related problem of graph matchability: Given a latent alignment between the vertex sets of two graphs, can graph matching uncover this alignment in the presence of shuffled vertex labels? This problem arises in a variety of contexts, from network de-anonymization and privatization to multi-network hypothesis testing [38] to multimodality graph embedding methodologies [11]. Many existing results are concerned with recovering a latent alignment present across random graph models where each of A and B have identical marginal distributions, and exciting advancements on the threshold of matchable versus unmatchable graphs have been made across many random graph settings, including: the homogeneous correlated Erdős-Renyi model (see, for example, [46,40,4,18]), the correlated stochastic blockmodel setting (see, for example, [45,38]), the ρ-correlated heterogeneous Erdős-Renyi model (see, for example, [39,41]), and in the correlated heterogeneous Erdős-Renyi model with varying edge correlations (see, for example, [49,42]).…”

“…This problem arises in a variety of contexts, from network de-anonymization and privatization to multi-network hypothesis testing [38] to multimodality graph embedding methodologies [11]. Many existing results are concerned with recovering a latent alignment present across random graph models where each of A and B have identical marginal distributions, and exciting advancements on the threshold of matchable versus unmatchable graphs have been made across many random graph settings, including: the homogeneous correlated Erdős-Renyi model (see, for example, [46,40,4,18]), the correlated stochastic blockmodel setting (see, for example, [45,38]), the ρ-correlated heterogeneous Erdős-Renyi model (see, for example, [39,41]), and in the correlated heterogeneous Erdős-Renyi model with varying edge correlations (see, for example, [49,42]). In the non-identically distributed model setting, the work in [14,15,16] provide theoretic phase transitions on matchability in the (A, B) ∼Erdős-Rényi(p,q, ) model (i.e., A ∼Erdős-Rényi(n,p), B ∼Erdős-Rényi(n,q) and the edge correlation across graphs in provided by the constant ; see Definition 2).The above results range from providing theoretic phase transitions on matchability [14,15,38] to providing nearly efficient methods for achieving matchability from an algorithmic perspective [18,4,16,22].…”