Partial Recovery of Erdðs-Rényi Graph Alignment via k-Core Alignment

Abstract

We determine information theoretic conditions under which it is possible to partially recover the alignment used to generate a pair of sparse, correlated Erdős-Rényi graphs. To prove our achievability result, we introduce the k-core alignment estimator. This estimator searches for an alignment in which the intersection of the correlated graphs using this alignment has a minimum degree of k. We prove a matching converse bound. As the number of vertices grows, recovery of the alignment for a fraction of the vertices tending to one is possible when the average degree of the intersection of the graph pair tends to infinity. It was previously known that exact alignment is possible when this average degree grows faster than the logarithm of the number of vertices. *

Matchability of heterogeneous networks pairs

“…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 . In each setting, the results showed that for sufficiently dense, sufficiently correlated graphs, δ GMP -matchability is almost surely achieved.…”

“…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]. While they have served to establish a novel theoretical understanding of the matchability problem, in each case the transition from matchable to unmatchable graphs is defined in terms of decreasing across-graph correlation and within-graph sparsity.…”

“…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]. While they have served to establish a novel theoretical understanding of the matchability problem, in each case the transition from matchable to unmatchable graphs is defined in terms of decreasing across-graph correlation and within-graph sparsity.…”

Multiplex graph matching matched filters

“…This is the multiplex analogue of graph matchability, i.e., uncovering conditions under which oracle graph matching will recover a latent vertex alignment. Here, that alignment is represented by H being an errorful version of G[m]; see, for example, [14,20,33,32,26,21,12,11,5,13] for a litany of graph matchability results in the monoplex setting.…”