Testing network correlation efficiently via counting trees

Abstract

We propose a new procedure for testing whether two networks are edge-correlated through some latent vertex correspondence. The test statistic is based on counting the co-occurrences of signed trees for a family of non-isomorphic trees. When the two networks are Erdős-Rényi random graphs G(n, q) that are either independent or correlated with correlation coefficient ρ, our test runs in n 2+o(1) time and succeeds with high probability as n → ∞, provided that n min{q, 1 − q} ≥ n −o(1) and ρ 2 > α ≈ 0.338, where α is Otter's constant so that the number of unlabeled trees with K edges grows as (1/α) K . This significantly improves the prior work in terms of statistical accuracy, running time, and graph sparsity.

Detection threshold for correlated Erdős-Rényi graphs via densest subgraphs

“…Despite the fact that Erdős-Rényi Graph perhaps does not quite capture important features for any network arising from realistic problems, (similar to most problems on networks) it is plausible that a complete understanding for the case of Erdős-Rényi Graphs forms an important and necessary step toward the much more ambitious goal of mathematically understanding graph detection and matching problems for realistic networks arising from applications (note that for many applications it remains a substantial challenge to propose a reasonable underlying random graph model). Along this line, many progress has been made recently, including information-theoretic analysis [10,9,24,43,42] and proposals for various efficient algorithms [34,44,26,25,17,38,4,13,6,11,12,30,16,21,15,27,28]. Out of these references, the ones closely related to our work include (the aforementioned) [43] and [42] which studied the information-theoretic threshold for the matching problem, as well as [28] which obtained an efficient algorithm for detection when the correlation between the two graphs is above a certain constant.…”

“…Along this line, many progress has been made recently, including information-theoretic analysis [10,9,24,43,42] and proposals for various efficient algorithms [34,44,26,25,17,38,4,13,6,11,12,30,16,21,15,27,28]. Out of these references, the ones closely related to our work include (the aforementioned) [43] and [42] which studied the information-theoretic threshold for the matching problem, as well as [28] which obtained an efficient algorithm for detection when the correlation between the two graphs is above a certain constant. As of now, a huge informationcomputation gap remains for both detection and matching problems, and it is a major challenge to completely understand the phase transition for the computational complexity for either detection or matching problems.…”

Aligning random graphs with a sub-tree similarity message-passing algorithm

“…In the latter perspective, the rather modest dependency of s c and s algo on λ in the investigated range, and the shape of the curve in figure 9, could lead to the conjecture that s c (λ) reaches a strictly positive value in the limit λ → ∞. Let us further mention what is at the moment an intriguing numerical coincidence; the authors of [32] studied the detection problem associated to graph alignment, namely the hypothesis testing question of, given a pair of graphs, distinguishing their generation probability between the correlated ER law and the product of two independent ER laws with the same marginals as the correlated one. They defined an estimator based on the correlation of the number of trees embedded as subgraphs in the observed pairs of graphs, and characterized the range of parameters for which this estimator achieves asymptotically a vanishing probability of error under both hypotheses; in the constant-degree regime this happens for all λ > 0 whenever s > √ α, where α is the Otter's constant [33] that governs the rate of growth of the number of unlabelled trees with the number of vertices (the results of [32] actually cover also denser regimes with degrees diverging with n).…”

“…Let us further mention what is at the moment an intriguing numerical coincidence; the authors of [32] studied the detection problem associated to graph alignment, namely the hypothesis testing question of, given a pair of graphs, distinguishing their generation probability between the correlated ER law and the product of two independent ER laws with the same marginals as the correlated one. They defined an estimator based on the correlation of the number of trees embedded as subgraphs in the observed pairs of graphs, and characterized the range of parameters for which this estimator achieves asymptotically a vanishing probability of error under both hypotheses; in the constant-degree regime this happens for all λ > 0 whenever s > √ α, where α is the Otter's constant [33] that governs the rate of growth of the number of unlabelled trees with the number of vertices (the results of [32] actually cover also denser regimes with degrees diverging with n). The numerical value of this threshold is √ α ≈ 0.5817, indicated with an horizontal dot-dashed line in figure 9, just slightly below the typical values of s c we observed, which makes this constant √ α a possible candidate for the conjectured limit of s c (λ) when λ → ∞.…”

Matching recovery threshold for correlated random graphs

“…Despite the fact that Erdős-Rényi Graph perhaps does not quite capture important features for any network arising from realistic problems, it is nevertheless reasonable to start our (presumably long) journey of completely understanding the information-computation phase transition for graph matching problems from a clean, simple and in some sense canonical random graph model such as Erdős-Rényi. Along this line, many progress has been made recently, including information-theoretic analysis [12,11,26,42,41] and proposals for various efficient algorithms [34,43,28,27,19,37,5,16,7,13,14,31,18,23,17,29,30].…”