Detection threshold for correlated Erdős-Rényi graphs via densest subgraphs

Abstract

The problem of detecting edge correlation between two Erdős-Rényi random graphs on n unlabeled nodes can be formulated as a hypothesis testing problem: under the null hypothesis, the two graphs are sampled independently; under the alternative, the two graphs are independently sub-sampled from a parent graph which is Erdős-Rényi G(n, p) (so that their marginal distributions are the same as the null). We establish a sharp information-theoretic threshold when p = n −α+o(1) for α ∈ (0, 1] which sharpens a constant factor in a recent work by Wu, Xu and Yu. A key novelty in our work is an interesting connection between the detection problem and the densest subgraph of an Erdős-Rényi graph.

Matching recovery threshold for correlated random graphs

“…As in many other problems of this type, an information-theoretic phase transition is easier and usually will also guide the study on the transition for computational complexity. Together with previous works [42,41,15] (which were naturally inspired by earlier works such as [12,11,26]), it seems now we have achieved a fairly satisfying understanding on the information-theoretic transition and we hope that this may be of help for future study on computational aspects.…”

“…for the inverse function of ̺. Building on insights from [42] and using Proposition 1.2, we have established the following detection threshold in [15] as stated in the next proposition. Define P as the law of a pair of independent Erdős-Rényi graphs G(n, ps) on V and V, respectively.…”

“…We have learned from [42,41] important insights on information thresholds for graph matching problems. An additional ingredient we realized is the connection to the densest subgraph, which allowed us to improve [42] and establish the sharp detection threshold as in [15]. The densest subgraph problem arose in the study of load balancing problem [25] and much progress has been made on densest subgraphs for random graphs [8,20,24,21].…”

“…It is perhaps not surprising that the partial recovery threshold coincides with the detection threshold. On the contrary, what may be unexpected at the first glance is that given [15] it still requires substantial amount of non-trivial work to prove impossibility of partial recovery as the default folklore assumption is that detection is easier than recovery. Indeed, detection is easier than exact recovery since with possibility of exact recovery one should be able to detect the correlation by examining the intersection of the two graphs under this (estimated) matching.…”

“…In this section, we prove Theorem 1.1-(1.2), where λ ≥ λ * + ε for some arbitrary and fixed ε > 0. The construction of the estimator π naturally takes inspiration from the detection statistics as in [15,Section 2]. For instance, we may simply define π to be the maximizer for…”