Partial Recovery in the Graph Alignment Problem

Abstract

In this paper, we consider the graph alignment problem, which is the problem of recovering, given two graphs, a one-to-one mapping between nodes that maximizes edge overlap. This problem can be viewed as a noisy version of the well-known graph isomorphism problem and appears in many applications, including social network deanonymization and cellular biology. Our focus here is on partial recovery, i.e., we look for a one-to-one mapping which is correct on a fraction of the nodes of the graph rather than on all of them, and we assume that the two input graphs to the problem are correlated Erdős-Rényi graphs of parameters (n, q, s). Our main contribution is then to give necessary and sufficient conditions on (n, q, s) under which partial recovery is possible with high probability as the number of nodes n goes to infinity. In particular, we show that it is possible to achieve partial recovery in the nqs = Θ(1) regime under certain additional assumptions.

Settling the Sharp Reconstruction Thresholds of Random Graph Matching

“…Interestingly, our proof relies on establishing an asymptotically matching lower bound to the mutual information I(π; A, B). This significantly deviates from the existing results in [HM20] based on Fano's inequality: P {overlap( π, π) ≤ δ} ≥ 1 − I(π;A,B)+1 log(n!/m) with m = |{π : overlap(π, π) ≥ δ}|, followed by applying the simple bound (6).…”

“…Capitalizing on this key finding and applying the Chernoff bound together with a union bound over π ′ yields our tight conditions. We remark that the partial recovery results in [HM20] are obtained by analyzing an estimator slightly different from the MLE and the same MGF bound (15) is used. However, there are two major differences that led to the looseness of their results.…”

“…It is shown in [CKMP19] that in the sparse regime p = n −Ω(1) , almost exact recovery is information-theoretically possible if and only if nps 2 = ω(1). The more recent work [HM20] further investigates partial recovery. It is shown that if nps 2 ≥ C(δ) max 1, log n log(1/p) , then there exists an exponential-time estimator π that achieves overlap ( π, π) ≥ δ with high probability, where C(δ) is some large constant that tends to ∞ as δ → 1; conversely, if…”

“…In contrast, approximate recovery are far less well understood. Apart from the sharp condition for almost exact recovery in the sparse regime p = n −Ω(1) [CKMP19], only upper and lower bounds are known for partial recovery [HM20]. See Section 1.2 for a detailed review of these previous results.…”

Testing correlation of unlabeled random graphs

“…It is shown that the partial recovery can be attained in polynomial time by a neighborhood tree matching algorithm in the sparse graph regime where nps ∈ (1, λ 0 ] for some constant λ 0 close to 1 and s ∈ (s 0 , 1] for some constant s 0 close to 1. More recently, the partial recovery is shown to be information-theoretically impossible if nps 2 log 1 p = o(1) when p = o(1) [HM20]. In contrast to the aforementioned work focusing on the estimation problem, our paper studies the hypothesis testing aspect of graph matching.…”