Exact alignment recovery for correlated Erd\H{o}s-R\'enyi graphs

We consider the problem of perfectly recovering the vertex correspondence between two correlated Erdős-Rényi (ER) graphs on the same vertex set. The correspondence between the vertices can be obscured by randomly permuting the vertex labels of one of the graphs. We determine the information-theoretic threshold for exact recovery, i.e. the conditions under which the entire vertex correspondence can be correctly recovered given unbounded computational resources. Graph alignment is the problem finding a matching between the vertices of the two graphs that matches, or aligns, many edges of the first graph with edges of the second graph. Alignment is a generalization of graph isomorphism recovery to non-isomorphic graphs. Graph alignment can be applied in the deanonymization of social networks, the analysis of protein interaction networks, and computer vision. Narayanan and Shmatikov successfully deanonymized an anonymized social network dataset by graph alignment with a publicly available network . In order to make privacy guarantees in this setting, it is necessary to understand the conditions under which graph alignment recovery is possible. We consider graph alignment for a randomized graphpair model. This generation procedure creates a “planted” alignment: there is a ground-truth relationship between the vertices of the two graphs. Pedarsani and Grossglauser  were the first to approach the problem of finding informationtheoretic conditions for alignment recovery. They established conditions under which exact recovery of the planted alignment is possible, The authors improved on these conditions and also established conditions under which exact recover is impossible . In this paper, we close the gap between these results and establish the precise threshold for exact recovery in sparse graphs. As a special case, we recover a result of Wright  about the conditions under which an Erdős-Rényi graph has a trivial automorphism group. I. MODEL A. The alignment recovery problem We consider the following problem. There are two correlated graphs Ga and Gb, both on the vertex set [n] = {0, 1, . . . , n−1}. By correlation we mean that for each vertex pair e, presence or absence of e ∈ E(Ga), or equivalently The material in this paper was presented in part at the ACM SIGMETRICS conference, Antibes Juan-les-Pins, France, June 2016 . Daniel Cullina is with the Department of Electrical, Princeton University, Princeton, New Jersey 08540 (email: dcullina@princeton.edu). Negar Kiyavash is with the Department of Electrical and Computer Engineering, the Department of Industrial and Enterprise Systems Engineering, and the Coordinated Science Laboratory, University of Illinois at ChampaignUrbana, Urbana, Illinois 61801 (email: kiyavash@illinois.edu). the indicator variable Ga(e), provides some information about Gb(e). The true vertex labels of Ga are removed and replaced with meaningless labels. We model this by applying a uniformly random permutation Π to map the vertices of Ga to the vertices of its anonymized version. The anonymized graph is Gc, where Gc({Π(i),Π(j)}) = Ga({i, j}) for all i, j ∈ [n], i 6= j. The original vertex labels of Gb are preserved and Gc and Gb are revealed. We would like to know under what conditions it is possible to discover the true correspondence between the vertices of Ga and the vertices of Gb. In other words, when can the random permutation Π be exactly recovered with high probability? In this context, an achievability result demonstrates the existence of an algorithm or estimator that exactly recovers Π with high probability. A converse result is an upper bound on the probability of exact recovery that applies to any estimator. B. Correlated Erdős-Rényi graphs To fully specify this problem, we need to define a joint distribution over Ga and Gb. In this paper, we will focus on Erdős-Rényi (ER) graphs. We discuss some of the advantages and drawbacks of this model in Section II-F. We will generate correlated Erdős-Rényi graphs as follows. Let Ga and Gb be graphs on the vertex set [n]. We will think of (Ga, Gb) as a single function with codomain {0, 1}2: (Ga, Gb)(e) = (Ga(e), Gb(e)). The random variables (Ga, Gb)(e), e ∈ ( [n] 2 ) , are i.i.d. and (Ga, Gb)(e) =    (1, 1) w.p. p11 (1, 0) w.p. p10 (0, 1) w.p. p01 (0, 0) w.p. p00. Call this distribution ER(n,p), where p = (p11, p10, p01, p00). Note that the marginal distributions of Ga and Gb are Erdős-Rényi and so is the distribution of the intersection graph Ga ∧ Gb: Ga ∼ ER(n, p10 + p11), Gb ∼ ER(n, p01 + p11), and Ga ∧Gb ∼ ER(n, p11). When p11 > (p10+ p11)(p01+ p11), we say that the graphs Ga and Gb have positive correlation. Observe that p11 − (p10 + p11)(p01 + p11) = p11p00 − p01p10 so p11p00 > p10p01 is an equivalent, more symmetric condition for positive correlation. In this paper, we only address the case of positive correlation.