Settling the Sharp Reconstruction Thresholds of Random Graph Matching

Abstract

This paper studies the problem of recovering the hidden vertex correspondence between two edge-correlated random graphs. We focus on the Gaussian model where the two graphs are complete graphs with correlated Gaussian weights and the Erdős-Rényi model where the two graphs are subsampled from a common parent Erdős-Rényi graph G(n, p). For dense graphs with p = n −o(1) , we prove that there exists a sharp threshold, above which one can correctly match all but a vanishing fraction of vertices and below which correctly matching any positive fraction is impossible, a phenomenon known as the "all-or-nothing" phase transition. Even more strikingly, in the Gaussian setting, above the threshold all vertices can be exactly matched with high probability. In contrast, for sparse Erdős-Rényi graphs with p = n −Θ(1) , we show that the all-or-nothing phenomenon no longer holds and we determine the thresholds up to a constant factor. Along the way, we also derive the sharp threshold for exact recovery, sharpening the existing results in Erdős-Rényi graphs [CK16, CK17].The proof of the negative results builds upon a tight characterization of the mutual information based on the truncated second-moment computation in [WXY20] and an "area theorem" that relates the mutual information to the integral of the reconstruction error. The positive results follows from a tight analysis of the maximum likelihood estimator that takes into account the cycle structure of the induced permutation on the edges.

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

“…In the easy phase partial recovery is achievable by a polynomial-time algorithm, in the hard phase it is achievable but in an a priori exponential time, while in the impossible phase the information contained in the graphs is insufficient to recover a constant fraction of the hidden permutation, even without bounds on the computational power employed. The evidences in favor of this shape of the phase diagram are on the one hand the numerical results presented in this paper, the boundary of the easy phase corresponding to the threshold s algo (λ), and on the other hand the various bounds previously obtained in the literature: λs < 1 is a sufficient condition to be in the impossible phase [17], for λs > 4 partial recovery is information-theoretically possible [16], hence this regime corresponds to an easy or hard phase. As the lowerbound on s c given by theorem 5 in [18] crosses the line λs = 4 for λ large enough, a hard phase must appear in this regime.…”

“…This implies that for λs ≤ 1 partial recovery is information-theoretically infeasible, i.e., for this set of parameters it is not possible to recover π , not even partially. On the other hand, in [16] it is proven instead that for λs > 4 partial recovery is information-theoretically feasible (improving on a previous bound in [19]). It has also been shown in [15,18] that there exists a polynomial-time feasible phase in the region λs > 1 (for large enough values of s).…”

“…On the other hand, it is impossible to exactly recover π for λ = O(1) [11]. For this reason the authors of [15][16][17][18] focused in this regime on the possibility of a partial recovery of the true labelling. In [17] it is shown that the fraction of correctly matched pairs of vertices is upper-bounded by c(λs), the largest non-negative solution of the equation 1 − c = e −λsc , for any statistical estimator.…”

“…Then two graphs are obtained from it by randomly removing edges from the parent one with probability 1 − s, independently for each of the two copies, and reshuffling the vertex labels of one of the graphs. The information-theoretical possibility of exact recovery in this ensemble has been characterized in [21] for the regime of degrees logarithmic in n. Another variant of the problem concerns the alignment of correlated random matrices, which corresponds to the case of weighted complete graphs [12,13,16,22,23]: one draws a pair of correlated random matrices (A, C) such that, independently for all i < j, A ij and C ij are standard Gaussian variables with correlation coefficient s ∈ [0, 1]. The observed pair is obtained by reshuffling the rows and columns of one of the matrices, C ij = B π (i) π (j) , and the goal is to recover the uniformly random permutation π from the observation of (A, B).…”

“…In [12,13] it is shown that a spectral (polynomial-time) algorithm exactly recovers π if √ 1 − s 2 = O(ln −1 n). On the other hand, in [16,23] it is proven that it is information-theoretically possible to recover π if s 2 ≥ c ln n n for some c > 4: this suggests that in this problem there might be a wide hard phase in which polynomial-time algorithms cannot exactly recover π .…”