2021/01/29 by Wu, Yihong, Xu, Jiaming, Yu, Sophie H. · 4 citations
#05C60 #05C80 #62B10 #68Q87 #94A15 #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Machine Learning (stat.ML) #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.2102.00082
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. The proof of the negative results builds upon a tight characterization of the mutual information based on the truncated second-moment computation 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.