Efficient random graph matching via degree profiles

• 成果类型: Article
• 署名作者: Ding, Jian; Ma, Zongming; Wu, Yihong; Xu, Jiaming
• 署名单位: University of Pennsylvania; Yale University; Duke University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-020-00997-4
• 发表日期: 2021
• 页码: 29-115
• 关键词: INEQUALITIES; alignment

摘要

Random graph matching refers to recovering the underlying vertex correspondence between two random graphs with correlated edges; a prominent example is when the two random graphs are given by Erdos-Renyi graphs G(n, d/n). This can be viewed as an average-case and noisy version of the graph isomorphism problem. Under this model, the maximum likelihood estimator is equivalent to solving the intractable quadratic assignment problem. This work develops an (O) over tilde (nd(2) + n(2))-time algorithm which perfectly recovers the true vertex correspondence with high probability, provided that the average degree is at least d = Omega(log(2) n) and the two graphs differ by at most delta = O(log(-2)(n)) fraction of edges. For dense graphs and sparse graphs, this can be improved to delta = O(log(-2/3)(n)) and delta = O(log(-2)(d)) respectively, both in polynomial time. The methodology is based on appropriately chosen distance statistics of the degree profiles (empirical distribution of the degrees of neighbors). Before this work, the best known result achieves delta = O(1) and n(o)(1) <= d <= n(c) for some constant c with an n(O(log n))-time algorithm and delta = (O) over tilde((d/n)(4)) and d = (Omega) over tilde (n(4/5)) with a polynomial-time algorithm.