Efficient random graph matching via degree profiles

Abstract

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 Erdős-Rényi graphs $G(n,\frac{d}{n})$. 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 $\tilde{O}(n{d}^2+n^2)$-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 $n^{o(1)}\le d\le n^c$ for some constant $c$ with an $n^{O(\log n)}$-time algorithm \cite{barak2018nearly} and $\delta =\tilde{O}((d/n{)}^4)$ and $d=\tilde{\Omega }(n^{4/5})$ with a polynomial-time algorithm \cite{dai2018performance}.

arXiv commentProof of Theorem 4 expanded and revised

References in corpus (4)

• De-anonymizing Social Networks
• Seeded graph matching for correlated Erdős-Rényi graphs
• (Nearly) Efficient Algorithms for the Graph Matching Problem on Correlated Random Graphs
• Exact alignment recovery for correlated Erdős-Rényi graphs

Cited by in corpus (8)

• Spectral Graph Matching and Regularized Quadratic Relaxations II: Erdős-Rényi Graphs and Universality
• Spectral Graph Matching and Regularized Quadratic Relaxations I: The Gaussian Model
• Testing network correlation efficiently via counting trees
• Impossibility of Partial Recovery in the Graph Alignment Problem
• Sharp threshold for alignment of graph databases with Gaussian weights
• Graph Matching with Partially-Correct Seeds
• Random Graph Matching with Improved Noise Robustness
• Graph matching beyond perfectly-overlapping Erdős--Rényi random graphs