Seeded Graph Matching: Efficient Algorithms and Theoretical Guarantees

In this paper, a new information theoretic framework for graph matching is introduced. Using this framework, the graph isomorphism and seeded graph matching problems are studied. The maximum degree algorithm for graph isomorphism is analyzed and sufficient conditions for successful matching are rederived using type analysis. Furthermore, a new seeded matching algorithm with polynomial time complexity is introduced. The algorithm uses `typicality matching' and techniques from point-to-point communications for reliable matching. Assuming an Erdos-Renyi model on the correlated graph pair, it is shown that successful matching is guaranteed when the number of seeds grows logarithmically with the number of vertices in the graphs. The logarithmic coefficient is shown to be inversely proportional to the mutual information between the edge variables in the two graphs

Graph matching beyond perfectly-overlapping Erdős–Rényi random graphs

Graph matching is a fruitful area in terms of both algorithms and theories. Given two graphs G1=(V1,E1) and G2=(V2,E2), where V1 and V2 are the same or largely overlapped upon an unknown permutation π∗, graph matching is to seek the correct mapping π∗. In this paper, we exploit the degree information, which was previously used only in noiseless graphs and perfectly-overlapping Erdős–Rényi random graphs matching. We are concerned with graph matching of partially-overlapping graphs and stochastic block models, which are more useful in tackling real-life problems. We propose the edge exploited degree profile graph matching method and two refined variations. We conduct a thorough analysis of our proposed methods’ performances in a range of challenging scenarios, including coauthorship data set and a zebrafish neuron activity data set. Our methods are proved to be numerically superior than the state-of-the-art methods. The algorithms are implemented in the R (A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, 2020) package GMPro (GMPro: graph matching with degree profiles, 2020)