Abstract
Graph matching is a fruitful area in terms of both algorithms and theories. Given two graphs G_1 = (V_1, E_1) and G_2 = (V_2, E_2), where V_1 and V_2 are the same or largely overlapped upon an unknown permutation pi ^*, graph matching is to seek the correct mapping pi ^*. 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).
Highlights
Graph matching has been an active area of research for decades
Before we formally state the problem framework and our proposed methods, we present some real data analysis based on the coauthorship data set generated from the coauthor relationship between authors who published papers in the Annals of Statistics (AoS), Biometrika, the Journal of American Statistician Association (JASA) and the Journal of Royal Statistical Society, Series B (JRSSB)
As explained in Remark 3 in Ding et al (2018), the time complexity of calculating W is of order O(n3ρ + n2), where n and ρ represent the magnitude of the network sizes and the upper bound on the connectivity probabilities in both networks, respectively
Summary
Graph matching has been an active area of research for decades. Due to the advancements in collecting, storing and processing large volume of data, graph matching is going through a renaissance, with a surge of work on graph matching in different application areas. Narayanan and Shmatikov (2009) targeted at acquiring information from an anonymous graph of Twitter with the graph of Flickr as the auxiliary information; Kazemi et al (2016) seek the alignment of protein-protein interaction networks in order to uncover the relationships between different species; Haghighi et al (2005) constructed graphs based on texts relationship and developed a system for deciding whether a given sentence can be inferred from text by matching graphs
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