Clique Comparator: A Fundamental Operator for Finding a Concise Clique Summary

Maximal cliques are useful in many applications, e.g., community detection, bioinformatics, anomaly detection and graph visualization. Enumerating maximal cliques from a graph is a computationally challenging problem, as the output size can be exponentially large with respect to the vertex number. Such a large number of cliques typically overlap heavily, which brings information redundancy when being applied to aforementioned domains. This paper studies how to use a small set of maximal cliques, i.e., a summary, to summarize all the maximal cliques in a graph. The state-of-the-art suffers from inefficiency in updating the summary with progressively generated cliques, especially when the summary grows large. In this work, we identify the challenge of summary updating to be how to efficiently estimate the overlap between cliques in the summary and each newly found maximal clique. By exploiting the vertex order information, we propose the notion of clique comparator, and devise four types of operators to quickly identify clique overlap in less costly manners. We conduct extensive experiments on six real-world datasets to verify the effectiveness of our approach, which reduces unnecessary clique intersection calculations by at least seven orders of magnitude and achieves a speedup of 2.5 3.1 times compared to the state-of-the-art.