Fast Maximal Clique Enumeration for Real-World Graphs

Abstract

Maximal Clique Enumeration (MCE) is one of the most fundamental problems in graph theory, and it has extensive applications in graph data analysis. The state-of-art approach (called as $MCE_{degeneracy}$ in this paper) that solves MCE problem in real-world graphs first computes the degeneracy ordering of the vertices in a given graph, and then for each vertex, conducts the $BK_{pivot}$ algorithm in its neighborhood (called as degeneracy neighborhood in this paper). In real-world graphs, the process of degeneracy ordering produces a large number of dense degeneracy neighborhoods. But, the $BK_{pivot}$ algorithm, with its down-to-top nature, adds just one vertex into the result set at each level of recursive calls, and cannot efficiently solve the MCE problem in these dense degeneracy neighborhoods.

In this paper, we propose a new MCE algorithm, called as $BK_{rcd}$, to improve the efficiency of MCE in a dense degeneracy neighborhood by recursively conducting core decomposition in it. Contrary to $BK_{pivot}$, $BK_{rcd}$ is a top-to-down approach, that repeatedly chooses and “removes” the vertex with the smallest degree until a clique is reached. We further integrate $BK_{rcd}$ into $MCE_{degeneracy}$ to form a hybrid approach named as $MCE_{degeneracy}^{hybrid}$, that chooses $BK_{rcd}$ or $BK_{pivot}$ adaptively according to the structural properties of the degeneracy neighborhoods. Experimental results conducted in real-world graphs show that $MCE_{degeneracy}^{hybrid}$ achieves high overall performance improvements on the graphs. For example, $MCE_{degeneracy}^{hybrid}$ achieves 1.34$\times$ to 2.97$\times$ speedups over $MCE_{degeneracy}$ in web graphs taken in our experiments.