Ordering heuristics for k -clique listing

Abstract

Listing all k -cliques in a graph is a fundamental graph mining problem that finds many important applications in community detection and social network analysis. Unfortunately, the problem of k -clique listing is often deemed infeasible for a large k , as the number of k -cliques in a graph is exponential in the size k. The state-of-the-art solutions for the problem are based on the ordering heuristics on nodes which can efficiently list all k -cliques in large real-world graphs for a small k (e.g., k ≤ 10). Even though a variety of heuristic algorithms have been proposed, there still lacks a thorough comparison to cover all the state-of-the-art algorithms and evaluate their performance using diverse real-world graphs. This makes it difficult for a practitioner to select which algorithm should be used for a specific application. Furthermore, existing ordering based algorithms are far from optimal which might explore unpromising search paths in the k -clique listing procedure. To address these issues, we present a comprehensive comparison of all the state-of-the-art k -clique listing and counting algorithms. We also propose a new color ordering heuristics based on greedy graph coloring techniques which is able to significantly prune the unpromising search paths. We compare the performance of 14 various algorithms using 17 large real-world graphs with up to 3 million nodes and 100 million edges. The experimental results reveal the characteristics of different algorithms, based on which we provide useful guidance for selecting appropriate techniques for different applications.