Efficient $k$k-Clique Counting on Large Graphs: The Power of Color-Based Sampling Approaches

Xiaowei Ye, Rong-Hua Li, Qiangqiang Dai, Hongzhi Chen, Guoren Wang

Abstract: K$ -clique counting is a fundamental problem in network analysis which has attracted much attention in recent years. Computing the count of $k$ -cliques in a graph for a large $k$ (e.g., $k=8$ ) is often intractable as the number of $k$ -cliques increases exponentially w.r.t. (with respect to) $k$ . Existing exact $k$ -clique counting algorithms are often hard to handle large dense graphs, while sampling-based solutions either require a huge number of samples or consume very high storage space to achieve a satisfactory accuracy. To overcome these limitations, we propose a new framework to estimate the number of $k$ -cliques which integrates both the exact $k$ -clique counting technique and three novel color-based sampling techniques. The key insight of our framework is that we only apply the exact algorithm to compute the $k$ -clique counts in the sparse regions of a graph, and use the proposed color-based sampling approaches to estimate the number of $k$ -cliques in the dense regions of the graph. Specifically, we develop three novel dynamic programming based $k$ -color set sampling techniques to efficiently estimate the $k$ -clique counts, where a $k$ -color set contains $k$ nodes with $k$ different colors. Since a $k$ -color set is often a good approximation of a $k$ -clique in the dense regions of a graph, our sampling-based solutions are extremely efficient and accurate. Moreover, the proposed sampling techniques are space efficient which use near-linear space w.r.t. graph size. We conduct extensive experiments to evaluate our algorithms using 8 real-life graphs. The results show that our best algorithm is at least one order of magnitude faster than the state-of-the-art sampling-based solutions (with the same relative error 0.1%) and can be up to three orders of magnitude faster than the state-of-the-art exact algorithm on large graphs.