Colorful Star Motif Counting: Concepts, Algorithms and Applications

Hongchao Qin, Gao Sen, Rong-Hua Li, Hongzhi Chen, Ye Yuan, Guoren Wang

Abstract: A colorful star motif is a star-shaped graph where any two nodes have different colors. Counting the colorful star motif can help to analyze the structural properties of real-life colorful graphs, model higher-order clustering, and accelerate the mining of the densest subgraph exhibiting $h$-clique characteristics in graphs. In this manuscript, we introduce the concept of colorful $h$-star in a colored graph and proposes two higher-order cohesive subgraph models, namely colorful $h$-star core and colorful $h$-star truss. We show that the colorful $h$-stars can be counted and updated very efficiently using a novel dynamic programming (DP) algorithm. Based on the proposed DP algorithm, we develop a colorful $h$-star core decomposition algorithm which takes $O(h m)$ time, $O(h n+m)$ space; and a colorful $h$-star truss decomposition algorithm which takes $O(h m^{1.5})$ time, $O(hm)$ space, where $m$ and $n$ denote the number of edges and nodes of the graph respectively. Moreover, we also propose a graph reduction technique based on our colorful $h$-star core model to accelerate the computation of the approximation algorithm for $ h$-clique densest subgraph mining. The results of comprehensive experiments on 11 large real-world datasets demonstrate the efficiency, scalability and effectiveness of the proposed algorithms.