Discovering Hierarchical Subgraphs of K-Core-Truss

Abstract

Discovering dense subgraphs in a graph is a fundamental graph mining task, which has a wide range of applications in social networks, biology and graph visualization to name a few. Even the problems of computing most dense subgraphs (e.g., clique, quasi-clique, k-densest subgraph) are NP-hard, there exists polynomial time algorithms for computing k-core and k-truss. In this paper, we propose a novel dense subgraph, $\mathsf {k}$-$\mathsf {core}$-$\mathsf {truss}$, that leverages on a new type of important edges based on the concepts of k-core and k-truss. Compared with k-core and k-truss, $\mathsf {k}$-$\mathsf {core}$-$\mathsf {truss}$ can significantly discover the interesting and important structural information outside the scope of the k-core and k-truss. We study two useful problems of $\mathsf {k}$-$\mathsf {core}$-$\mathsf {truss}$ decomposition and $\mathsf {k}$-$\mathsf {core}$-$\mathsf {truss}$ search. In particular, we develop a $\mathsf {k}$-$\mathsf {core}$-$\mathsf {truss}$ decomposition algorithm to find all $\mathsf {k}$-$\mathsf {core}$-$\mathsf {truss}$ in a graph G by iteratively removing edges with the smallest $\mathsf {degree}$-$\mathsf {support}$. In addition, we propose a $\mathsf {k}$-$\mathsf {core}$-$\mathsf {truss}$ search algorithm to identify a particular $\mathsf {k}$-$\mathsf {core}$-$\mathsf {truss}$ containing a given query node such that the core-number k is the largest. Extensive experiments on several web-scale real-world datasets show the effectiveness and efficiency of the $\mathsf {k}$-$\mathsf {core}$-$\mathsf {truss}$ model and proposed algorithms.