Signed Clique Search in Signed Networks: Concepts and Algorithms

Abstract

Mining cohesive subgraphs from a network is a fundamental problem in network analysis. Most existing cohesive subgraph models are mainly tailored to unsigned networks. In this paper, we study the problem of seeking cohesive subgraphs in a signed network, in which each edge can be positive or negative, denoting friendship or conflict, respectively. We propose a novel model, called maximal $(\alpha, k)$ ( α , k ) -clique, that represents a cohesive subgraph in signed networks. Specifically, a maximal $(\alpha, k)$ ( α , k ) -clique is a clique in which every node has at most $k$ k negative neighbors and at least $\lceil \alpha k \rceil$ ⌈ α k ⌉ positive neighbors ( $\alpha \geq 1$ α ≥ 1 ). We show that the problem of enumerating all maximal $(\alpha, k)$ ( α , k ) -cliques in a signed network is NP-hard. To enumerate all maximal $(\alpha, k)$ ( α , k ) -cliques efficiently, we first develop an elegant signed network reduction technique to significantly prune the signed network. Then, we present an efficient branch and bound enumeration algorithm with several carefully-designed pruning rules to enumerate all maximal $(\alpha, k)$ ( α , k ) -cliques in the reduced signed network. In addition, we also propose an efficient algorithm with three novel upper-bounding techniques to find the maximum $(\alpha, k)$ ( α , k ) -clique in a signed network. The results of extensive experiments on five large real-life datasets demonstrate the efficiency, scalability, and effectiveness of our algorithms.