Maximum Signed Θ-Clique Identification in Large Signed Graphs

Abstract

The maximum clique problem, which is to find the clique with the largest size, can find many real-world applications and is notable for its capability of modeling many combinatorial problems. However, most existing research focuses on processing unsigned graphs, i.e., treat each connection equally. In real applications, edges of graphs are usually associated with signed information, i.e., positive or negative edges, and signed graph analysis has attracted great attentions in the recent. In this paper, we first analyze the disadvantages of existing signed clique models, and then propose a novel clique model, named signed <inline-formula><tex-math notation="LaTeX">$\theta$</tex-math></inline-formula> -clique. Given a signed graph <inline-formula><tex-math notation="LaTeX">$G$</tex-math></inline-formula> and a subgraph <inline-formula><tex-math notation="LaTeX">$S$</tex-math></inline-formula> , let <inline-formula><tex-math notation="LaTeX">$d^{+}_{S}(u)$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$d^{-}_{S}(u)$</tex-math></inline-formula> be the number of positive and negative neighbors of vertex <inline-formula><tex-math notation="LaTeX">$u$</tex-math></inline-formula> in <inline-formula><tex-math notation="LaTeX">$S$</tex-math></inline-formula> . We say a subgraph <inline-formula><tex-math notation="LaTeX">$S$</tex-math></inline-formula> is a signed <inline-formula><tex-math notation="LaTeX">$\theta$</tex-math></inline-formula> -clique if <inline-formula><tex-math notation="LaTeX">$i)$</tex-math></inline-formula> <inline-formula><tex-math notation="LaTeX">$S$</tex-math></inline-formula> is a clique and <inline-formula><tex-math notation="LaTeX">$ii)$</tex-math></inline-formula> each vertex <inline-formula><tex-math notation="LaTeX">$u$</tex-math></inline-formula> in <inline-formula><tex-math notation="LaTeX">$S$</tex-math></inline-formula> fulfills <inline-formula><tex-math notation="LaTeX">$d^{+}_{S}(u) - d^{-}_{S}(u) \geq \theta$</tex-math></inline-formula> . We show that the problem of identifying the maximum signed <inline-formula><tex-math notation="LaTeX">$\theta$</tex-math></inline-formula> -clique is NP-hard. Novel pruning techniques are proposed to reduce the searching space. In addition, efficient searching strategies are developed to scale for large graphs. Comprehensive experiments on 8 real-world datasets are conducted to demonstrate the effectiveness and efficiency of the proposed approaches.