Computing Maximum Structural Balanced Cliques in Signed Graphs

Abstract

Signed graphs have been used to capture the polarity of relationships between entities through positive and negative edge signs, indicating friendly and antagonistic relationships, respectively. In this paper, we focus on (structural) balanced cliques in signed graphs, where a clique, denoted by its vertex set <tex>$C$</tex>, is (structural) balanced if it can be uniquely partitioned into two sets <tex>$C_{L}$</tex> and <tex>$C_{R}$</tex> such that all negative edges in the clique are between <tex>$C_{L}$</tex> and <tex>$C_{R}$</tex>. We study the maximum balanced clique problem that aims to find the balanced clique <tex>$C^{\ast}$</tex> such that <tex>$\min\{\vert C_{L}^{\ast}\vert, \vert C_{R}^{\ast}\vert \}\geq\tau$</tex> for a user-given threshold <tex>$\tau$</tex> and <tex>$\vert C^{\ast}\vert$</tex> is the largest possible. We propose a novel graph reduction technique by transforming the maximum balanced clique problem over a signed graph <tex>$G$</tex> to a series of maximum dichromatic clique problems over small subgraphs of <tex>$G$</tex>. That is, for a vertex <tex>$u$</tex> in <tex>$G$</tex>, we first extract the subgraph <tex>$G_{u}$</tex> of <tex>$G$</tex> induced by vertex set <tex>$V_{L}\cup V_{R}$</tex>, where <tex>$V_{L}$</tex> is the union of <tex>$u$</tex> and its positive neighbors and <tex>$V_{R}$</tex> is <tex>$u$</tex>&#x0027;s negative neighbors. Then, we remove from <tex>$G_{u}$</tex> all negative edges between vertices of the same set (i.e., <tex>$V_{L}$</tex> or <tex>$V_{R}$</tex>) as well as remove all positive edges between V<inf>L</inf> and <tex>$V_{R}$</tex>; denote the resulting graph of discarding edge signs as <tex>$g_{u}$</tex>. We show that the maximum balanced clique containing <tex>$u$</tex> in <tex>$G$</tex> is the same as the maximum dichromatic clique (i.e., it has at least <tex>$\tau$</tex> vertices from each of <tex>$V_{L}$</tex> and <tex>$V_{R}$</tex>) containing <tex>$u$</tex> in <tex>$g_{u}$</tex>. Due to the small size and no edge signs in <tex>$g_{u}$</tex>, the maximum dichromatic clique containing <tex>$u$</tex> in <tex>$g_{u}$</tex> can be efficiently computed by exploiting the existing pruning and bounding techniques that are designed for the classic maximum clique problem on unsigned graphs. Furthermore, we extend our techniques to the polarization factor problem which aims to find the largest <tex>$\tau$</tex> such that there is a balanced clique <tex>$C$</tex> with <tex>$\min\{\vert C_{L}\vert, \vert C_{R}\vert \}\geq\tau$</tex>, and to the generalized maximum balanced clique problem that reports a maximum balanced clique for each <tex>$\tau\geq 0$</tex>. Experimental studies on large real signed graphs demonstrated the efficiency and effectiveness of our techniques.