Efficient Antagonistic $k$k-Plex Enumeration in Signed Graphs

A signed graph is a graph where each edge receives a sign, positive or negative. The signed graph model has been used in many real applications, such as protein complex discovery and social network analysis. Finding cohesive subgraphs in signed graphs is a fundamental problem. A <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-plex is a common model for cohesive subgraphs in which every vertex is adjacent to all but at most <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> vertices within the subgraph. In this paper, we propose the model of size-constrained antagonistic <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-plex in a signed graph. The proposed model guarantees that the resulting subgraph is a <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-plex and can be divided into two sub-<inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-plexes, both of which have positive inner edges and negative outer edges. This paper aims to identify all maximal antagonistic <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-plexes in a signed graph. Through rigorous analysis, we show that the problem is NP-Hardness. We propose a novel framework for maximal antagonistic <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-plexes utilizing set enumeration. Efficiency is improved through pivot pruning and early termination based on the color bound. Preprocessing techniques based on degree and dichromatic graphs effectively narrow the search space before enumeration. Extensive experiments on real-world datasets demonstrate our algorithm’s efficiency, effectiveness, and scalability.