Counting Small Balanced (p,q)-bicliques in Signed Bipartite Graphs

2026-05-05

AI Summary

Counting balanced $(p,q)$-bicliques of fixed size in signed bipartite graphs is crucial for uncovering higher-order structural patterns, yet existing methods struggle to do so efficiently. This work presents the first systematic study of this problem and introduces two novel algorithms: BBWC, based on signed wedge expansion, and BBVP, which integrates vertex pruning. By synergistically combining balance theory from signed graphs with subgraph enumeration techniques, the authors devise expansion and pruning strategies under signed constraints, substantially enhancing computational efficiency. Experimental results on large-scale real-world datasets demonstrate that BBVP achieves an average speedup of 636× over SBCList++ when $p = q = 3$, effectively overcoming the limitations of prior approaches that are confined to local motifs or non-fixed-size structures.

Abstract

Two disjoint sets of entities and their relationship can be modelled as a bipartite graph. Real-life examples include drug-target interaction in biological networks, user-item relationships in e-commerce networks, etc. Motif-based analysis is essential for understanding the structure of large-scale networks, and bipartite graphs are no exception. In contrast to unsigned graphs, motif analysis in signed bipartite graphs has received limited attention. The smallest non-trivial motif in a signed bipartite graph is a balanced (2,2)-biclique, often called a balanced butterfly, which captures only local patterns and cannot reveal higher-order relationships. Bipartite motifs have been studied in the literature in the context of signed bipartite graphs, such as maximal biclique, bitruss, and so on. None of these works addresses bipartite motifs with fixed-sized vertex sets, which are often relevant in practical situations. In this work, we study the balanced (p,q)-biclique counting problem for small values of p and q. As a baseline, we first adapt and extend the state-of-the-art BCList++ algorithm for unsigned bipartite graphs to incorporate edge signs, which we call SBCList++. We then propose two efficient algorithms: BBWC, a wedge-centric approach that enforces balance constraints during enumeration, and BBVP, a vertex-based pruning approach that directly enumerates feasible vertex sets. Extensive experiments on large real-world datasets demonstrate that the vertex-based pruning algorithm, BBVP, significantly outperforms the baseline, achieving an average speedup of 636$\times$ over SBCList++ (where p=q=3).

Problem

• signed bipartite graphs
• balanced (p,q)-biclique
• motif counting
• butterfly
• higher-order relationships

Innovation

• signed bipartite graph
• balanced (p,q)-biclique
• motif counting
• vertex-based pruning
• wedge-centric enumeration