Fast Biclique Counting on Bipartite Graphs: A Node Pivot-based Approach

TL;DR

This work addresses the NP-hard problem of counting (p, q)-bicliques in bipartite graphs by introducing NPivoter, a node-pivot based framework that unifies existing methods and extends counting capabilities to local and range variants. Central to NPivoter are node-pivots, a minimum non-neighbor candidate partition strategy, and a cost estimator that adaptively selects between node-split and edge-split strategies, enabling tight worst-case bounds and practical efficiency. Theoretical results show a favorable worst-case complexity of O(|E|2^{n_{max}/2}) for NPC under the MNCP strategy, along with empirical demonstrations on 12 real-world datasets where NPivoter outperforms BCList++ and EPivoter by up to two orders of magnitude. The approach also supports efficient local counting per node and simultaneous range counting, making it well-suited for applications in graph analytics, GNNs, and query optimization. The combination of a general framework, advanced partitioning, and adaptive optimization yields a scalable and versatile solution for large bipartite networks.

Abstract

Counting the number of $(p, q)$-bicliques (complete bipartite subgraphs) in a bipartite graph is a fundamental problem which plays a crucial role in numerous bipartite graph analysis applications. However, existing algorithms for counting $(p, q)$-bicliques often face significant computational challenges, particularly on large real-world networks. In this paper, we propose a general biclique counting framework, called \npivot, based on a novel concept of node-pivot. We show that previous methods can be viewed as specific implementations of this general framework. More importantly, we propose a novel implementation of \npivot based on a carefully-designed minimum non-neighbor candidate partition strategy. We prove that our new implementation of \npivot has lower worst-case time complexity than the state-of-the-art methods. Beyond basic biclique counting, a nice feature of \npivot is that it also supports local counting (computing bicliques per node) and range counting (simultaneously counting bicliques within a size range). Extensive experiments on 12 real-world large datasets demonstrate that our proposed \npivot substantially outperforms state-of-the-art algorithms by up to two orders of magnitude.

Figures (7)

• Figure 1: An example graph
• Figure 2: A running example of our proposed $\mathsf{NPivoter}$
• Figure 3: Running time of various algorithms
• Figure 4: Memory overhead of all algorithms
• Figure 5: Effect of the cost estimator

Theorems & Definitions (9)

• definition 1
• definition 2: node-pivot
• theorem 1
• theorem 2
• definition 3: Minimum non-neighbor candidate partition
• theorem 3
• theorem 4
• theorem 5
• definition 4: $(p,q)$-core alphabetacoreCIKMalphabetacoreWWW