1-planar graph

Algorithm

Combinatorics

Discrete mathematics

Dense graph

Graph

Complete bipartite graph

Mathematical analysis

Enumeration

Bipartite graph

Line graph

Mathematics

Bounded function

Vertex (graph theory)

Proceedings of the VLDB Endowment

Maximal bicliques are effective to reveal meaningful information hidden in bipartite graphs. Maximal biclique enumeration (MBE) is challenging since the number of the maximal bicliques grows exponentially w.r.t. the number of vertices in a bipartite graph in the worst case. However, a large bipartite graph is usually very sparse, which is against the worst case and may lead to fast MBE algorithms. The uncharted opportunity is taking advantage of the sparsity to substantially improve the MBE efficiency for large sparse bipartite graphs. We observe that for a large sparse bipartite graph, a vertex u may converge to a few vertices in the same vertex set as u via its neighbours, which reveals that the enumeration scope for a vertex could be very small. Based on this observation, we propose novel concepts: unilateral coreness for individual vertices, unilateral order for each vertex set and unilateral convergence (ζ) for a large sparse bipartite graph, ζ could be a few thousand for a large sparse bipartite graph with hundreds of million edges. Using the unilateral order, every vertex with τ unilateral coreness only needs to check at most 2 τ combinations so that all maximal bicliques can be enumerated and τ is bounded by ζ, which leads to a novel MBE algorithm running in O * (2 ζ ). We then propose a batch-pivots technique to eliminate all enumerations resulting in non-maximal bicliques, which guarantees that every maximal biclique is reported in O (ζ e )-delay, where e is the number of edges. We devise novel data structures that allow storing subgraphs at omissible space for further speeding up MBE. Extensive experiments are conducted on synthetic and real large datasets to justify that our proposed algorithm is faster and more scalable than the existing algorithms.

Efficient maximal biclique enumeration for large sparse bipartite graphs