[Paper Review] A Change-Sensitive Algorithm for Maintaining Maximal Bicliques in a Dynamic Bipartite Graph

Apurba Das, Srikanta Tirthapura|arXiv (Cornell University)|Jul 26, 2017

Gene Regulatory Network Analysis24 references3 citations

TL;DR

This paper presents a change-sensitive algorithm, DynamicBC, for efficiently maintaining maximal bicliques in a dynamic bipartite graph under edge insertions or deletions. The algorithm's time complexity is proportional to the actual change in the set of maximal bicliques, achieving near-optimal performance with provable bounds on change magnitude, and outperforms baselines by orders of magnitude in practice.

ABSTRACT

We consider the maintenance of maximal bicliques from a dynamic bipartite graph that changes over time due to the addition or deletion of edges. When the set of edges in a graph changes, we are interested in knowing the change in the set of maximal bicliques (the "change"), rather than in knowing the set of maximal bicliques that remain unaffected. The challenge in an efficient algorithm is to enumerate the change without explicitly enumerating the set of all maximal bicliques. In this work, we present (1) near-tight bounds on the magnitude of change in the set of maximal bicliques of a graph, due to a change in the edge set (2) a "change-sensitive" algorithm for enumerating the change in the set of maximal bicliques, whose time complexity is proportional to the magnitude of change that actually occurred in the set of maximal bicliques in the graph. To our knowledge, these are the first algorithms for enumerating maximal bicliques in a dynamic graph, with such provable performance guarantees. Our algorithms are easy to implement, and experimental results show that their performance exceeds that of current baseline implementations by orders of magnitude.

Motivation & Objective

• To address the challenge of efficiently maintaining maximal bicliques in a bipartite graph that evolves via edge additions or deletions.
• To quantify the maximum possible change in the set of maximal bicliques due to a single edge update.
• To design an algorithm whose runtime scales with the actual magnitude of change, rather than worst-case graph size.
• To enable practical, scalable maintenance of maximal bicliques in large dynamic graphs, especially when changes are sparse.

Proposed method

• The algorithm maintains a dynamic representation of the graph and tracks changes in biclique structure after edge updates.
• It uses a subsumption detection mechanism, SubBC, to identify maximal bicliques in the old graph that are no longer maximal after edge additions.
• New maximal bicliques are discovered by exploring neighborhoods of newly added edges and expanding potential biclique candidates.
• The algorithm employs a signature-based representation (e.g., 64-bit murmur hash) to reduce memory overhead while preserving correctness.
• It supports incremental processing with configurable batch sizes for edge updates, enabling trade-offs between overhead and performance.
• A size threshold $ s $ can be applied to filter out small bicliques, reducing computation cost for large-scale graphs.

Experimental results

Research questions

• RQ1 What is the maximum possible change in the number of maximal bicliques when a single edge is added to a bipartite graph?
• RQ2 Can we design an algorithm whose runtime depends only on the actual number of changed bicliques, rather than the graph size?
• RQ3 How does the magnitude of change in maximal bicliques scale with the number of edge updates?
• RQ4 Can we efficiently detect both newly formed and subsumed maximal bicliques after an edge update?

Key findings

• The maximum change in the set of maximal bicliques due to a single edge addition is bounded by $ 3g(n-2) \approx 1.5 \times 2^{n/2} $, which is exponential in $ n $, and this bound is tight.
• The algorithm's time complexity is proportional to the actual number of changed bicliques, making it change-sensitive and efficient for sparse updates.
• Experimental results show that DynamicBC outperforms baseline methods by orders of magnitude in both runtime and scalability on graphs with millions of edges.
• Using biclique signatures reduces memory usage significantly, especially for large bicliques, and enables execution on memory-constrained datasets.
• Increasing the size threshold $ s $ for bicliques reduces total computation time substantially, with up to 50% reduction observed in large graphs like epinion-rating-init.
• Computation time increases with batch size due to the exponential dependence of subsumed biclique detection on batch size, but small batches incur high overhead.