Scalable Large Near-Clique Detection in Large-Scale Networks via Sampling

Extracting dense subgraphs from large graphs is a key primitive in a variety of graph mining applications, ranging from mining social networks and the Web graph to bioinformatics [41]. In this paper we focus on a family of poly-time solvable formulations, known as the k-clique densest subgraph problem (k-Clique-DSP) [57]. When k=2, the problem becomes the well-known densest subgraph problem (DSP) [22, 31, 33, 39]. Our main contribution is a sampling scheme that gives densest subgraph sparsifier, yielding a randomized algorithm that produces high-quality approximations while providing significant speedups and improved space complexity. We also extend this family of formulations to bipartite graphs by introducing the (p,q)-biclique densest subgraph problem ((p,q)-Biclique-DSP), and devise an exact algorithm that can treat both clique and biclique densities in a unified way. As an example of performance, our sparsifying algorithm extracts the 5-clique densest subgraph --which is a large-near clique on 62 vertices-- from a large collaboration network. Our algorithm achieves 100% accuracy over five runs, while achieving an average speedup factor of over 10,000. Specifically, we reduce the running time from ∼2 107 seconds to an average running time of 0.15 seconds. We also use our methods to study how the k-clique densest subgraphs change as a function of time in time-evolving networks for various small values of k. We observe significant deviations between the experimental findings on real-world networks and stochastic Kronecker graphs, a random graph model that mimics real-world networks in certain aspects. We believe that our work is a significant advance in routines with rigorous theoretical guarantees for scalable extraction of large near-cliques from networks.