Detecting Bicliques in GF[q]

We consider the problem of finding planted bicliques in random matrices over GF [q]. That is, our input matrix is a GF [q]-sum of an unknown biclique (rank-1 matrix) and a random matrix. We study different models for the random graphs and characterize the conditions when the planted biclique can be recovered. We also empirically show that a simple heuristic can reliably recover the planted bicliques when our theory predicts that they are recoverable. Existing methods can detect bicliques of O( √ N), while it is NP-hard to find the largest such clique. Real graphs, however, are typically extremely sparse and seldom contain such large bicliques. Further, the noise can destroy parts of the planted biclique. We investigate the practical problem of how small a biclique can be and how much noise there can be such that we can still approximately correctly identify the biclique. Our derivations show that with high probability planted bicliques of size logarithmic in the network size can be detected in data following the Erdős-Rényi model and two bipartite variants of the Barabási-Albert model.