Bipartite Communities

For a given graph, $G$, let $A$ be the adjacency matrix, $D$ is the diagonal matrix of degrees, $L' = D - A$ is the combinatorial Laplacian, and $L = D^{-1/2}L'D^{-1/2}$ is the normalized Laplacian. Recently, the eigenvectors corresponding to the smallest eigenvalues of $L$ and $L'$ have been of great interest because of their application to community detection, which is a nebulously defined problem that essentially seeks to find a vertex set $S$ such that there are few edges incident with exactly one vertex of $S$. The connection between community detection and the second smallest eigenvalue (and the corresponding eigenvector) is well-known. The $k$ smallest eigenvalues have been used heuristically to find multiple communities in the same graph, and a justification with theoretical rigor for the use of $k \geq 3$ eigenpairs has only been found very recently. The largest eigenpair of $L$ has been used more classically to solve the MAX-CUT problem, which seeks to find a vertex set $S$ that maximizes the number of edges incident with exactly one vertex of $S$. Very recently Trevisan presented a connection between the largest eigenvalue of $L$ and a recursive approach to the MAX-CUT problem that seeks to find a "bipartite community" at each stage. This is related to Kleinberg's HITS algorithm that finds the largest eigenvalue of $A^TA$. We will provide a justification with theoretical rigor for looking at the $k$ largest eigenvalues of $L$ to find multiple bipartite communities in the same graph, and then provide a heuristic algorithm to find strong bipartite communities that is based on the intuition developed by the theoretical methods. Finally, we will present the results of applying our algorithm to various data-mining problems.