Graph explorer

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 t