Discovering bands from graphs

Discovering the underlying structure of a given graph is one of the fundamental goals in graph mining. Given a graph, we can often order vertices in a way that neighboring vertices have a higher probability of being connected to each other. This implies that the edges form a band around the diagonal in the adjacency matrix. Such structure may rise for example if the graph was created over time: each vertex had an active time interval during which the vertex was connected with other active vertices. The goal of this paper is to model this phenomenon. To this end, we formulate an optimization problem: given a graph and an integer \(K\), we want to order graph vertices and partition the ordered adjacency matrix into \(K\) bands such that bands closer to the diagonal are more dense. We measure the goodness of a segmentation using the log-likelihood of a log-linear model, a flexible family of distributions containing many standard distributions. We divide the problem into two subproblems: finding the order and finding the bands. We show that discovering bands can be done in polynomial time with isotonic regression, and we also introduce a heuristic iterative approach. For discovering the order we use Fiedler order accompanied with a simple combinatorial refinement. We demonstrate empirically that our heuristic works well in practice.