Identifying Graph Clusters using Variational Inference and links to Covariance Parameterisation

Finding clusters of well-connected nodes in a graph is useful in many domains, including Social Network, Web and molecular interaction analyses. From a computational viewpoint, finding these clusters or graph communities is a difficult problem. We consider the framework of Clique Matrices to decompose a graph into a set of possibly overlapping clusters, defined as well-connected subsets of vertices. The decomposition is based on a statistical description which encourages clusters to be well connected and few in number. The formal intractability of inferring the clusters is addressed using a variational approximation which has links to meanfield theories in statistical mechanics. Clique matrices also play a natural role in parameterising positive definite matrices under zero constraints on elements of the matrix. We show that clique matrices can parameterise all positive definite matrices restricted according to a decomposable graph and form a structured Factor Analysis approximation in the non-decomposable case.