Bayesian fuzzy clustering of colored graphs

With the increasing availability of interaction data stemming form fields as diverse as systems biology, telecommunication or social sciences, the task of mining and understanding the underlying graph structures becomes more and more important. Here we focus on data with different types of nodes; we subsume this meta information in the color of a node. An important first step is the unsupervised clustering of nodes into communities, which are of the same color and highly connected within but sparsely connected to the rest of the graph. Recently we have proposed a fuzzy extension of this clustering concept, which allows a node to have membership in multiple clusters. The resulting gradient descent algorithm shared many similarities with the multiplicative update rules from nonnegative matrix factorization. Two issues left open were the determination of the number of clusters of each color, as well as the non-defined integration of additional prior information. In this contribution we resolve these issues by reinterpreting the factorization in a Bayesian framework, which allows the ready inclusion of priors. We integrate automatic relevance determination to automatically estimate group sizes. We derive a maximum-a-posteriori estimator, and illustrate the feasibility of the approach on a toy as well as a protein-complex hypergraph, where the resulting fuzzy clusters show significant enrichment of distinct gene ontology categories.