Graph decomposition for community identification and covariance constraints
Seminar Room 1, Newton Institute
An application in large databases is to find well-connected clusters of nodes in an undirected graph where a link represents interaction between objects. For example, finding tight-knit communities in social networks, identifying related product-clusters in collaborative filtering, finding genes which collaborate in different biological functions. Unlike graph-partitioning, in this scenario an object may belong to more than one community -- for example, a person might belong to more than one group of friends, or a gene may be active in more than one gene-network. I'll discuss an approach to identifying such overlapping communities based on extending the incidence matrix decomposition of a graph to a clique-decomposition. Clusters are then identified by approximate variational (mean-field) inference in a related probabilistic model. The resulting decomposition has the side-effect of enabling a parameteristion of positive definite matrices under zero-constraints on entries in the matrix. Provided the graph corresponding to the constraints is decomposable all such matrices are reachable by this parameterisation. In the non-decomposable case, we show how the method forms an approximation of the space and relate it to more standard latent variable parameterisations of zero-constrained covariances.
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