ABSTRACT

Large datasets, which are naturally modeled as a network or graph, arise in almost every field of human endeavor. For example, Facebook is a social network, where nodes are users, with edges corresponding to friendships. In gene networks, nodes represent genes with connections corresponding to their co-expression. In ecological networks, the nodes are animal species, with edges determined according to who eats whom. A major focus of research for network or graph data has been on identifying community membership of the nodes. However, what is often more important for scientific purposes is examining the nature and evolution of edge and membership probabilities, for instance changes in gene features of individuals as a function of some unknown factor, like a disease. The focus on using other measured features of nodes and edges could add, in decisive ways, to the information available from observed edges or interactions between nodes. These could be disease symptoms or test results, or demographic information of users in social networks. Statistical inference in such models, despite its importance, has only just begun to be studied. There are both theoretical and computational challenges, due both to the complexity of models fitted, and the size of data sets. The research will lead to the development of algorithms for fitting models and statistical measures of confidence, with potential applications to many fields.

The research is focused on block models for graphs, when node or edge covariates are present. When formulated, these models are no longer block models, but models whose membership probabilities depend upon covariates and whose connection probabilities depend both on block membership and individual covariates. Fitting algorithms involve alternating between fitting block and covariate parameters. Variational (mean field) approaches which effectively lead to semi-parametric model fitting with nK membership "nuisance" parameters, with n representing the number of nodes and K the number of communities, are examined. As these approaches have been found by the PIs to be unstable for large n, the PIs have already begun to investigate the theoretical and practical aspects of divide and conquer algorithms where many subgraphs are independently fit. The PIs will study the statistical properties, both asymptotically and through simulations, and develop practicable and computationally stable methods for large, relatively sparse graphs.

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