Abstract
We present a Bayesian nonparametric Poisson factorization model for modeling dense network data with an unknown and potentially growing number of overlapping communities. The construction is based on completely random measures and allows the number of communities to either increase with the number of nodes at a specified logarithmic or polynomial rate, or be bounded. We develop asymptotics for the number and size of the communities of the network and derive a Markov chain Monte Carlo algorithm for targeting the exact posterior distribution for this model. The usefulness of the approach is illustrated on various real networks.
Highlights
Nonnegative matrix factorization (NMF) methods (Paatero and Tapper 1994; Lee and Seung 2001) aim to find a latent representation of a positive n × m matrix A as a sum of K nonnegative factors
We focus on the application to network analysis, where m = n and the n × n count matrix A, the adjacency matrix, represents the number of directed or undirected interactions between n individuals; the latent factors may be interpreted as latent and potentially overlapping communities (Ball et al 2011), such as sport team members or other social activities circles
Zhou et al (2012), Gopalan et al (2014) and Zhou (2015) proposed Bayesian nonparametric approaches that allow the number of latent factors to be estimated from the data, and to grow unboundedly with the size n of the matrix
Summary
Nonnegative matrix factorization (NMF) methods (Paatero and Tapper 1994; Lee and Seung 2001) aim to find a latent representation of a positive n × m matrix A as a sum of K nonnegative factors. We consider binary data where the matrix represents the existence or absence of a directed or undirected link between individuals. Poisson factorization approaches require the user to set the number K of latent factors, which is typically assumed to be independent of the sample size n. To address this problem, Zhou et al (2012), Gopalan et al (2014) and Zhou (2015) proposed Bayesian nonparametric approaches that allow the number of latent factors to be estimated from the data, and to grow unboundedly with the size n of the matrix.
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