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A0519
Title: Estimating network memberships by simplex vertices hunting Authors:  Tracy Ke - University of Chicago (United States) [presenting]
Abstract: Consider an undirected mixed-membership network with $n$ nodes and $K$ communities. For each node $i$, we model the membership by a Probability Mass Function (PMF) $\pi_{i} = (\pi_{i}(1), \pi_{i}(2), \ldots$, $\pi_{i}(K))'$, where $\pi_{i}(k)$ is the probability that node $i$ belongs to community $k$. We call node $i$ `pure' if $\pi_i$ is degenerate and `mixed' otherwise. The primary interest is to estimate $\pi_i$, $1 \leq i \leq n$. We propose Mixed-SCORE as a new spectral method for mixed membership estimation. At the heart of Mixed-SCORE is a (tall by very skinny) matrix of entry-wise ratios, formed by dividing by the first few eigenvectors of the network adjacency matrix over the leading eigenvector of the same matrix in an entry-wise fashion. The main surprise is that the rows of the entry-wise ratio matrix form a cloud of points in a low-dimensional space with the silhouette of a simplex, which simplex carries all information we need for estimating the memberships. We apply Mixed SCORE to four network data sets (a coauthorship and a citee network for statisticians, a political book network, and a football network) and obtain meaningful results. We propose a Degree-Corrected Mixed Membership (DCMM) model, and use it to solidify our discoveries with delicate spectral analysis and Random Matrix Theory.