Estimating Mixed Memberships with Sharp Eigenvector Deviations

Abstract

We consider the problem of estimating community memberships of nodes in a network, where every node is associated with a vector determining its degree of membership in each community. Existing provably consistent algorithms often require strong assumptions about the population, are computationally expensive, and only provide an overall error bound for the whole community membership matrix. This paper provides uniform rates of convergence for the inferred community membership vector of each node in a network generated from the Mixed Membership Stochastic Blockmodel (MMSB); to our knowledge, this is the first work to establish per-node rates for overlapping community detection in networks. We achieve this by establishing sharp row-wise eigenvector deviation bounds for MMSB. Based on the simplex structure inherent in the eigen-decomposition of the population matrix, we build on established corner-finding algorithms from the optimization community to infer the community membership vectors. Our results hold over a broad parameter regime where the average degree only grows poly-logarithmically with the number of nodes. Using experiments with simulated and real datasets, we show that our method achieves better error with lower variability over competing methods, and processes real world networks of up to 100,000 nodes within tens of seconds.

References in corpus (5)

• Entrywise Eigenvector Analysis of Random Matrices with Low Expected Rank
• Detecting Overlapping Communities in Networks Using Spectral Methods
• Mixed Membership Estimation for Social Networks
• Bayesian estimation from few samples: community detection and related problems
• Consistent Estimation of Mixed Memberships with Successive Projections

Cited by in corpus (6)

• Mixed Membership Estimation for Social Networks
• Unified ${\ell }_{2\to \infty }$ Eigenspace Perturbation Theory for Symmetric Random Matrices
• Consistency of Spectral Clustering on Hierarchical Stochastic Block Models
• A Spectral Method for Identifiable Grade of Membership Analysis with Binary Responses
• Subspace Estimation from Unbalanced and Incomplete Data Matrices: ${\ell }_{2,\infty }$ Statistical Guarantees
• Provable Overlapping Community Detection in Weighted Graphs