2017/09/17 by Jiashun Jin, Zheng Tracy Ke, Jin, Jiashun +1
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Random Matrices and Applications #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1709.05603
openalex publication_date 2017/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider an undirected network with n nodes and K perceivable communities, where some nodes may have mixed memberships. We assume that for each node 1 ≤ i ≤ n, there is a probability mass function πi defined over \1, 2, …, K\ such that πi(k) = the weight of node i on community k, 1 ≤ k ≤ K. The goal is to estimate \πi, 1 ≤ i ≤ n\ (i.e., membership estimation). We model the network with the \it degree-corrected mixed membership (DCMM) model \citeMixed-SCORE. Since for many natural networks, the degrees have an approximate power-law tail, we allow \it severe degree heterogeneity in our model. For any membership estimation \πi, 1 ≤ i ≤ n\, since each πi is a probability mass function, it is natural to measure the errors by the average ℓ1-norm (1)/(n) ∑i = 1n ‖ πi - πi‖1. We also consider a variant of the ℓ1-loss, where each ‖πi - πi‖1 is re-weighted by the degree parameter θi in DCMM (to be introduced). We present a sharp lower bound. We also show that such a lower bound is achievable under a broad situation. More discussion in this vein is continued in our forthcoming manuscript. The results are very different from those on community detection. For community detection, the focus is on the special case where all πi are degenerate; the goal is clustering, so Hamming distance is the natural choice of loss function, and the rate can be exponentially fast. The setting here is broader and more difficult: it is more natural to use the ℓ1-loss, and the rate is only polynomially fast.