ADJUSTED CHI-SQUARE TEST FOR DEGREE-CORRECTED BLOCK MODELS

• 成果类型: Article
• 署名作者: Zhang, Linfan; Amini, Arash a.
• 署名单位: University of California System; University of California Los Angeles
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/23-AOS2329
• 发表日期: 2023
• 页码: 2366-2385
• 关键词: GOODNESS-OF-FIT; community detection

摘要

We propose a goodness-of-fit test for degree-corrected stochastic block models (DCSBM). The test is based on an adjusted chi-square statistic for measuring equality of means among groups of n multinomial distributions with d(1), ... , d(n) observations. In the context of network models, the num-ber of multinomials, n, grows much faster than the number of observations, di, corresponding to the degree of node i, hence the setting deviates from classical asymptotics. We show that a simple adjustment allows the statistic to converge in distribution, under null, as long as the harmonic mean of {di} grows to infinity. When applied sequentially, the test can also be used to deter-mine the number of communities. The test operates on a compressed version of the adjacency matrix, conditional on the degrees, and as a result is highly scalable to large sparse networks. We incorporate a novel idea of compress-ing the rows based on a (K + 1)-community assignment when testing for K communities. This approach increases the power in sequential applications without sacrificing computational efficiency, and we prove its consistency in recovering the number of communities. Since the test statistic does not rely on a specific alternative, its utility goes beyond sequential testing and can be used to simultaneously test against a wide range of alternatives outside the DCSBM family. In particular, we prove that the test is consistent against a general family of latent-variable network models with community structure. We show the effectiveness of the approach by extensive numerical experi-ments with simulated and real data. In particular, applying the test to the Facebook-100 data set, a collection of one hundred social networks, we find that a DCSBM with a small number of communities (say < 25) is far from a good fit in almost all cases.