# Adjusted chi-square test for degree-corrected block models

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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 $d_1,\dots,d_n$ observations. In the context of network models, the number of multinomials, $n$, grows much faster than the number of observations, $d_i$, 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 $\{d_i\}$ grows to infinity.

When applied sequentially, the test can also be used to determine the number of communities. The test operates on a (row) 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 compressing 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.

The test can also be easily applied to Poisson count arrays in clustering or biclustering applications, as well as bipartite and directed networks. We show the effectiveness of the approach by extensive numerical experiments with simulated and real data. In particular, applying the test to the Facebook-100 dataset, 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. Despite the lack of fit, we show that the statistic itself can be used as an effective tool for exploring community structure, allowing us to construct a community profile for each network.

https://arxiv.org/abs/2012.15047