  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 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. In particular, we prove that the test is consistent against a
general family of latent-variable network models with community structure.
