  We consider the two-sample testing problem for networks, where the goal is to
determine whether two sets of networks originated from the same stochastic
model. Assuming no vertex correspondence and allowing for different numbers of
nodes, we address a fundamental network testing problem that goes beyond simple
adjacency matrix comparisons. We adopt the stochastic block model (SBM) for
network distributions, due to their interpretability and the potential to
approximate more general models. The lack of meaningful node labels and vertex
correspondence translate to a graph matching challenge when developing a test
for SBMs. We introduce an efficient algorithm to match estimated network
parameters, allowing us to properly combine and contrast information within and
across samples, leading to a powerful test. We show that the matching
algorithm, and the overall test are consistent, under mild conditions on the
sparsity of the networks and the sample sizes, and derive a chi-squared
asymptotic null distribution for the test. Through a mixture of theoretical
insights and empirical validations, including experiments with both synthetic
and real-world data, this study advances robust statistical inference for
complex network data.
