  Popular network models such as the mixed membership and standard stochastic
block model are known to exhibit distinct geometric structure when embedded
into $\mathbb{R}^{d}$ using spectral methods. The resulting point cloud
concentrates around a simplex in the first model, whereas it separates into
clusters in the second. By adopting the formalism of generalised random
dot-product graphs, we demonstrate that both of these models, and different
mixing regimes in the case of mixed membership, may be distinguished by the
persistent homology of the underlying point distribution in the case of
adjacency spectral embedding. Moreover, despite non-identifiability issues, we
show that the persistent homology of the support of the distribution and its
super-level sets can be consistently estimated. As an application of our
consistency results, we provide a topological hypothesis test for
distinguishing the standard and mixed membership stochastic block models.
