  The stochastic blockmodel (SBM) models the connectivity within and between
disjoint subsets of nodes in networks. Prior work demonstrated that the rows of
an SBM's adjacency spectral embedding (ASE) and Laplacian spectral embedding
(LSE) both converge in law to Gaussian mixtures where the components are curved
exponential families. Maximum likelihood estimation via the
Expectation-Maximization (EM) algorithm for a full Gaussian mixture model (GMM)
can then perform the task of clustering graph nodes, albeit without appealing
to the components' curvature. Noting that EM is a special case of the
Expectation-Solution (ES) algorithm, we propose two ES algorithms that allow us
to take full advantage of these curved structures. After presenting the ES
algorithm for the general curved-Gaussian mixture, we develop those
corresponding to the ASE and LSE limiting distributions. Simulating from
artificial SBMs and a brain connectome SBM reveals that clustering graph nodes
via our ES algorithms can improve upon that of EM for a full GMM for a wide
range of settings.
