  Spectral embedding of network adjacency matrices often produces node
representations living approximately around low-dimensional submanifold
structures. In particular, hidden substructure is expected to arise when the
graph is generated from a latent position model. Furthermore, the presence of
communities within the network might generate community-specific submanifold
structures in the embedding, but this is not explicitly accounted for in most
statistical models for networks. In this article, a class of models called
latent structure block models (LSBM) is proposed to address such scenarios,
allowing for graph clustering when community-specific one dimensional manifold
structure is present. LSBMs focus on a specific class of latent space model,
the random dot product graph (RDPG), and assign a latent submanifold to the
latent positions of each community. A Bayesian model for the embeddings arising
from LSBMs is discussed, and shown to have a good performance on simulated and
real world network data. The model is able to correctly recover the underlying
communities living in a one-dimensional manifold, even when the parametric form
of the underlying curves is unknown, achieving remarkable results on a variety
of real data.
