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

Latent structure blockmodels for Bayesian spectral graph clustering