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.

网络邻接矩阵的谱嵌入通常会产生近似于低维子流形结构周围的节点表示。特别是，当图是从潜在位置模型生成时，预计会出现隐藏的子结构。此外，网络中社区的存在可能会在嵌入中生成社区特定的子流形结构，但这在大多数网络统计模型中并未明确说明。在本文中，提出了一类称为潜在结构块模型（LSBM）的模型来解决这种情况，当存在特定于社区的一维流形结构时，允许进行图聚类。LSBM 专注于特定类别的潜在空间模型，即随机点积图 (RDPG)，并为每个社区的潜在位置分配一个潜在子流形。讨论了由 LSBM 产生的嵌入的贝叶斯模型，并显示其在模拟和现实世界网络数据上具有良好的性能。该模型即使在底层曲线的参数形式未知的情况下也能够正确地恢复生活在一维流形中的底层社区，在各种真实数据上取得了显着的效果。