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.
翻译:网络外观嵌入网络的相邻性矩阵往往产生大约围绕低维次元结构的节点表示。 特别是, 当图形由潜伏位置模型生成时, 预计将会出现隐藏的子结构。 此外, 网络内社区的存在可能会在嵌入过程中产生社区特有的子层结构, 但大多数网络的统计模型中并未明确说明这一点。 在本篇文章中, 提出了一组称为潜伏结构块模型( LSBM ) 的模型, 以应对这些假设情景, 允许在社区特有的一维多元结构出现时进行图形组合。 LSBMS 侧重于一个特定的潜伏空间模型, 随机点产品图( RDPG), 并给每个社区的潜在位置指定一个潜在的子层。 讨论来自 LSBMS 的嵌入模式, 并显示其在模拟和真实世界网络数据上表现良好。 该模型能够正确恢复生活在一维方形中的基本社区, 即使基本曲线的对称形式未知, 在各种真实数据上取得显著的结果 。