Abstract

We connect two random graph models, the Popularity Adjusted Block Model (PABM) and the Generalized Random Dot Product Graph (GRDPG), by demonstrating that the PABM is a special case of the GRDPG in which communities correspond to mutually orthogonal subspaces of latent vectors. This insight allows us to construct new algorithms for community detection and parameter estimation for the PABM, as well as improve an existing algorithm that relies on Sparse Subspace Clustering. Using established asymptotic properties of Adjacency Spectral Embedding for the GRDPG, we derive asymptotic properties of these algorithms. In particular, we demonstrate that the absolute number of community detection errors tends to zero as the number of graph vertices tends to infinity. Simulation experiments illustrate these properties. Supplementary materials for this article are available online.

摘要

我们通过证明 PABM 是 GRDPG 的一个特例来连接两个随机图模型，即流行度调整块模型 (PABM) 和广义随机点积图 (GRDPG)，其中社区对应于潜在向量的相互正交子空间。这种洞察力使我们能够为 PABM 构建用于社区检测和参数估计的新算法，并改进依赖于稀疏子空间聚类的现有算法。使用 GRDPG 的邻接谱嵌入的已建立渐近特性，我们推导出这些算法的渐近特性。特别是，我们证明了随着图顶点的数量趋于无穷大，社区检测错误的绝对数量趋于零。模拟实验说明了这些特性。