Unsupervised node clustering (or community detection) is a classical graph learning task. In this paper, we study algorithms, which exploit the geometry of the graph to identify densely connected substructures, which form clusters or communities. Our method implements discrete Ricci curvatures and their associated geometric flows, under which the edge weights of the graph evolve to reveal its community structure. We consider several discrete curvature notions and analyze the utility of the resulting algorithms. In contrast to prior literature, we study not only single-membership community detection, where each node belongs to exactly one community, but also mixed-membership community detection, where communities may overlap. For the latter, we argue that it is beneficial to perform community detection on the line graph, i.e., the graph's dual. We provide both theoretical and empirical evidence for the utility of our curvature-based clustering algorithms. In addition, we give several results on the relationship between the curvature of a graph and that of its dual, which enable the efficient implementation of our proposed mixed-membership community detection approach and which may be of independent interest for curvature-based network analysis.

翻译：无监督节点聚类（或社区检测）是一项经典的图学习任务。本文研究利用图的几何结构识别形成聚类或社区的密集连接子结构的算法。我们的方法实现了离散里奇曲率及其相关的几何流，在这些几何流作用下，图的边权重演化以揭示其社区结构。我们考虑了多种离散曲率概念，并分析了相应算法的实用性。与先前文献不同，我们不仅研究了每个节点仅属于一个社区的单一隶属关系社区检测，还研究了社区可能重叠的混合隶属关系社区检测。对于后者，我们认为在线图（即图的对偶图）上进行社区检测是有益的。我们为基于曲率的聚类算法的效用提供了理论和实证证据。此外，我们给出了关于图的曲率与其对偶图曲率之间关系的若干结果，这些结果使我们提出的混合隶属关系社区检测方法能够高效实现，并且对于基于曲率的网络分析可能具有独立的研究价值。