Modular and hierarchical community structures are pervasive in real-world complex systems. A great deal of effort has gone into trying to detect and study these structures. Important theoretical advances in the detection of modular have included identifying fundamental limits of detectability by formally defining community structure using probabilistic generative models. Detecting hierarchical community structure introduces additional challenges alongside those inherited from community detection. Here we present a theoretical study on hierarchical community structure in networks, which has thus far not received the same rigorous attention. We address the following questions: 1) How should we define a hierarchy of communities? 2) How do we determine if there is sufficient evidence of a hierarchical structure in a network? and 3) How can we detect hierarchical structure efficiently? We approach these questions by introducing a definition of hierarchy based on the concept of stochastic externally equitable partitions and their relation to probabilistic models, such as the popular stochastic block model. We enumerate the challenges involved in detecting hierarchies and, by studying the spectral properties of hierarchical structure, present an efficient and principled method for detecting them.

翻译：模块化与层级社区结构在现实世界的复杂系统中普遍存在。大量研究致力于检测和探索这些结构。在模块化检测领域的重要理论进展包括：通过概率生成模型正式定义社区结构，从而识别可检测性的基本极限。检测层级社区结构在继承社区检测挑战的同时引入了额外困难。本文首次对网络中层级社区结构进行严格的理论研究，我们探讨以下问题：1）如何定义社区层级？2）如何判断网络中存在层级结构的充分证据？3）如何高效检测层级结构？我们引入基于随机外部公平划分的概念及其与概率模型（如流行的随机块模型）的关系，提出了层级结构的定义。我们列举了检测层级结构面临的挑战，并通过研究层级结构的谱特性，提出了一种高效且基于原理的检测方法。