We study community detection on Markovian random networks outside of the Stochastic Block Model (SBM) framework. Specifically, we consider a random network growth process which generates $K$ separate preferential attachment trees and connects them with Erdős--Rényi edges, so that each tree represents a community and each node inherits the label of the tree to which it belongs. This model is able to produce many features of real world networks that are improbable under SBM, such as power law degree distribution and the existence of chains and hubs. Given only the final graph, without any knowledge of the growth process, we seek to recover the unobserved community membership of the nodes. We first prove that it is impossible for any algorithm to consistently recover the community label of all the nodes. However, we design algorithms which are provably able to recover the community labels of subsets of central nodes, for several different notions of node centrality such as arrival time or degree. Our procedure consists of two stages where, in the first stage, we classify high degree nodes and then, in the second stage, extend the community assignments to the remaining vertices. Numerical experiments and a real data application on a coauthorship network demonstrate the effectiveness of our proposed approach.

翻译：我们研究了马尔可夫随机网络上超越随机块模型（SBM）框架的社群检测问题。具体而言，我们考虑一个随机网络增长过程，该过程生成$K$棵独立的优先依附树，并通过埃尔德什–雷尼（Erdős–Rényi）边将它们连接起来，使得每棵树代表一个社群，每个节点继承其所属树的标签。该模型能够生成SBM下不可能出现的真实世界网络的许多特征，例如幂律度分布以及链和枢纽的存在。仅基于最终图，在没有任何增长过程知识的情况下，我们寻求恢复节点未观察到的社群成员身份。我们首先证明，任何算法都无法一致地恢复所有节点的社群标签。然而，我们设计了能够可证明地恢复中心节点子集（基于到达时间或度数等几种不同的节点中心性概念）的社群标签的算法。我们的过程包括两个阶段：第一阶段分类高度数节点，第二阶段将社群分配扩展到剩余顶点。数值实验以及在合著网络上的实际数据应用证明了我们提出方法的有效性。