Detecting and Localizing Anomalous Cliques in Inhomogeneous Networks using Egonets

Cliques, or fully connected subgraphs, are among the most important and well-studied graph motifs in network science. We consider the problem of finding a statisti- cally anomalous clique hidden in a large network. There are two parts to this problem: (1) detection, i.e., determining whether an anomalous clique is present, and (2) localization, i.e., determining which vertices of the network constitute the detected clique. While this problem has been extensively studied under the homogeneous Erdos-Renyi model, little progress has been made beyond this simple setting, and no existing method can perform detection and localization in inhomogeneous networks within finite time. To address this gap, we first show that in homogeneous networks, the anomalousness of a clique depends solely on its size. This property does not carry over to inhomogeneous networks, where the identity of the vertices forming the clique plays a critical role, and a smaller clique can be more anomalous than a larger one. Building on this insight, we propose a unified method for clique detection and localization based on a class of subgraphs called egonets. The proposed method generalizes to a wide variety of inhomogeneous network models and is naturally amenable to parallel computing. We establish the theoretical properties of the proposed method and demonstrate its empirical performance through simulation studies and application to two real world networks.