Modularity of the ABCD random graph model with community structure

Abstract

Abstract The Artificial Benchmark for Community Detection (ABCD) graph is a random graph model with community structure and power-law distribution for both degrees and community sizes. The model generates graphs with similar properties as the well-known LFR (Lancichinetti, Fortunato, Radicchi) one, and its main parameter $\xi$ can be tuned to mimic its counterpart in the LFR model, the mixing parameter $\mu$. In this article, we investigate various theoretical asymptotic properties of the ABCD model. In particular, we analyse the modularity function, arguably, the most important graph property of networks in the context of community detection. Indeed, the modularity function is often used to measure the presence of community structure in networks. It is also used as a quality function in many community detection algorithms, including the widely used Louvain algorithm.