Modularity of the ABCD Random Graph Model with Community Structure

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 one, and its main parameter ξ can be tuned to mimic its counterpart in the LFR model, the mixing parameter µ.In this paper, we investigate various theoretical asymptotic properties of the ABCD model. In particular, we analyze 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.

Preferential attachment hypergraph with high modularity

“…In these models, not only the degrees but also the sizes of communities follow a power-law. Some asymptotic results for the modularity of ABCD were given recently in Kamiński et al (2022). Both LFR and ABCD are, however, static graphs (the number of nodes must be given in advance at the generation phase).…”