Self-similarity of Communities of the ABCD Model

TL;DR

This paper analyzes the self-similar properties of the ABCD graph model, revealing that community degree distributions mirror the overall graph, which aids in estimating internal edges and simplifying graph generation.

Contribution

It demonstrates the self-similarity in the ABCD model's community structure and provides methods to estimate internal edges and handle multi-edges efficiently.

Findings

Community degree distribution matches the overall graph asymptotically.

Ability to estimate edges, self-loops, and multi-edges within communities.

Insights into graph rewiring impacts on model uniformity.

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 similar to the well-known LFR model but it is faster and can be investigated analytically. In this paper, we show that the ABCD model exhibits some interesting self-similar behaviour, namely, the degree distribution of ground-truth communities is asymptotically the same as the degree distribution of the whole graph (appropriately normalized based on their sizes). As a result, we can not only estimate the number of edges induced by each community but also the number of self-loops and multi-edges generated during the process. Understanding these quantities is important as (a) rewiring self-loops and multi-edges to keep the graph simple is an expensive part of the algorithm, and (b) every rewiring causes the underlying configuration models to deviate slightly from uniform simple graphs on their corresponding degree sequences.