Self-similarity of Communities of the ABCD Model
This provides incremental insights for researchers in network science and community detection, helping optimize algorithms by understanding expensive rewiring steps.
The paper tackles the problem of analyzing the self-similarity properties of the ABCD graph model, showing that the degree distribution of ground-truth communities asymptotically matches that of the whole graph, enabling estimation of edges, self-loops, and multi-edges.
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.