# Self-similarity of Communities of the ABCD Model

arXiv: https://arxiv.org/abs/2312.00238 · HTML: https://mlsanity.com/paper/2312.00238/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 32 references that are in the map.
- First posted on arXiv 30 Nov 2023.

## Builds on (most cited first, 6 of 6)

- [Scale-free networks are rare](https://mlsanity.com/paper/1801.03400/) (2018)
- [Artificial Benchmark for Community Detection (ABCD): Fast Random Graph Model with Community Structure](https://mlsanity.com/paper/2002.00843/) (2020)
- [Modularity of the ABCD Random Graph Model with Community Structure](https://mlsanity.com/paper/2203.01480/) (2022)
- [Hypergraph Artificial Benchmark for Community Detection (h-ABCD)](https://mlsanity.com/paper/2210.15009/) (2022)
- [Properties and Performance of the ABCDe Random Graph Model with Community Structure](https://mlsanity.com/paper/2203.14899/) (2022)
- [Artificial Benchmark for Community Detection with Outliers (ABCD+o)](https://mlsanity.com/paper/2301.05749/) (2023)

## Built on by (most cited first, 2 of 2)

- [Modularity Based Community Detection in Hypergraphs](https://mlsanity.com/paper/2406.17556/) (2024)
- [Twinning Complex Networked Systems: Data-Driven Calibration of the mABCD Synthetic Graph Generator](https://mlsanity.com/paper/2602.02044/) (2026)

## Abstract (arXiv)

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.