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.

ABCD图人工基准是一个带有社区结构和幂律分布的随机图模型。该模型生成类似于著名的LFR模型的图，但速度更快，并且可以通过分析进行研究。本文证明ABCD模型具有一些有趣的自相似行为，即地面真实社区的度分布渐近地与整个图的度分布相同（根据其大小适当标准化）。因此，我们不仅可以估计每个社区引起的边数，还可以估计在过程中生成的自环和多边数。了解这些数量很重要，因为（a）重连自环和多边数以保持图的简单性是算法中的一个昂贵部分，（b）每次重连都会导致底层配置模型与相应度序列上的统一简单图略有偏离。

ABCD模型中的社群自相似性