Upper bounds for number of removed edges in the Erased Configuration Model

TL;DR

This paper investigates the number of edges removed in the erased configuration model for simple graphs, providing upper bounds and revealing three scaling regimes based on degree distribution, especially for power-law networks.

Contribution

It offers the first quantitative bounds on the number of removed edges in the model, linking graph size and degree distribution to finite-size effects.

Findings

Upper bounds for removed edges depending on graph size

Identification of three scaling regimes for power-law degree distributions

Enhanced understanding of finite-size effects in network models

Abstract

Models for generating simple graphs are important in the study of real-world complex networks. A well established example of such a model is the erased configuration model, where each node receives a number of half-edges that are connected to half-edges of other nodes at random, and then self-loops are removed and multiple edges are concatenated to make the graph simple. Although asymptotic results for many properties of this model, such as the limiting degree distribution, are known, the exact speed of convergence in terms of the graph sizes remains an open question. We provide a first answer by analyzing the size dependence of the average number of removed edges in the erased configuration model. By combining known upper bounds with a Tauberian Theorem we obtain upper bounds for the number of removed edges, in terms of the size of the graph. Remarkably, when the degree distribution follows a power-law, we observe three scaling regimes, depending on the power law exponent. Our results provide a strong theoretical basis for evaluating finite-size effects in networks.