Generating simple random graphs with prescribed degree distribution

TL;DR

This paper compares four methods for generating simple undirected graphs with a specified degree distribution, analyzing their theoretical properties and limitations for large graphs.

Contribution

It introduces and compares four novel methods for generating graphs with prescribed degree distributions, highlighting their theoretical guarantees and differences.

Findings

All methods produce correct degree distribution in large graphs

Methods vary in assumptions and operational order

Each method has specific conditions for accuracy

Abstract

Let $F$ be a probability distribution with support on the non-negative integers. Four methods for generating a simple undirected graph with (approximate) degree distribution $F$ are described and compared. Two methods are based on the so called configuration model with modifications ensuring a simple graph, one method is an extension of the classical Erd\H{o}s-R\'{e}nyi graph where the edge probabilities are random variables, and the last method starts with a directed random graph which is then modified to a simple undirected graph. All methods are shown to give the correct distribution in the limit of large graph size, but under different assumptions on the degree distribution $F$ and also using different order of operations.