Generating graphs randomly

TL;DR

This paper discusses algorithms for efficiently generating random graphs with specific degree sequences, enabling researchers to compare real networks to typical models within the same family.

Contribution

It introduces and analyzes several algorithms for uniformly sampling simple graphs with a given degree sequence, addressing computational efficiency and theoretical rigor.

Findings

Algorithms for uniform graph sampling are effective and computationally feasible.

Multiple methods for sampling graphs with fixed degree sequences are compared.

The paper provides rigorous analysis of the sampling algorithms.

Abstract

Graphs are used in many disciplines to model the relationships that exist between objects in a complex discrete system. Researchers may wish to compare a network of interest to a "typical" graph from a family (or ensemble) of graphs which are similar in some way. One way to do this is to take a sample of several random graphs from the family, to gather information about what is "typical". Hence there is a need for algorithms which can generate graphs uniformly (or approximately uniformly) at random from the given family. Since a large sample may be required, the algorithm should also be computationally efficient. Rigorous analysis of such algorithms is often challenging, involving both combinatorial and probabilistic arguments. We will focus mainly on the set of all simple graphs with a particular degree sequence, and describe several different algorithms for sampling graphs from this family uniformly, or almost uniformly.