Unifying Directed and Undirected Random Graph Models

TL;DR

This paper develops mathematical tools to relate directed and undirected random graph models, establishing conditions for their equivalence and providing coupling techniques for approximate comparisons.

Contribution

It introduces a framework for translating between directed and undirected graph models and extends known relationships to broader classes of random graphs.

Findings

Identifies probability spaces where directed and undirected models are equivalent.

Provides coupling techniques for approximate equivalence.

Extends relationships between directed models to undirected counterparts.

Abstract

In this paper we explore mathematical tools that can be used to relate directed and undirected random graph models to each other. We identify probability spaces on which a directed and an undirected graph model are equivalent, and investigate which graph events can subsequently be translated between equivalent models. We finally give coupling techniques that can be used to establish an approximate equivalence between directed and undirected random graph models. As an application of these tools, we give conditions under which two broad classes of random graph models are equivalent. In one of these classes the presence of edges/arcs is determined by independent Bernoulli random variables, while in the other class a fixed number of edges/arcs is placed in between vertices according to some probability measure. We finally use these equivalences to extend a previously established relationship between the directed versions of these model classes to their undirected counterparts.