Edge-exchangeable graphs and sparsity

TL;DR

This paper introduces edge exchangeability as a new framework for random graph models, allowing for sparse graphs with sub-quadratic edge growth, unlike traditional node exchangeable models constrained by the Aldous-Hoover Theorem.

Contribution

It defines edge exchangeability for graphs, demonstrating its ability to model sparse graphs and relating it to existing exchangeability concepts in combinatorics.

Findings

Edge exchangeability enables sparse graph models.

Contrasts with node exchangeability constrained by Aldous-Hoover.

Provides a natural connection to clustering and partitions.

Abstract

A known failing of many popular random graph models is that the Aldous-Hoover Theorem guarantees these graphs are dense with probability one; that is, the number of edges grows quadratically with the number of nodes. This behavior is considered unrealistic in observed graphs. We define a notion of edge exchangeability for random graphs in contrast to the established notion of infinite exchangeability for random graphs --- which has traditionally relied on exchangeability of nodes (rather than edges) in a graph. We show that, unlike node exchangeability, edge exchangeability encompasses models that are known to provide a projective sequence of random graphs that circumvent the Aldous-Hoover Theorem and exhibit sparsity, i.e., sub-quadratic growth of the number of edges with the number of nodes. We show how edge-exchangeability of graphs relates naturally to existing notions of exchangeability from clustering (a.k.a. partitions) and other familiar combinatorial structures.