Sampling perspectives on sparse exchangeable graphs

TL;DR

This paper clarifies the relationships between sampling, exchangeability, and graph limits in sparse exchangeable graphs, introducing a new notion of sampling convergence and linking exchangeability to a natural sampling scheme.

Contribution

It introduces sampling convergence as a new graph limit concept for sparse graphs and connects exchangeability to a sampling scheme within the graphex framework.

Findings

Sampling convergence generalizes left convergence for sparse graphs

Exchangeability is equivalent to invariance under the sampling scheme

Provides a unified view of network modeling, graph limits, and exchangeability

Abstract

Recent work has introduced sparse exchangeable graphs and the associated graphex framework, as a generalization of dense exchangeable graphs and the associated graphon framework. The development of this subject involves the interplay between the statistical modeling of network data, the theory of large graph limits, exchangeability, and network sampling. The purpose of the present paper is to clarify the relationships between these subjects by explaining each in terms of a certain natural sampling scheme associated with the graphex model. The first main technical contribution is the introduction of sampling convergence, a new notion of graph limit that generalizes left convergence so that it becomes meaningful for the sparse graph regime. The second main technical contribution is the demonstration that the (somewhat cryptic) notion of exchangeability underpinning the graphex framework is equivalent to a more natural probabilistic invariance expressed in terms of the sampling scheme.