Limits of Sparse Configuration Models and Beyond: Graphexes and Multi-Graphexes

TL;DR

This paper explores the limits of sparse random graph models using the concept of graphexes, establishing convergence conditions for various models and extending the framework to multigraphs.

Contribution

It introduces sampling convergence for multigraph sequences and characterizes the limits of key sparse graph models using graphexes.

Findings

Established sampling convergence for multigraph sequences.

Derived necessary and sufficient conditions for model convergence.

Identified the limit objects as augmented exchangeable random graph models.

Abstract

We investigate structural properties of large, sparse random graphs through the lens of "sampling convergence" (Borgs et. al. (2017)). Sampling convergence generalizes left convergence to sparse graphs, and describes the limit in terms of a "graphex". We introduce a notion of sampling convergence for sequences of multigraphs, and establish the graphex limit for the configuration model, a preferential attachment model, the generalized random graph, and a bipartite variant of the configuration model. The results for the configuration model, preferential attachment model and bipartite configuration model provide necessary and sufficient conditions for these random graph models to converge. The limit for the configuration model and the preferential attachment model is an augmented version of an exchangeable random graph model introduced by Caron and Fox (2017).

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