Multigraph limit of the dense configuration model and the preferential attachment graph

TL;DR

This paper characterizes the limit objects of dense configuration models and preferential attachment graphs using graph limit theory, exchangeability, and urn models, providing a deeper understanding of their asymptotic structure.

Contribution

It introduces a novel application of dense graph limits to the configuration model and preferential attachment graphs, connecting exchangeability and urn models for explicit constructions.

Findings

Limit objects for dense configuration models are characterized.

Explicit constructions for preferential attachment graph limits are provided.

Connections between graph limits, exchangeability, and urn models are established.

Abstract

The configuration model is the most natural model to generate a random multigraph with a given degree sequence. We use the notion of dense graph limits to characterize the special form of limit objects of convergent sequences of configuration models. We apply these results to calculate the limit object corresponding to the dense preferential attachment graph and the edge reconnecting model. Our main tools in doing so are (1) the relation between the theory of graph limits and that of partially exchangeable random arrays (2) an explicit construction of our random graphs that uses urn models.