# On edge exchangeable random graphs

## TL;DR

This paper analyzes a model of edge exchangeable random graphs, exploring their asymptotic properties, degree distributions, and convergence behaviors, demonstrating the model's ability to produce various types of sparse and dense graphs.

## Contribution

It provides a detailed study of the asymptotic properties of edge exchangeable random graphs, including examples with different density and convergence characteristics.

## Key findings

- The model can produce dense, sparse, and extremely sparse graphs.
- One example yields a power-law degree distribution.
- The model exhibits various convergence behaviors, including convergence to graphons and generalized graphons.

## Abstract

We study a recent model for edge exchangeable random graphs introduced by Crane and Dempsey; in particular we study asymptotic properties of the random simple graph obtained by merging multiple edges. We study a number of examples, and show that the model can produce dense, sparse and extremely sparse random graphs. One example yields a power-law degree distribution. We give some examples where the random graph is dense and converges a.s. in the sense of graph limit theory, but also an example where a.s. every graph limit is the limit of some subsequence. Another example is sparse and yields convergence to a non-integrable generalized graphon defined on $(0,\infty)$.