An example of graph limits of growing sequences of random graphs

TL;DR

This paper explores the limits of growing random graphs created by sequentially adding vertices with random degrees, establishing conditions under which these graphs converge to a kernel, and linking the growth process to graph limit theory.

Contribution

It introduces a framework connecting sequential graph growth models with graph limits via kernels, including conditions for convergence and distributional equivalence.

Findings

Identifies conditions for graph convergence to a kernel.

Shows equivalence between growing graph distributions and kernel-defined random graphs.

Includes examples like ER and threshold graphs.

Abstract

We consider a class of growing random graphs obtained by creating vertices sequentially one by one: at each step, we choose uniformly the neighbours of the newly created vertex; its degree is a random variable with a fixed but arbitrary distribution, depending on the number of existing vertices. Examples from this class turn out to be the ER random graph, a natural random threshold graph, etc. By working with the notion of graph limits, we define a kernel which, under certain conditions, is the limit of the growing random graph. Moreover, for a subclass of models, the growing graph on any given n vertices has the same distribution as the random graph with n vertices that the kernel defines. The motivation stems from a model of graph growth whose attachment mechanism does not require information about properties of the graph at each iteration.