Limits of randomly grown graph sequences

TL;DR

This paper investigates the limits of sequences of randomly grown graphs, proving their convergence and characterizing their limit objects using graph limit theory, with implications for models like internet growth.

Contribution

It demonstrates the almost sure convergence of various randomly growing graph sequences and determines their limit objects within the graph limit framework.

Findings

Proved almost sure convergence of several random graph sequences

Identified limit objects for these sequences

Developed methods to estimate cut distance and subgraph densities

Abstract

Motivated in part by various sequences of graphs growing under random rules (like internet models), convergent sequences of dense graphs and their limits were introduced by Borgs, Chayes, Lov\'asz, S\'os and Vesztergombi and by Lov\'asz and Szegedy. In this paper we use this framework to study one of the motivating class of examples, namely randomly growing graphs. We prove the (almost sure) convergence of several such randomly growing graph sequences, and determine their limit. The analysis is not always straightforward: in some cases the cut distance from a limit object can be directly estimated, in other case densities of subgraphs can be shown to converge.