Subgraph densities and scaling limits of random graphs with a prescribed modular decomposition

TL;DR

This paper studies large random graphs with specific modular structures, estimating subgraph counts and showing their convergence to a Brownian limit object in the graphon space.

Contribution

It introduces a method to analyze subgraph densities and proves convergence of these graphs to a Brownian limit, advancing understanding of their asymptotic behavior.

Findings

Estimated number of subgraph copies in large graphs

Proved convergence to a Brownian limit object

Developed combinatorial and analytical tools for graph analysis

Abstract

We consider large uniform labeled random graphs in different classes with prescribed decorations in their modular decomposition. Our main result is the estimation of the number of copies of every graph as an induced subgraph. As a consequence, we obtain the convergence of a uniform random graph in such classes to a Brownian limit object in the space of graphons. Our proofs rely on combinatorial arguments, computing generating series using the symbolic method and deriving asymptotics using singularity analysis.