The structure of typical clusters in large sparse random configurations

TL;DR

This paper links the structure of clusters in large sparse random graphs generated by the configuration model to Galton-Watson processes, providing a probabilistic explanation for coagulation dynamics with limited aggregations.

Contribution

It establishes a hydrodynamical limit theorem connecting cluster shapes in the configuration model to Galton-Watson process distributions.

Findings

Cluster structures resemble Galton-Watson distributions

Limit theorem for empirical measures of cluster shapes

Probabilistic explanation for coagulation with limited aggregations

Abstract

The initial purpose of this work is to provide a probabilistic explanation of a recent result on a version of Smoluchowski's coagulation equations in which the number of aggregations is limited. The latter models the deterministic evolution of concentrations of particles in a medium where particles coalesce pairwise as time passes and each particle can only perform a given number of aggregations. Under appropriate assumptions, the concentrations of particles converge as time tends to infinity to some measure which bears a striking resemblance with the distribution of the total population of a Galton-Watson process started from two ancestors. Roughly speaking, the configuration model is a stochastic construction which aims at producing a typical graph on a set of vertices with pre-described degrees. Specifically, one attaches to each vertex a certain number of stubs, and then join pairwise the stubs uniformly at random to create edges between vertices. In this work, we use the configuration model as the stochastic counterpart of Smoluchowski's coagulation equations with limited aggregations. We establish a hydrodynamical type limit theorem for the empirical measure of the shapes of clusters in the configuration model when the number of vertices tends to $\infty$. The limit is given in terms of the distribution of a Galton-Watson process started with two ancestors.