Convergence Rates for the Degree Distribution in a Dynamic Network Model

TL;DR

This paper analyzes the convergence rate of the degree distribution in a dynamic network model where individuals evolve via a birth-death process and form connections based on a social index, providing bounds on how quickly the distribution approaches its limit.

Contribution

It derives a convergence rate for the degree distribution towards a mixed Poisson distribution in a stochastic network model, using finite-time degree distribution analysis and approximation bounds.

Findings

Established an explicit convergence rate for the degree distribution.

Derived finite-time degree distribution formulas.

Provided an upper bound for total variation distance to the asymptotic distribution.

Abstract

In the stochastic network model of Britton and Lindholm [Dynamic random networks in dynamic populations. Journal of Statistical Physics, 2010], the number of individuals evolves according to a supercritical linear birth and death process, and a random social index is assigned to each individual at birth, which controls the rate at which connections to other individuals are created. We derive a rate for the convergence of the degree distribution in this model towards the mixed Poisson distribution determined by Britton and Lindholm based on heuristic arguments. In order to do so, we deduce the degree distribution at finite time and derive an approximation result for mixed Poisson distributions to compute an upper bound for the total variation distance to the asymptotic degree distribution.