Random networks with preferential growth and vertex death

TL;DR

This paper models a dynamic random network with vertex birth and death, deriving the asymptotic fitness distribution under various preferential attachment and death rate functions, revealing different distribution types including power law and stretched exponential.

Contribution

It introduces a continuous-time network model incorporating vertex death, deriving explicit asymptotic fitness distributions for various functional forms, expanding understanding of network evolution dynamics.

Findings

Power law distribution for linear preferential attachment with constant death rate

Exponential decay when death rate is proportional to fitness

Stretched exponential distribution for nonlinear death rates

Abstract

A dynamic model for a random network evolving in continuous time is defined where new vertices are born and existing vertices may die. The fitness of a vertex is defined as the accumulated in-degree of the vertex and a new vertex is connected to an existing vertex with probability proportional to a function $b$ of the fitness of the existing vertex. Furthermore, a vertex dies at a rate given by a function $d$ of its fitness. Using results from the theory of general branching processes, an expression for the asymptotic empirical fitness distribution $\{p_k\}$ is derived and analyzed for a number of specific choices of $b$ and $d$. When $b(i)=i+\alpha$ and $d(i)=\beta$ -- that is, linear preferential attachment for the newborn and random deaths -- then $p_k\sim k^{-(2+\alpha)}$. When $b(i)=i+1$ and $d(i)=\beta(i+1)$, with $\beta<1$, then $p_k\sim (1+\beta)^{-k}$, that is, if also the death rate is proportional to the fitness, then the power law distribution is lost. Furthermore, when $b(i)=i+1$ and $d(i)=\beta(i+1)^\gamma$, with $\beta,\gamma<1$, then $\log p_k\sim -k^\gamma$ -- a stretched exponential distribution. The momentaneous in-degrees are also studied and simulations suggest that their behaviour is qualitatively similar to that of the fitnesses.