Random Birth-and-Death Networks

TL;DR

This paper introduces a random birth-and-death network model (RBDN) that captures network size fluctuations and derives exact degree distribution solutions, revealing different tail behaviors depending on the parameter p.

Contribution

The study develops the RBDN model and provides exact solutions for degree distributions using stochastic process and generating function methods, analyzing tail behaviors.

Findings

Degree distribution tails are Poisson for 0<p<=1/2.

Degree distribution tails are exponential as p approaches 1.

Network size fluctuations are characterized for different p values.

Abstract

In this paper, a baseline model termed as random birth-and-death network model (RBDN) is considered, in which at each time step, a new node is added into the network with probability p (0<p <1) connect it with m old nodes uniformly, or an existing node is deleted from the network with probability q=1-p. This model allows for fluctuations in size, which may reach many different disciplines in physics, ecology and economics. The purpose of this study is to develop the RBDN model and explore its basic statistical properties. For different p, we first discuss the network size of RBDN. And then combining the stochastic process rules (SPR) based Markov chain method and the probability generating function method, we provide the exact solutions of the degree distributions. Finally, the characteristics of the tail of the degree distributions are explored after simulation verification. Our results show that the tail of the degree distribution for RBDN exhibits a Poisson tail in the case of 0<p<=1/2 and an exponential tail as p approaches to 1.