Average degree under different network sizes for random birth-and-death networks

Abstract

In the social and biological networks,each agent experiences a birth-and-death process.These evolving networks may exhibit some unique characteristics.Recently,the birth-and-death networks have gradually caught attention,and thus far,most of these studies on birth-and-death networks have focused on the calculations of the degree distributions and their properties.In this paper,a kind of random birth-and-death network (RBDN) with reducing network size is discussed,in which at each time step,with probability p(0pq=1-p.Unlike the existing literature,this study is to calculate the average degrees of the proposed networks under different network sizes.First,for the reducing RBDN,the steady state equations for each node's degree are given by using the Markov chain method based on stochastic process rule,and then the recursive equations of average degree for different network sizes are obtained according to these steady state equations.Second,by means of the recursive equations,we explore four basic properties of average degrees as follows:1) the average degrees are limited,2) the average degrees are strictly monotonically increasing,3) the average degrees are convergent to 2mq,and 4) the sum of each difference between the average degree and 2mq is a bounded number.Theoretical proofs for these four properties are also provided in this paper.Finally,on the basis of these properties,a generation function approach is employed to obtain the exact solutions of the average degrees for various network sizes.In addition to the theoretical derivations to the average degrees,computer simulation is also used to verify the correctness of exact solutions of the average degrees and their properties.Furthermore,we use numerical simulation to study the relationship between the average degree and node increasing probability p.Our simulation results show as follows:1) with the increasing of p,the convergent speed of the average degree to 2mq is increasing;2) with the increasing of m,the convergent speed of the average degree to 2mq is decreasing.In conclusion,for the proposed RBDN model,the main contributions of this study include 1) providing the recursive equations of the average degrees under different network sizes,2) investigating the basic properties for the average degrees,and 3) obtaining the exact solutions of the average degrees.