Research On The Degree Distribution Of Random Birth-and-Death Networks With Network Size Decline

The study of complex network has been in the phase of vigorous development in the past 20 years,its researchers come from the fields of physics,mathematics,computer science,infectious diseases and so on.Random networks with complex topological structure are common in Nature,such as communication network,power network,Internet network,road traffic network and interpersonal relationship network.In order to make these networks to better serve humans,the comprehensive research on complex network is needed.However,the study of network model,statistical characteristics and dynamic behavior of complex networks is an extremely important part of research on complex network,which has received more and more attention,but its research is far from perfect.Recently,study on evolving network with network node births and deaths and its statistical characteristics has caught much attention.Compared with the networks with network size(the number of nodes in the network)growing or keep unchanged,the decreasing networks are less studied.In addition,degree distribution as the main physical quantity of the statistical characteristics of complex network has an important role in understanding the real network.Therefore,we take the random birth-and-death network model as the research object in this thesis,the situation of network size decrease is discussed.Especially,we take degree distribution as the key content of this article,in-depth study the steady state degree distribution,average degree and their mathematical properties.Although several papers have explored the steady state equations of the degree distribution on decreasing random birth-and-death network,have proposed random process rule based on Markov chain to calculate the degree distribution,and have given the general expression of the degree distribution,there is no theoretical proof in the case of m(29)3.Moreover,the study of average degree is in its infancy.Therefore,in the steady state,the degree distribution and the average degree have what kind of natures,and how to solve these statistical characteristics are the emphasis of this thesis.In this work,first we give a theoretical proof for the general expression of the degree distribution,and discuss its tail characteristics.Second,on the basis of the existing literatures,we make a thorough study on the average degree,obtain the concreteexpression of the average degree and explore its mathematical properties,especially the convergence.Furthermore,to alleviate the large error problem of calculating the steady state average degree directly when the network size is large,we adopt numerical method to obtain approximate expression of the average degree to approximate its solution.Finally,computer simulations are carried out to verify these results.