The Research Of Edge-Markovian Independent Random Graph Process And Random Branching Tree

In the late of1950s and early of1960s, Erd s and Rényi established the theory ofrandom graph. The theory of random graph had a great development in the next halfcentury and widely applied in the areas of nature science and social science. In therecent years, because of the rasing of the research of the complex network, randomtheory draws much attentions of many researchers, such as mathematician, physicistand so on. Many new results of research are presented. In this dissertation, weresearch the edge-Markovian independent random graph processes and randombranching processes and give the mathematics models of random graph processes ofthese two classes and discuss their topological properties. We organize thisdissertation as follows:1. Some basic concepts and propositions are sorted and discussed, such as the limitof network sequence, distance between networks, measurable graph space, and so on.By the concepts and methods of the classic random processes, the statistics propertiesof Bernoulli random graph sequence, stationary random graph processes and Markovrandom graph processes are discussed.2. For edge-Markovian independent random graph processes, we derive thetransition probability function, prove its stability, obtain its stationary distribution;Wegeneralize the concept of isolated node and establish the isolated node of randomgraph process; When the initial distribution of the edge-Markovian independentrandom graph processes is stationary distribution, we give the distribution sequence ofthe number of isolated node, prove the probability of no isolated node;We discuss itsconnection,and obtain the probability of two nodes which are connected, estimate thelower bound of the connection probability for random graph and prove that therandom graph is almost connected everywhere if the number of node is infinite.3. We establish the birth and death random branching tree model whose birth rateand death rate are dependent on the age period. In the model of birth rate which isdependent on the age period, we discuss the distribution of the total I-generationnode’s number by two difference expressions. We give the concept of node falsedegree,prove the distribution of living(node degree) and dead(node false degree)I-generation node’s number are the Possion distribution, and obtain the probabilities of all keeping alive or dead of I-generation node; We discuss the monotone functions,random terminate Possion processes, and prove the1-generation node countingprocess is the random terminate Possion processes.We obtain the conditionaldistribution function of a arbitrary node’s first birth age,prove that order statistic ofthe related of n nodes’ birth age is subject to order statistic of n independent randomvariable of uniformly distribution.We also give the concept of period isolated nodeand give the probabilities that node is a period isolated node at sometime or in someperiod.