General Birth-death Process And Random Branching Tree Evolution

In the last15years, the study of complex networks is developing very fast. Due toits dynamics and complexity, it is hard to do the theoretic research and we have not manyefective theory methods. As the major theory method of studying network evolution,the random graph process draws the favour of many scholars. So far, the main methodof constructing random graph process is based on the the discrete time and Markov evo-lution. Given the the current state of graph process, during a unit time one link (orone node) is added (or deleted) to realize the graph process constructing. This arti-cle, breaking through the traditional way of constructing random graph process, basedon the biological mechanism of asexual reproduction, constructs the random branchingtree evolution based on birth death process, using node birth-rate function and death-ratefunction to figure the topology evolution characteristics of random graph. This paper,viareal values stochastic process with continuous time characterize certain graph topolo-gy evolution characteristics to build graph process, pioneers new ideas about the studyof random graphs, and provides continuous time Markov random graph or non-Markovrandom graph process paradigm, and enriches the research contents of the random graphprocess, provides a more objective biological reproduction or epidemic model(The tra-ditional biological reproduction or epidemic model(branching process and birth-deathprocess) mainly study the problem such as population size and extinction probability,but ignored the internal structure of biological networks and relationships between indi-viduals). This paper using theory tools such as Markov process, graph theory and thetheory of diferential equations, study the existence theorem of random branching tree,make analytical analysis for such as graph topology structure, age structure and the re-productive structure etc.. Detailed content as follows:Chapter one briefly described the research background,and Outlines the main re-search contents, the organization arrangement, main innovation and research signifi-cance.Chapter two study general birth-death process. The generating function of thenumber of ofspring is mainly studied and the distribution of the number of ofspring isgot.Chapter3research the evolution of Markov birth-death branching tree (randombranching tree evolution based on the birth-death process).(1) the evolution of Markovbirth-death branching tree is constructed, namely the existence theorem of graph processis show;(2) make analytical analysis for some characteristic variables of graph topologystructure such as: degree(out-degree, in-degree etc.), the number of real nodes or imagi-nary nodes or Isolated nodes,the number of real nodes with diferent degree(out-degree,imaginary out-degree etc.), the number of real connected components, the number ofreal connected components which.s root node is any generation node,the number of n-odes in one connected component;(3) get the extinction probability of linear birth-death process using the elementary and concise method;(4) make analytical analysis for somecharacteristic variables of age structure and reproductive structure such as: the numberof real nodes at diferent age, the number of nodes born at appropriate age, the numberof real nodes born at appropriate age, the number of nodes born at over-age, the numberof real nodes born at over-age, the number of nodes dead at diferent age, the numberof nodes in any generation, the number of nodes at diferent age in any generation, thenumber of nodes with diferent degree in any generation;(5) study the distribution of pro-ductive age such as:first-born age and last-born age, productive age order statistic;(6)discuss the possible extension of the Markov birth death branching tree and show theexistence of the random branching tree evolution with age-dependent birth rate.In chapter four, the Markov birth-death branching trees evolution is extended tothe branching trees model with nodes being multiparous(In the branching trees modelwith nodes being multiparous, it is assumed that the number of son-nodes born at eachdelivery is a random variable). The existence of the model is shown, and it is studiedthat the number of alive nodes, the number of dead nodes, the number of connectedcomponents, the number of son-nodes of any node at any age or at dying moment.In chapter five, this paper research prospect is presented. In the Markov birth-deathbranching trees model, the arrival process of the child-nodes is homogeneous Poissonprocess. So It can be extended as that the arrival process of the child-nodes is a renewalprocess or other point process.