Study On Complex Networks And PERT Networks

Complex networks exist in every aspect of nature and society, the most commons are the Information Networks (such as World Wide Web, Internet, Computer Sharing System and E-mail Networks), Transportation Networks (such as Rail Networks, Highway Networks), Power Grid, Technology Networks and Social Networks. Thus a lot of scientists devote to reveal the essence of the topological structures, functions, rules of evolvement and dynamical behaviours, and have gained fruitful results in the fundamental theory of complex networks. A lot of developed countries have published the roadmaps for complex networks. In this paper, we give an unified solution of the steady-state degree distribution of the growing and evolving networks by the theory of the Markov Chain, and obtain the analytic solution of the distribution of the completion time of PERT networks with independently and generally distributed activity durations by the theory of Markov Skeleton Process. The main results are as follows:(1) The development and research of complex networks were reviewed in brief, the current research contents of complex networks was summarized, and researches in this paper was listed. See chapter1.(2) We give a general summary of the definition of the complex networks and some statistical parameters. And give a systemic review of some important and most studied complex network models. Then, we have a description of several methods for solving the steady-state degree distribution. See chapter2.(3) First, Based on the concepts and techniques of Markov chain theroy, the rigorous proof for the existence of the steady-state degree distribution of the BA model was given, and re-derives the exact and analytical formulas of the distribution. Then we abstract a kind of stochastic processes from the evolving processes of growing networks, this process is called growing network Markov chains. Thus the existence and the formulas of the steady-state degree distribution of growing networks are transformed to the corresponding problems of growing network Markov chains. With this method we get a rigorous, exact and unified solution of the steady-state degree distribution for the growing networks. See chapter3.(4) First, we propose a simple evolving network with link additions as well as removals, from the perspective of Markov chain, we give the exact solution of the steady-state degree distribution. Then proposed the theory of evolving network Markov chains, with this method, we finally obtain a rigorous, exact and unified solution of the steady-state degree distribution for the evolving networks. See chapter4.(5) We consider the PERT networks with independently and generally distributed activity durations. For any path in this network, we select all the activities related to this path such that the completion time of the sub-network is equal to the completion time of this path. We model this sub-network as a Markov skeleton process using the elapsed time as the supplementary variables, the state space is related to the sub-network structure. Then we obtain an analytic solution of the completion time using the backward equations, which generalized the overall planing method advocated by Mr. Hua Luogeng.