Degree Distribution Of A Class Of Preferred Growth System

This paper mainly devotes to studying a class of preferential growth systems. Based on the Markov chain theory, the author gets the rigorous proof for the existence of the steady-state degree distribution and drives the exact analytic formulas of the distribution, then proves the system has scale-free property. The full text is composed of six parts, the concrete structure is as follows:Chapter one introduces the background and existing work of the growth system as well as the framework of the paper.Chapter two introduces the elementary theories which are needed in the paper, mainly include: network measure characteristics; the evolved history of complex network which begins with regular net to stochastic graph theory then to complex network; three important structure features in complex networks (small-world effect, scale-free property and networks navigability);different tools of computing degree distribution.Chapter three considers a preferential growth model where balls and boxes are both added one by one to the system. A new ball can either join in the new box (with probability q) or join in an already existing box with a probability proportional to the size thereof. Based on the Markov chain theory, the author gets the rigorous proof for the existence of the steady-state degree distribution and obtains the exact solution, then proves the model has scale-free property. At the end, the author gives the result of computing simulation of degree distribution.Considering there are m balls added to the system at each time step, chapter four promotes the preferential growth model to an general situation. The m boxes can either join in the new box (with probability q) or dependently join in an already existing box with a probability proportional to the size thereof. The author gets the rigorous proof for the existence of the degree distribution and obtain the exact solution.Chapter five promotes the preferential growth model to the other general situation. Considering there are m balls added to the system at each time step, m -s balls join in the new box, s balls can either join in the new box (with probability 1-p) or dependently join in an already existing box with a probability proportional to the size thereof. The author gets the rigorous proof for the existence of the degree distribution and obtain the exact solution.Chapter six summarizes the mainly work of this paper, and carry on this result to further discusses.