Asymptotic Degree Distributions In Several Complex Network Models

The study of random networks has drawn more and more attention.Random networks also exist widely in the real world,such as the Internet,social networks,biological networks,geographic networks,and citation networks of scientists.As the study of random networks deepens,several common properties of random networks can be revealed,such as small-world property,scale-free,and high clustering.This thesis aims to study random networks’ degree distribution and related properties.First,we study a random pseudofractal network model and explore the degree distribution of this model.We find that the degree distribution of the model converges a power-law distribution with an exponent of 3 in probability.According to its structural properties,we find that its cluster coefficient converges to a constant in probability.Furthermore,we explore the maximum degree of this model and extend it to random k-tree and Apollonian network models.Second,based on the random pseudofractal network models,we proposed evolving pseudofractal network models:in n-th step,each edge in the model can produce new edges with probability q_n,where q_n are evolutionary parameters.If all the evolutionary parameters are equal to a constant,we obtain the probability generating function of the asymptotic degree distribution.If the evolutionary parameters q_n satisfy certain conditions,we then prove that the asymptotic degree distribution in our model obeys the power law with exponent 3.On this basis,finally,we obtain its clustering coefficient.Finally,we discuss the asymptotic degree distribution of random Apollonian networks with multiple color edges.In the N-color case,we denote a given vertex’s(generalized)degree by d.If the colors of edges are chosen randomly,we find that the proportion of the number of edges with a given color k converges to a random variable a.s.We also provide a recurrence equation for its asymptotic degree distribution.Furthermore,we also introduce the perturbation into the color selection.By stochastic approximation,we yield that the proportion of the number of edges with color k converges to a constant a.s.Similarly,we also obtain the recurrence equation for the asymptotic degree distribution of the second model.