Bayesian Inference Of Power-Law Distribution In Scale-Free Networks

Power-law distribution phenomenon exists widely in nature and social life,and the laws of many network behavior data also reflect the characteristics of power-law distribution,as one of the basic characteristics of complex networks,thus their study has broad and far-reaching significance.And with the development of the research,how to better estimate the scale parameters in the power-law model based on the heavy-tailed property of the power-law distribution has become a research topic of great significance,and also a problem faced by many network science researchers.At present,there are two commonly used methods.One is to first draw the graph of probability density function in the log-log coordinate,then estimate the slope of the line by the least square method,and obtain the scale parameters of the power-law distribution;The other is the maximum likelihood estimation(MLE)based on KS(Kolmogorov-Smirnov)statistics and maximum likelihood ratio proposed by Clauset et al.,combined with the goodness of fit test.In this paper,we propose another method for estimating power law parameters,which is bayesian inference based on markov chain sampling and Hastings-Metropolis algorithm.The work of this paper is mainly based on the following two aspects: One is how to generate a random graph network with vertex degree distribution obeying power-law.Among the numerous random graph models,we choose a generalized random graph with vertex weights,when the vertex weight makes it obey the power-law distribution form(equation 2.5),now has the following conclusion: the degrees of m uniformly chosen vertices in [n] converge in distribution to the mixed poisson distribution and is asymptotically independent.At the same time,the degree distribution of the generated generalized random graph also has the form of power-law distribution,so the Scale-free network whose degree distribution obeys power-law distribution can be generated.The second is to make bayesian inference on the parameters of the power-law model based on the MCMC sampling method.The rationale is to sample from the conditional probability and estimate the scale parameters through repeated iteration.The fourth chapter is data analysis.The MCMC method is applied to simulated network and real network data respectively,In the simulation network,we simulate six scale-free networks with degree distribution obeying power-law,make bayesian estimation according to the Hastings-Metropolis sampling algorithm,and compare with the maximum likelihood estimation proposed by Clauset et al.,the conclusion is that the estimation result of MCMC method is more close to the real parameter value,which verifies the superiority of MCMC method.Secondly,the method of MCMC parameter estimation was applied to four real network data,and the goodness of fit test was performed on the estimated results based on KS(kolmogorov-smirnov)statistics.Generate power-law distribution data according to the given parameter,made a power-law distribution fitting the observed data validation,finally has two sets of network data through the inspection,which further confirmed the feasibility and e?ectiveness of MCMC method.