Generation And Parameter Estimation Of 3-parameter Exponential Random Graphs

In this paper,we mainly study the generation of 3-parameter exponential random graph model and estimate the corresponding parameters.Firstly,we introduce the definition and the symbol mark of random graph and some classical random graph models,but with the complexity of the real network,these classical random graph models are insufficient to describe the characteristics of real social networks,then a special subset of exponential random graph model is derived: Markov random graph model(a model that satisfies the Markov dependency hypothesis).However,there are some degenerate problems in the Markov random graph model(the distribution is concentrated on the small subset),and the larger the network,the higher the density,the more serious the degradation,until Snijders proposed a new description of the exponential random graph model,successfully replaced the previous Markov random graph model.In recent years,through the study of exponential random graph model,this paper introduces the definition and properties of exponential random graph model,the relationships among participants in the network are considered as random variables,the assumption of the correlation between random variables determines the general form of exponential random graph model,followed by the interpretation of the relevant parameters.In this paper,we mainly focus on the 3-parameter exponential random graph model,and give the probability distribution of the model,and the expression of the three parameters and the significance of the parameters.This paper then introduces the Hastings-Metropolis algorithm(H-M algorithm)and the Gibbs sampling,using Gibbs sampling to simulate the adjacency matrix of 3-parameter exponential random graph model,however the simulation results show that: the size of the graph should not be too large to be properly chosen.In this paper,due to the complexity and the uncertainty of the regularization constant ?,the conventional maximum likelihood estimation method is invalid,so the method of maximum pseudo likelihood estimation is introduced.Then the nonlinear regression model is transformed into linear regression model by Logistic regression,select the appropriate initial parameters,the probability distribution of the exponential random graph model is transformed into a linear expression which can be solved by using the maximum pseudo likelihood estimation method.Finally,the corresponding parameters are estimated by Newton-Raphon iterative method,numerical simulation.