3-参数指数随机图的生成与参数估计

发布时间：2018-07-10 05:10

本文选题：3-参数指数随机图模型 + Logistic回归　；参考：《吉林大学》2017年硕士论文

【摘要】：本文主要研究了3-参数指数随机图模型的生成及其相应参数的估计.首先介绍了随机图的定义、符号标记和一些经典的随机图模型,但是随着现实网络的复杂性,这些经典随机图模型不足以刻画现实社会网络的特性,继而引出指数随机图模型的一个特殊子集:Markov随机图模型(即满足Markov依赖假设的模型).但是通过实例探究,Markov随机图模型存在着退化问题(分布比较集中于小子集),而且网络越大、密度越高,这种退化现象越严重,直到Snijders等人提出了关于指数随机图模型的新的描述,成功替代之前的Markov随机图模型.近些年,通过学者们对指数随机图模型的探究,介绍了指数随机图模型的定义和性质,将网络中参与者之间的关系看作随机变量,而随机关系变量间的相关性这一假设决定了指数随机图模型的一般形式,紧接着又给出相关参数的解释.本文着重研究的是3-参数指数随机图模型,给出了模型的概率分布,以及三参数的具体表达形式和参数意义.本文随后介绍了Hastings-Metropolis算法(H-M算法)和Gibbs抽样,用Gibbs抽样模拟图分布生成3-参数指数随机图模型的邻接阵,而多次模拟的对比结果表明:图的规模不应过大,要适当选择.本文由于模型中正则化常数ψ的复杂性、未知性,使得常规的极大似然估计方法失效,因此引入伪极大似然估计的方法.紧接着运用Logistic回归将非线性回归模型转化为线性回归模型,选取合适的参数初始值,利用伪极大似然估计方法将指数随机图模型的概率分布转变为可以求解的线性表达式进行求解.最后结合Newton-Raphon迭代法估计相应参数,进行数值模拟.
[Abstract]:In this paper, we study the generation of 3-parameter exponential random graph model and the estimation of its corresponding parameters. This paper first introduces the definition of random graph, symbol marking and some classical random graph models, but with the complexity of real network, these classical random graph models are not sufficient to describe the characteristics of real social network. Then a special subset of the exponential random graph model: Markov random graph model (that is, the model satisfying the Markov dependence hypothesis) is introduced. But there is a degradation problem in Markov stochastic graph model (distribution is concentrated on small set), and the larger the network, the higher the density, and the more serious the degradation phenomenon is. Until Snijders et al proposed a new description of exponential random graph model, the Markov random graph model was successfully replaced. In recent years, through the research of exponential random graph model, the definition and properties of exponential random graph model are introduced, and the relationship between the participants in the network is regarded as a random variable. The hypothesis of the correlation between random relation variables determines the general form of exponential random graph model, and then gives the explanation of related parameters. This paper focuses on the 3-parameter exponential random graph model, and gives the probability distribution of the model, as well as the concrete expression form and parameter meaning of the three-parameter. Then the Hastings-Metropolis algorithm (H-M algorithm) and Gibbs sampling are introduced. The adjacent matrix of 3-parameter exponential random graph model is generated by Gibbs sampling simulation graph distribution. The comparison results of several simulations show that the scale of graph should not be too large and should be properly selected. In this paper, due to the complexity and uncertainty of regularization constant 蠄 in the model, the conventional maximum likelihood estimation method is invalidated, so the pseudo maximum likelihood estimation method is introduced. Then the nonlinear regression model is transformed into the linear regression model by Logistic regression, and the appropriate initial parameters are selected. The probabilistic distribution of exponential random graph model is transformed into a linear expression which can be solved by using pseudo maximum likelihood estimation method. Finally, the Newton-Raphon iterative method is used to estimate the corresponding parameters.
【学位授予单位】：吉林大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：F224

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