【作者】 赵晨；

【导师】 晏挺；

【作者基本信息】 华中师范大学 ， 应用统计学， 2021， 硕士

【摘要】 隶属网络普遍存在于网络数据中。但是,目前对隶属网络形成机制的统计研究却相对较少。隶属网络是由两组不同的节点组成的双模网络:其中一组节点用“演员”表示,另一组节点用“事件”表示。它们之间的边表示“演员”与“事件”的隶属关系。在本文中,我们利用概率元(Probit)分布对隶属网络顶点度的异质性进行建模分析,研究了概率元隶属网络模型中的统计推断问题,主要内容如下:第一、在隶属网络g(m,n)中,我们假设第i个演员有一个参数αi,其中n个演员用{1,…,n}表示;第j个事件有一个参数βj,其中m个事件表示为{1,...,m}。我们通过概率元分布对演员i和事件j所关联的边进行了建模,如果演员i和事件j有关联,那么xi,j=1;如果演员i和事件j没有关联,那么xi,j=0。我们假设所有边都是相互独立分布的伯努利随机变量,且概率分布为:P(xi.j=1)=Φ(αi+βj),其中Φ(·)是一个标准正态随机变量的累积分布函数。第二、我们证明了矩估计量的一致相合性,具体表述为:在一定条件下,如果真实参数θ*∈Rm+n-1满足‖θ*‖∞≤(?),其中τ是一个常数,并且m/n=O(1),那么当n趋于无穷时,依概率趋于1,(?)的矩估计量存在且满足第三、我们证明了矩估计量的渐近正态性,具体表述为:在一定条件下,对任意固定的k≥1,且m/n=O(1),随着演员个数n →∞,向量θ-θ*的前k个元素渐近服从多元标准正态分布。通过数值模拟和一个真实网络数据分析验证了我们理论的合理性。更多还原

【Abstract】 Affiliation network is a common form of network model.Therefore,it is very important to study the formation mechanism of affiliation network.Affiliation network is one kind of two-mode social network with two different sets of nodes(namely,a set of actors and a set of social events)and edges representing the affiliation of the actors with the social events.In this paper,we use the Probit model to model and analyze the degree heterogeneity of the affiliation networks,and study the statistical inference problem in the Probit network model.The main results are as follows:Firstly,in this subsection,we model the directed edge that actor i is affiliated with event j through the probit distribution.In the affiliation network G(m,n),we assume that all edges are mutually independently distributed as Bernoulli random variables with success probability:P(xi,j=1)=Φ(αi+βi),where Φ is the cumulative distribution function of a standard normal random variable.Secondly,under the condition of the number of nodes n→∞,we obtained the uniform consistency of the moment estimator and proved that,specifically expressed as Assume that θ*∈Rm+n-1 with ‖θ*‖∞≤(?),where τ is a constant.If m/n=O(1),then as n goes to infinity,with probability approaching one,the moment estimator θ exists and satisfiesThirdly,under the condition of the number of nodes n→∞,we obtained the asymptotic normality of the moment estimator and proved that,specifically expressed as:Assume that A～Pθ*.If ‖θ*‖∞≤(?),where τ is a constant.For any fixed k≥1,as n→∞,the vector consisting of the first k elements of θ-θ*is asymptotically multivariate normal.Numerical simulation and real network data are used to verify the theoretical results.更多还原