Estimation Of Network Connections And Clustering Statistics Researc

With the development of data collection ability,network data has attracted more and more attention of statisticians.Network data mainly focuses on the correlation among data.We define data as nodes and the correlation between data as edges or links.We explore the properties of each node with links.Due to the important role of links in network data,network link models have been a hot issue in network data.Large amount of data and complex relationship are the important characteristics of network data.Each node has homogeneity and heterogeneity at the same time with these two characteristics.Homogeneity indicates that the properties of nodes are the same,while heterogeneity indicates that each node has its own independent properties.Characterizing the homogeneity and heterogeneity of each node is the key to the research of network model.In recent years,another characteristic of network data is diversification.Two-mode network,ego network and other special networks have received more attention.In this doctoral dissertation,we model the network connection of one-mode network and two-mode network,taking into account the heterogeneity and homogeneity of nodes.As a whole,this paper is divided into two parts.In the first part,we mainly use mixture model and autoregressive model to study the mixture network composed of one-mode network and two-mode network.In the second part,we study one-mode network based on heterogeneous fixed effect model,and add nonparametric covariates and homogeneous structure to the model.In the first chapter,we mainly introduce the research background and significance of the topic,literature review and the framework of the article.In the second chapter,we mainly introduce the basic tools in this paper,including some basic definitions of network data,mixture model,homogeneity pursuit and model inference.In the third chapter,we study the GRMM,which studies mixture network composed of two-mode network and single-mode network.The basic idea of the model is that the links within the actor set in the dual-mode network will affect the connection between the two actor sets.We use the autoregressive model to express the influence of the one-mode network links,and combine with Rasch model to add the influence to the link generation of two-mode network.At the same time,we use Rasch model to describe the homogeneity of nodes,and use autoregressive term to express the heterogeneity of nodes.GRMM combines the homogeneity and heterogeneity of nodes.We propose a modified EM algorithm to estimate the model,and make a series of inferences based on the model,including model selection,autoregressive parameter test and link prediction.Finally,theoretical proof,numerical simulation and actual data analysis prove the effectiveness of our model.The fourth chapter is to pursuit the homogeneity based on the traditional heterogeneous fixed effect model.Heterogeneous fixed effect model can well describe the properties of each node,but it often ignores the similarity between nodes.Using homogeneity pursuit method,we can study the internal similarity from the data.We prove the improvement of the convergence of parameter estimation by considering the homogeneous structure in theory,and prove our theoretical results by using a numerical model.Finally,this method is applied to the New England lawyer dataset and Lastfm dataset.The fifth chapter adds the influence of covariates to the fourth chapter model.The covariates of nodes play an important role in the analysis of links.In Chapter 5,we consider the nonparametric effect of covariates on the probability of link generation.We also show the advantages of the model through theoretical proof,numerical simulation and actual data analysis.The sixed chapter considers dynamic network.We establish a dynamic network model with homogeneous structure.And we use a three-step method to detect the homogeneous structure.We prove the CLT of the each parameters in this model.Finally,in the seventh chapter,we summarize this paper and discuss some potential research directions.