Statistical Modeling And Estimation For Longitudinal Social Network Data

With the rapid development of social networks,a large amount of network data is generated every day,and the research on network data is becoming more and more abundant.Network data includes data reflecting the structure of the network and data of nodes in the network.Studying network data can help us better analyze the relationships between individuals,including the correlation between individuals and how they affect each other.In most cases,these relationships are not static but change over time.Therefore,research on dynamic networks,or broader,longitudinal network data,has become a popular direction for network data research in recent years.Applying network data to classic statistical models is one of the common methods of network data analysis,such as network vector autoregressive(NAR)model.From the perspective of the time series model,the(NAR)model assumes that the performance of each individual in the network at a certain moment will not only be affected by its own,but also by other individuals connected to it in the network.The model can explain the interaction between individuals through the network,but considers that each individual has the same degree of influence on other individuals.Therefore,this doctoral thesis considers statistical inference of network vector autoregressive models with individual effects.In addition to strengthening classic models,structural analysis of dynamic networks is more common in network data analysis.A network can usually be expressed in the form of a matrix where scholars explore the inherent relationships of the network through matrix decomposition.A dynamic network can be expressed in the form of a third-order tensor.Naturally,the method of analyzing dynamic networks by tensor decomposition also came into being.This doctoral dissertation also considers the probit tensor factorization model and the hybrid probit tensor factorization model that connect the latent tensor variables and the network through the probit function.Specifically,this paper is mainly divided into two parts.The first part is based on the network vector autoregressive model.Based on the existing statistical theories and methods,a network vector autoregressive model with individual effects is proposed.The parameter estimation and properties are studied.The second part is based on the newly proposed probit tensor factorization model.First,a fusion penalty is added in the loss function to detect change points.Then a hybrid probit tensor factorization model is proposed to improve the estimation effect of the original model.The first chapter mainly introduces the research background and significance of this article,literature review,research content,and the writing framework of the article.The second chapter mainly introduces the basic concepts and tools needed in the following chapters,including the definition and representation of networks,some penalty functions,and several forms of tensor decomposition.Chapter 3 mainly studies the network vector autoregressive model.Aiming at the problem of using the network structure to optimize the traditional autoregressive model,we proposes a network vector autoregression with individual effects.It is considered that the performance of nodes in the network at a certain moment is affected by several factors,namely: covariates that do not change over time,the performance of the node itself at the previous moment,and the performances of other nodes connected to this node at the previous moment.And due to the fact that the effects of nodes on the entire network are different,we think that different nodes behave differently to other nodes,that is,each individual has an individual effect.To estimate the parameters in the new model,we uses the idea of cohesion penalty.Considers that connected individuals have similar effects,cohesion penalty punishes the difference between individual effects,so as to obtain estimates of individual effects and all other parameters.We also provide the out-sample prediction,the selection method of the tuning parameters in the estimation process,and some theoretical properties.Finally,we verifies the superiority and stability of the proposed model through two simulation experiments,and demonstrates the effectiveness of the proposed method by analyzing data of well-known international trade.Chapter 4 mainly studies the probit tensor factorization model.This model was originally used to establish a multi-relationship network structure.We extend it to the field of dynamic networks.It considers that dynamic networks are built on a hidden tensor variable connected by the probit function.Through tensor factorization,we can analyze the individual features of each node that do not change with time,and the dynamic factors that change with time.Based on the data of the same up and down relationship between stocks during the financial crisis around 2008,we used fusion penalty on the hidden factor matrix in the probit tensor factorization model to find out the time point when the network structure changed.We also conducted some simulations to compare the estimated results with or without penalty.Chapter 5 still studies the probit tensor factorization model.Considering that there are observable factors in real data,such as the classic three-factor model in the stock market,which can provide three factors about market information,we propose a hybrid probit tensor factorization model.The model splits the original hidden tensor into two parts,one of which is completely unobservable as in Chapter4,and the other contains an observable factor matrix.We also propose a new EM algorithm to estimate the parameters in the model.Through comparison of simulation experiments,we find that the new model has higher accuracy.Finally,we analyzed the dynamic network data of stocks during the financial crisis and reached some new conclusions.In Chapter 6 of this article,we have summarized the methods proposed in the entire doctoral dissertation and considered some potential research directions and areas for improvement.