Large-scale Three-mode Network Autoregressive Model

Network data is the data born in the process of networking in human society,and with the development of Internet technology,the network structure is becoming increasingly complex.Many scholars have tried to build network models for network data,but most network models believe that nodes in the network are homogeneous,that is,there is no difference between nodes.Therefore,they propose a series of network autoregressive models,but none of them can well describe the network data with node heterogeneity.Subsequently,scholars gradually put forward the concept of network data with node heterogeneity,such as community network,grouped network and multimode network,and then the heterogeneity of nodes can be considered into the construction of the model.In recent years,many studies on network data have been based on spatial autoregressive models and network autoregressive(NAR)models.Subsequently,by considering the heterogeneity of nodes in the network,scholars have successively proposed the grouped NAR model,community NAR model,and two-mode NAR model.A two-mode network is a network that contains two types of nodes and edges exist only between nodes of different types.Two-mode network data can be described by two-mode NAR model.However,with the development of society,there are many three-mode network data,which cannot be depicted by the tow-mode NAR model,and then the specific form of the three-mode NAR model is given on the basis of the two-mode NAR model,and further analysis and research are carried out.In this paper,we introduce the basic definition of the three-mode NAR model,give the correlation theorems and proofs for the reversibility and parameter recognizability of the model,and give two estimation methods,quasimaximum likelihood estimation and conditional least squares estimation,and the asymptotic normality and coincidence of their parameters.Secondly,in the simulation stage,the parameters of the model were estimated by two estimation methods.By generating network data that conforms to the characteristics of three-mode networks,numerical simulations are carried out under different conditions of weight matrix setting,error distribution and parameter discussion.By giving a large number of simulation results,the accuracy and computational efficiency of the two estimation methods are compared,the corresponding conclusions are given through the comparison results,and finally two three-mode network datas on film scoring is introduced for the actual case analysis.