Research On Some Network Statistical Problems Combined With Node Information

Statistical analysis of network data is one of the most active and frontier fields in recent decades.There are a large number of complex systems around us that can be represented by network graph.Such as social networks,metabolic networks and protein-protein interaction network etc.We can profile attribute features in the network structure by modeling the network data,for example the heterogeneity of degrees and homogeneity of covariates.This helps us to understand,describe,and predict the complex systems.Therefore,statistical models are important tool for studying the generation mechanism of network structure.Based on the existing theoretical research,combined with the node information in different networks to construct a model.This paper studies the problems of the consistency and asymptotical normality distribution of parameter estimators under different conditions in different network models.The thesis consists of three parts,details are as follows:In the first part,we consider the statistical inference of the two-mode network model under the difference privacy condition.We give a relaxed edge differential privacy mechanism.We add nonnegative Laplace noise to the degree sequence of the network when released it to satisfy our mechanism.We established the consistency and asymptotic normality of the moment estimators under our mechanism.We propose an efficient algorithm to obtain the optimal solutions under the L1 error.Numerical studies and the real data of UC irvine forum network data analysis demonstrate out theoretical findings.In the second part,we consider the statistical inference problem of a generalβ-model with covariates.We introduce the general model for modelling the degree heterogeneity and homphily by using an n-dimensional node parameter β and a pdimensional homophily parameter γ.We obtained the consistency and asymptotical normality of the moment estimators under relaxed conditions.Two specific model studies demonstrate our theoretical findings.In the third part,we concerned the asymptotical properties of the maximum likelihood estimation in a sparse p0 model with covariates.We proved the consistency and asymptotical normality of the maximum likelihood estimations when the number of node goes to infinity.Numerical studies and the lawyer friendship network data analysis demonstrate our theoretical findings.