Asymptotic Theory For Several Random Graph Models

With the advent of the information age,more and more network data are col-lected and stored.It also has aroused widespread concern of scholars.The vertex degrees are one of the most important indexes to measure network characteristics.Therefore,many random graph models based on the vertex degrees are proposed.This thesis mainly studies three aspects,including the asymptotic theory of gen-eralized p0 models,the asymptotic theory of affiliation network models based on the vertex degrees,and the differential privacy statistical inference on generalized ?models.First,we study the asymptotic theory of generalized p0 models.In the gener-alized p0 models,we assume that the probability mass or density function of the edge weights ai,j(=a)from vertex i pointing to vertex j is f((?i+?j)a),where f(·)is a known probability mass or density function.In the model,?i denotes the out-degree parameter of vertex i;?j denotes the in-degree parameter of vertex j.The p0 model is a special case when f(·)is a logistic distribution.In the generalized p0 models,the number of parameters increases as the size of network grows and there-fore,asymptotic inference is nonstandard.We adopt moment estimation to infer the degree parameters based on vertex degrees.When the number of vertices goes infinity,we establish a unified theoretical framework in which the c onsistency of the moment estimator and the central limit theorem hold.We apply it to two special distributions of f(·).Numerical simulations verify the rationality of our results,and a real data application is carried out to illustrate their effects.Second,we study the asymptotic theory of affiliation network models based on vertex degrees.Affiliation networks contain two types of vertices:a set of actors{1,…,m} and a set of events {1,…,n}.Edges between vertex i and vertex j denote the affiliation relationships between actors and events.We assume that the probability mass or density function of the edge weights xi,j(=a)between vertex i and vertex j is f((?i+?j)a),where f(·)is a known proba.bility mass or density function.The adjacency matrix of the affiliation network is X=(Xi,j)m×n,and the adjacency matrix of the generalized p0 model is A=(ai,j)n×n.The differences between them are as follows:(1)X is a rectangular matrix and A is a square matrix;(2)xi,j is equal to xj,i,while ai,j and aj,i may be different.In this model,?i is the degree parameter of actor i measuring the activeness of actors,and ?j is the degree parameter of event j measuring the popularity of events.We use moment estimation to infer the degree parameters.We establish a unified theoretical framework that the consistency of the moment estimator and the central limit theorem hold as the numbers of actors and events both go to infinity.We apply it to some common distributions of f(·).Simulation studies and a real data verify our conclusions.Third,we study the differential privacy statistical inference on generalized ?models.In the generalized ? models,we assume that the edge ai,j(=a)with the probability is ea(?i+?j)[?k=0q-1 ek(?i+?j)]-1,where q? 2 is a fixed constant.The ?model is a special case of this model with q=2.In the model,the degree sequence is the sufficient statistic.Therefore.we add some discrete Laplace noises into the degree sequence to achieve the edge differential privacy.Based on the noisy degree sequence,we estimate unknown parameters via the moment equations.When the number of vertices goes to infinity,we obtain the conditions in which the consistency of the moment estimators for the differential privacy parameter and the central limit theorem hold.Numerical simulations verify the rationality of our results.Besides,we illustrate their utility via the analysis of two real data examples.