Degree-based Moment Estimation For Ordered Networks

Network data are common in a wide variety of fields; examples include social networks, collaboration networks and biological information networks and so on.However, most of literatures focus on the analysis of binary edges ?e.g., appear or disappear?. The network edges sometimes take weighted, even ordered values?e.g.,the measurement of the relationship between people from worst to best in social networks?. At present, there still lacks of a rigorous analysis of ordered networks whose edges are ordered random variables. In this paper, we will study the asymp-totic property of the moment estimator based on the degrees of vertices for ordered networks. In particular, Yan et al. ?2016b? established a unified asymptotic theory of the moment estimator in the degree-based moment equations for a class of undi-rected random graph models parameterized by the strength of vertices when the number of parameters goes to infinity. Built on their results, we derive the uniform consistency and asymptotic normality of the moment estimator based on the degrees of ordered networks.Main results of this thesis are given as follows:Theorem 1 If e5?r-1?Qn=o????n/log n?= then as n ??, with probability approaching one, the moment estimator ? exists and satisfies||???-?||? =Op(e3?r-1Qn???log n/n?=op?1?,where ||x||?=max1?i?n|xi| and Qn = maxi?j??i+?j?.Theorem 2 If e?9?r-1?Qn? =o?n1/2/log n?,then for any fixed k?1,as n ??, the vector consisting of the first k elements of ?B-1?1/2????-?? is asymptotically standard multivariate normal, where ?B-1?1/2= diag(v₁₁^1/2,...,v_nn^1/2) and vii =?j?ivij.In the real data examples, the parameters of the degree sequence are estimated in the TRAIN dataset constructed by Jose A. Rodriguez and DHHS Collaboration Network dataset. The results show that the degrees of the vertices qualitatively agrees with the corresponding parameters estimated, and the accuracy of the results is further verified to illustrate asymptotic results.