An Exponential Family of Probability Distributions for Dynamic Directed Graphs and Some Asymptotic Properties of the Maximum Likelihood Estimators in These Models

• Activity Number: 355
• Type: Contributed
• Date/Time: Tuesday, July 31, 2012 : 10:30 AM to 12:20 PM
• Sponsor: Section on Statistical Graphics
• Author(s): Subhomoy Ghosh*+ and Arka P. Ghosh
• Companies: Iowa State University and Iowa State University
• Keywords: p_{1} model ; preferential attachment model ; directed random graph ; consistency

Abstract

After Erdos and Renyi, analyzing sequence of finite random graphs through preferential attachment process has become a major subject of study till date. On the other hand, after Holland and Leinhardt's (1981) p_{1} model for the analysis of binary directed graph data in network studies, various types of descriptive models have been developed based on "static" graphs, consisting of fixed set of nodes.This paper introduces a dynamic version of Exponential Random Graph model, that immitates Preferential Attachment Model in terms of the generative nature. A parametric random graph model for this dynamic version of ERGM (denoted by p_{1}^{\infty}) is proposed. This model includes existing p_{1}model as special case. A substantive rationale for the proposed model is described. Some statistical asymptotic properties of the Maximum Likelihood Estimator in this evolving p_{1}^{\infty} model are studied. An iterative scaling algorithm is given for fitting the model parameters by maximum likelihood at each layer. The process is illustrated through an empirical example.