Multi-Mode and Evolutionary Networks

Abstract

In this dissertation, I present advanced mathematical methods underpinning networks, graphs and matrices. I develop a methodology to manipulate multi-mode high-dimensional networks and operate a mechanism for storing and performing matrix arithmetics on such networks and graphs. Additionally, I introduce the concept of having infinite networks and matrices and expand the literature involving traditional networks and matrices. Furthermore, I build up a model to estimate missing edges and vertices in a graph using covariate information and similarities among actors. The covariates are the exogenous attributes of entities, which could be numerical as well as categorical attributes. The model can be applied to social networks in addition to other networks. I then utilize the mathematical model to estimate missing vertices in a graph, a process that can be achieved through matrix transformation. In the next stage, I present a method to predict the emergence of new actors in a network based on stochastic processes and suggest a model of preferential attachment. Finally, I apply quantitative methods to examine evolving networks. Ultimately, I examine the structure of real networks and model their behavior. I perform a comprehensive analysis and simulation on applications in the social networks field, which includes coauthorship social networks (social networks of coauthors of scholarly publications), road fatal crashes networks in the United States, and news documents networks.