Structural and Dynamical Properties of Complex Networks.

Abstract

Recent years have witnessed a substantial amount of interest within the physics community in the properties of networks. Techniques from statistical physics coupled with the widespread availability of computing resources have facilitated studies ranging from large scale empirical analysis of the worldwide web, social networks, biological systems, to the development of theoretical models and tools to explore the various properties of these systems.

Following these developments, in this dissertation, we present and solve for a diverse set of new problems, investigating the structural and dynamical properties of both model and real world networks.We start by defining a new metric to measure the stability of network structure to disruptions, and then using a combination of theory and simulation study its properties in detail on artificially generated networks; we then compare our results to a selection of networks from the real world and find good agreement in most cases. In the following chapter, we propose a mathematical model that mimics the structure of popular file-sharing websites such as Flickr and CiteULike and demonstrate that many of its properties can solved exactly in the limit of large network size. The remaining part of the dissertation primarily focuses on the dynamical properties of networks. We first formulate a model of a network that evolves under the addition and deletion of vertices and edges, and solve for the equilibrium degree distribution for a variety of cases of interest. We then consider networks whose structure can be manipulated by adjusting the rules by which vertices enter and leave the network. We focus in particular on degree distributions and show that, with some mild constraints, it is possible by a suitable choice of rules to arrange for the network to have any degree distribution we desire. In addition we define a simple local algorithm by which appropriate rules can be implemented in practice. Finally, we conclude our dissertation with a game theory model on social networks that tracks the dynamical evolution of a group of interacting agents such as diplomats or political lobbyists seeking to rise to a position of influence, by balancing competing interests.