The structure and dynamics of networks

Abstract

Networks have become a general concept to model the structure of arbitrary relationships among entities. The framework of a network introduces a fundamentally new approach apart from ‘classical’ structures found in physics to model the topology of a system. In the context of networks fundamentally new topological effects can emerge and lead to a class of topologies which are termed ‘complex networks’. The concept of a network successfully models the static topology of an empirical system, an arbitrary model, and a physical system. Generally networks serve as a host for some dynamics running on it in order to fulfill a function. The question of the reciprocal relationship among a dynamical process on a network and its topology is the context of this Thesis. This context is studied in both directions. The network topology constrains or enhances the dynamics running on it, while the reciprocal interaction is of the same importance. Networks are commonly the result of an evolutionary process, e.g. protein interaction networks from biology. Within such an evolution the dynamics shapes the underlying network topology with respect to an optimal achievement of the function to perform. Answering the question what the influence on a dynamics of a particular topological property has requires the accurate control over the topological properties in question. In this Thesis the degree distribution, twopoint correlations, and clustering are the studied topological properties. These are motivated by the ubiquitous presence and importance within almost all empirical networks. An analytical framework to measure and to control such quantities of networks along with numerical algorithms to generate them is developed in a first step. Networks with the examined topological properties are then used to reveal their impact on two rather general dynamics on networks. Finally, an evolution of networks is studied to shed light on the influence the dynamics has on the network topology.