Structure and dynamics of complex networks: Network epidemics and a geometric robustness measure

As new technologies continue to find their way into everyday life, the world becomes more and more connected. Airplanes and other means of transportation provide global connections in the physical world, while the omnipresence of the Internet means that information is shared around the globe, easier than ever before. But not only these man-made systems aredistinctly connected, other complex systems like the human brain, or metabolic networks are successfully being studied from the perspective of their constituting connections. The combining concept in all these examples is the structure of the problem at hand: each system consists of interacting elementary components at the lowest level, from which a network structure emerges at the global level. The study of such networked systems, their observed features and the wide range of related analysis tools is commonly referred to as NetworkScience.In this thesis, the specific problem of how diseases spread over networks is addressed. Better understanding this spreading behavior has significant practical importance, i.e. for the prediction and control of disease prevalence, and poses many interesting theoretical challenges. In the context of modeling epidemics on networks, we formulate the Universal Mean-Field Framework. This new and theoretically well-founded framework unifies and generalizes a number of existing approximate models, and brings forth new approaches to bound the approximations. Apart from the work on epidemics, some new insights are explored in the context of the connections between electrical circuits, networks and simplices (higher-dimensional triangles). These deep theoretical equivalences allow the tools and intuitions from electrical circuits and geometry to be used in the study of networks. A comprehensive introduction and discussion of the equivalent representations and their connections is given. Additionally, we derive a new formula for the volume of a hyperacute simplex and propose to use this volume as a network-robustness measure