The modulus and epidemic processes on graphs

Abstract

This thesis contains three chapters split into two parts. In the first chapter, the discrete p-modulus of families of walks is introduced and discussed from various perspectives. Initially, we prove many properties by mimicking the theory from the continuous case and use Arne Beurling's criterion for extremality to build insight and intuition regarding the modulus. After building an intuitive understanding of the p-modulus, we proceed to switch perspectives to that of convex analysis. From here, uniqueness and existence of extremal densities is shown and a better understanding of Beurling's criterion is developed before describing an algorithm that approximates the value of the p-modulus arbitrarily well. In the second chapter, an exclusively edge-based approach to the discrete transboundary modulus is described. Then an interesting application is discussed with some preliminary numerical results. The final chapter describes four different takes of the Susceptible-Infected (SI) epidemic model on graphs and shows them to be equivalent. After developing a deep understanding of the SI model, the epidemic hitting time is compared to a variety of different graph centralities to indicate successful alternative methods in identifying important agents in epidemic spreading. Numerical results from simulations on many real-world graphs are presented. They indicate the effective resistance, which coincides with the 2-modulus for connecting families, is the most closely correlated indicator of importance to that of the epidemic hitting time. In large part, this is suspected to be due to the global nature of both the effective resistance and the epidemic hitting time. Thanks to the equivalence between the epidemic hitting time and the expected distance on an randomly exponentially weighted graph, we uncover a deeper connection- the effective resistance is also a lower bound for the epidemic hitting time, showing an even deeper connection.