Spatial Marriage Problems and Epidemics

Sammanfattning: This thesis consists of four papers covering three different topics on the modeling of large real networks and phenomena thereon. In Papers I and II, we propose and study the properties of a bipartite version of the model introduced by Deijfen, Holroyd and Häggström for generating translation-invariant spatial random graphs with prescribed degree distribution. In particular, we focus our attention on spatial random graphs generated by a matching scheme based on the Gale-Shapley stable marriage problem. In paper III, we propose a random graph model for generating edge-weighted graphs with prescribed degree and weight distributions, and tunable degree-degree correlation. We then study a simple inhomogeneous epidemic model on such graphs, where the infection probabilities are functions of the edge-weights, and investigate how the epidemic threshold is affected by the degree-degree correlation. In paper IV, we study a simple stochastic model aimed at representing a competition between two virus strains in a population. A longstanding principle in ecology known as the competitive exclusion principle predicts that when one of the strains has even the slightest advantage over the other, the one with the advantage will either drive the competitor to extinction or lead to a transformation in the ecological niche. We investigate how long it will take for the strain to drive its competitor to extinction.