Random networks with weights and directions, and epidemics thereon

Abstract: Networks, consisting of nodes and of edges, can be used to model numerous phenomena, e.g, web pages linking to each other or interactions between people in a population. Edges can be directed, such as a one way link from one web page to another, or undirected (bi-directional), such as physical contacts between pairs of people, which potentially could spread an infection either way between them. Edges can also have weights associated with them, in this thesis corresponding to the probability that an infection is transmitted on the edge.Empirical networks are often only partially known, in the form of ego-centric network data where only a subset of the nodes and the number of adjacent edges of each node have been observed. This situation lends itself well to analysis through the undirected or partially directed configuration model - a random network model where the number of edges of each node (the degree) is given but where the way these edges are connected is random.The four papers in this thesis are concerned with the properties of the configuration model and with the usefulness of it with respect to its ability to model the spread of epidemics on empirical networks. Paper I proves the asymptotic convergence to a given degree distribution for the partially directed configuration model. In Paper II it is shown that epidemics on some empirical and theoretically constructed networks grow exponentially, similarly to what can be seen on the corresponding configuration models. Finally, in Papers III and IV, large population analytical results for the reproduction number, the probability of a large epidemic outbreak and the final size of such an outbreak are derived assuming a configuration model network with weighted and/or partially directed edges. These results are then evaluated on several large empirical networks upon which epidemics are simulated. We find that on some of these networks the analytical expressions are compatible with the results of the simulations. This makes the model useful as a tool for analyzing such networks.