Reaction-Diffusion Processes on Complex Networks

The availability of an unprecedented amount of large-scale data sets has allowed to discover complex features in many real-world networks. This boosted the attention on complex networks in di�erent disciplines: Mathematics, Medicine, Biology, Social Sciences, Computer Sciences and Physics. Recently, a lot of attention has been devoted to the study of dynamical processes occurring on complex networks. In this work we focused on the general framework of Reaction-Di�usion models on complex topologies. Using this approach we discussed two di�erent problems. The �rst one is based on the evaluation of the importance or centrality of nodes. In heterogeneous networks not all the nodes are the same. To sort out the di�erences is a relevant problem in data retrieval, biology and in general infrastructure management. The relative importance of units is not just a local feature. The centrality of a node is in fact related to the importance of the nodes that are connected to it and so on. Therefore we have a di�usion process, the di�usion of importance, which is encoded in the spectral properties of several kinds of matrices. Spectral centrality measures are accordingly de�ned and new results and the interpretations of these measures on directed, undirected and real networks are presented. The second problem discussed within the same framework is the epidemic spreading in homogeneous and heterogeneous networks. This is an extremely relevant problem for our society as demonstrated by the last H1N1 pandemic in 2009. Complex networks analysis is crucial to get enough insight on the epidemic processes to suggest efficient interventions policies and make forecasts. We introduced the general theory of epidemic spreading on networks. We presented new single population models in order to deal with the e�ects of social disruption due to the epidemic di�usion itself. We discussed the framework of metapopulation network models in which each population is considered as a node of a network. The populations are coupled by di�usion of individuals. Markovian di�usion is �rst considered and all the known results are reproduced and derived. A more realistic protocol considering origindestination matrices is introduced and analytically solved. In the last chapter we show how these models can be used in order to build a realist data driven model, GLEaM, which is a powerful tool to make global epidemic forecasts. The use of this model during the recent H1N1 pandemic is described and all the new methods and results obtained are discussed in detail.