EPIDEMIC PROCESSES ON ACTIVITY-DRIVEN NETWORKS WITH ATTRACTIVENESS

Complexity science is today a well-established field of study, in which physicists have long been playing a significant role. Many complex systems can be represented as networks: the Internet, social networks (both in the real and in the virtual world), airplane routes, scientific papers citations, neurons, road maps, and many others. Complexity in networks notably manifests itself in the high diversity of nodes' connectivity patterns, as for what concerns both the heterogeneity and the intensity of connections - a feature common to most real networks. Physicists have been contributing a great deal to the analysis of large-scale networks, where many of the tools developed in statistical physics have found an application. The study of complex networks is often coupled with the study of dynamical processes taking place over them, such as epidemic processes, random walks, and others. In this context, the time-evolution of the onsidered network itself may play a significant role and affect the outcome of the process studied. For this reason, recent years have seen an increasing interest in the subject of time-varying networks, also called temporal networks or dynamical networks in the literature. The activity driven model represents one attempt at providing an analytical framework simple enough to find application in a variety of contexts, particularly in social systems, and capable to connect the topological structure of a network to its temporal dynamics. The activity of a node in a time-varying network is a measure of its propensity to form connections, and networks of human agents tend to share the same activity distribution: a power-law. An analytical model aimed at representing different kinds of social networks (and possibly other species of networks) could be constructed starting indeed from the activity distribution, and this principle, as the name suggests, is precisely what underlies the activity-driven model. In our thesis we propose a new version of the model where, beyond the activity distribution, we let the network be characterized by an attractiveness distribution. The attractiveness of a node is a measure of its propensity to receive interactions from others, therefore it is to some extent the reciprocal of the activity, and a natural complement to it within the model; a non-constant attractiveness is observed in many real social networks, and the analysis of this feature will provide us with some insight on how dynamical process unfold on such kind of systems. The introduction of this new property leads to an appreciable modification of the contact dynamics, which we study by considering an SIS/SIR epidemic process taking place on the network, calculating the general expression of the epidemic threshold holding for any joint distribution of activity and attractiveness in the thermodynamic limit; we study in particular the case of power-law uncorrelated distributions, and the case of identical activity-attractiveness correlation. Analytical calculations, corroborated by simulations on synthetic networks, show how a high heterogeneity in the distributions (i.e. a large variance) makes it easier for the disease to propagate throughout the network, with respect to the case of the same epidemic process considered while taking place on the original activity-driven model. These findings fit nicely in the picture we already have of epidemic processes on networks, where the fact that heterogeneity facilitates the spreading is a well known and general feature common to both static and temporal representations. In general, our results confirm the importance of the role played by time structure in the formation of connectivity patterns. To equip the reader with the necessary tools to appreciate the results presented, the first part of the thesis is devoted to an introduction on the subject of dynamical processes on complex networks, both static and time-varying. The most important measures, properties, models, and their impact on different kind of processes are described