Diffusion processes on networks

Diffusion processes on network systems are ubiquitous. An epidemic outbreak in an interconnected population, the adoption of a new smartphone application by the members of a social community, and the spread of a genetically modified organism in a geographic area are just three examples of real-world phenomena that can be represented through the formalism of diffusion dynamics on networks. The main goal of this dissertation is to study the evolution of these diffusion processes and predict their outcome. Specifically, we aim to investigate whether the diffusion process reaches large part of the network and how long does the spreading process lasts, unveiling the influence of i) the topological structure of the networked system, and ii) the characteristics of the diffusive dynamics. In order to address this issues, we formulate a general and flexible theory for diffusion processes which can be tailored to suit the specific features of many real-world phenomena. The main strength of our theory is that the mathematical models designed in accordance to it satisfy Markov property, enabling us to analytically treat them. The main contribution of this dissertation, besides proposing a mathematical model for general diffusion processes on networks, is thus the development of a set of analytical techniques, which are used to gain new insights into the the effect of the network topology on the evolution of diffusive real-world phenomena and to develop effective control techniques for the system. After having introduced our general formulation and having developed the techniques mentioned above, we focus on the analysis of three relevant applications of our general theory, which exemplify different physical and social phenomena and include various distinct features into our framework. Specifically, we consider an epidemiological, a marketing, and a biological application. First, we deepen the analysis of a well established model for the diffusion of an infectious disease: the susceptible-infected-susceptible model. The techniques developed in our general theory enable us i) to gain new insights into the epidemic process, enhancing the characterization of its phase transition; and ii) to obtain short- and medium-term predictions of the evolution of the outbreak, from few empirical data. Second, we tailor our general theory to propose a novel model for the adoption of a new technological asset. The main modeling novelty is the presence of positive externalities, i.e., the indirect effect of the adoption of the asset by a user, which boosts its diffusion. Using the techniques developed in our theory, we are able to study the system, showing a complex bi-stability phenomenon: besides the success and failure regimes - which are similar to the endemic and fast extinction regimes in epidemic models, respectively - we witness the presence of an intermediate regime, where the outcome of the system depends on the initial condition. Third, we model evolutionary dynamics using our framework, enabling us to extend the theoretical analysis of these models and to study the effect of the introduction of an external control. We apply evolutionary dynamics to model the insertion of a mutant species in a geographic area to substitute the native one, e.g., the species of mosquitoes that transmit Zika virus. Our general analysis enables us to understand the effect of the topology and of the control policy adopted on the time required to spread the mutants and on the effort needed to achieve this goal, and allows also for the definition of an effective feedback control policy to speed up the diffusion process.