Title:

Longrange interactions in complex networks

An interaction in a complex network is any kind of information or process that can propagate between network units or components along network links. Complex networks, which represent the structural skeleton of our societal, technological and infrastructural systems, play a major role in the propagation of processes. These processes include for example the case of epidemic spreading, the diffusion process, synchronisation, the consensus process and many others. It is usually assumed that interactions in networks propagate only from a node to its nearest neighbours. This thesis is about interactions that can be transmitted from a node to others that are not directly connected to it. These types of interactions are here called longrange interactions (LRI). The thesis is about those longrange interactions in complex networks. We will focus on the case of infection or epidemic spreading in complex networks. An "infection", understood here in a very broad sense, can be propagated through the network of social contacts among individuals. These social contacts include both "close" contacts and "casual" encounters among individuals in transport, leisure, shopping, etc. Knowing the first through the study of the social networks is not a difficult task, but having a clear picture of the network of casual contacts is a very hard problem in a society of increasing mobility. Here we assume, on the basis of several pieces of empirical evidence, that the casual contacts between two individuals are a function of their social distance in the network of close contacts. Then, we assume that we know the network of close contacts and infer the casual encounters by means of nonrandom longrange (LR) interactions determined by the social proximity of the two individuals. This approach is then implemented in a susceptibleinfectedsusceptible (SIS) model accounting for the spread of infections in complex networ ks. A parameter called "conductance" controls the feasibility of those casual encounters. In a zero conductance network only contagion through close contacts is allowed. As the conductance increases the probability of having casual encounters also increases. We show here that as the conductance parameter increases, the rate of propagation increases dramatically and the infection is less likely to die out. This increment is particularly marked in networks with scalefree degree distributions, where infections easily become epidemics. We show that the epidemic threshold of the model is given by the inverse of the largest eigenvalue of the generalised graph matrix that represents all the social contacts in the network. We point out that, from a Statistical Mechanical point of view, the epidemic threshold is also seen as the negative of the inverse of the free energy of the network when the system is frozen at extremely low temperatures. The proposed model is able to reproduce the ageassortativity or homophily observed in many social networks. Our model provides a general framework for studying epidemic spreading in networks with arbitrary topology with and without casual contacts accounted for by means of LR interactions.
