Invited review: Epidemics on social networks

Since its first formulations almost a century ago, mathematical models for disease spreading contributed to understand, evaluate and control the epidemic processes.They promoted a dramatic change in how epidemiologists thought of the propagation of infectious diseases.In the last decade, when the traditional epidemiological models seemed to be exhausted, new types of models were developed.These new models incorporated concepts from graph theory to describe and model the underlying social structure.Many of these works merely produced a more detailed extension of the previous results, but some others triggered a completely new paradigm in the mathematical study of epidemic processes. In this review, we will introduce the basic concepts of epidemiology, epidemic modeling and networks, to finally provide a brief description of the most relevant results in the field.Comment: 17 pages, 13 figure

Stochastic resonance in a model of opinion formation on small-world networks

We analyze the phenomenon of stochastic resonance in an Ising-like system on a small-world network. The system, which is subject to the combined action of noise and an external modulation, can be interpreted as a stylized model of opinion formation by imitation under the effects of a ``fashion wave''. Both the amplitude threshold for the detection of the external modulation and the width of the stochastic-resonance peak show considerable variation as the randomness of the underlying small-world network is changed.Comment: 5 pages, 5 figures include

Complex structures in generalized small worlds

We propose a generalization of small world networks, in which the reconnection of links is governed by a function that depends on the distance between the elements to be linked. An adequate choice of this function lets us control the clusterization of the system. Control of the clusterization, in turn, allows the generation of a wide variety of topologies.Comment: 4 pages, 6 figures, RevTe

Game theory in models of pedestrian room evacuation

We analyze the pedestrian evacuation of a rectangular room with a single door considering a Lattice Gas scheme with the addition of behavioral aspects of the pedestrians. The movement of the individuals is based on random and rational choices and is affected by conflicts between two or more agents that want to advance to the same position. Such conflicts are solved according to certain rules closely related to the concept of strategies in Game Theory, cooperation and defection. We consider game rules analogous to those from the Prisoner's Dilemma and Stag Hunt games, with payoffs associated to the probabilities of the individuals to advance to the selected site. We find that, even when defecting is the rational choice for any agent, under certain conditions, cooperators can take advantage from mutual cooperation and leave the room more rapidly than defectors

Statistical fluctuations in pedestrian evacuation times and the effect of social contagion

Mathematical models of pedestrian evacuation and the associated simulation software have become essential tools for the assessment of the safety of public facilities and buildings. While a variety of models are now available, their calibration and test against empirical data are generally restricted to global, averaged quantities, the statistics compiled from the time series of individual escapes (" microscopic " statistics) measured in recent experiments are thus overlooked. In the same spirit, much research has primarily focused on the average global evacuation time, whereas the whole distribution of evacuation times over some set of realizations should matter. In the present paper we propose and discuss the validity of a simple relation between this distribution and the " microscopic " statistics, which is theoretically valid in the absence of correlations. To this purpose, we develop a minimal cellular automaton, with novel features that afford a semi-quantitative reproduction of the experimental " microscopic " statistics. We then introduce a process of social contagion of impatient behavior in the model and show that the simple relation under test may dramatically fail at high contagion strengths, the latter being responsible for the emergence of strong correlations in the system. We conclude with comments on the potential practical relevance for safety science of calculations based on " microscopic " statistics

Associative memory on a small-world neural network

We study a model of associative memory based on a neural network with small-world structure. The efficacy of the network to retrieve one of the stored patterns exhibits a phase transition at a finite value of the disorder. The more ordered networks are unable to recover the patterns, and are always attracted to mixture states. Besides, for a range of the number of stored patterns, the efficacy has a maximum at an intermediate value of the disorder. We also give a statistical characterization of the attractors for all values of the disorder of the network.Comment: 5 pages, 4 figures (eps