Abstract:
We present a unified mean-field theory, based on the single site approximation to the master-equation, for stochastic self-organized critical models. In particular, we analyze in detail the properties of sandpile and forest-fire (FF) models. In analogy with other non-equilibrium critical phenomena, we identify the order parameter with the density of ``active'' sites and the control parameters with the driving rates. Depending on the values of the control parameters, the system is shown to reach a subcritical (absorbing) or super-critical (active) stationary state. Criticality is analyzed in terms of the singularities of the zero-field susceptibility. In the limit of vanishing control parameters, the stationary state displays scaling characteristic of self-organized criticality (SOC). We show that this limit corresponds to the breakdown of space-time locality in the dynamical rules of the models. We define a complete set of critical exponents, describing the scaling of order parameter, response functions, susceptibility and correlation length in the subcritical and supercritical states. In the subcritical state, the response of the system to small perturbations takes place in avalanches. We analyze their scaling behavior in relation with branching processes. In sandpile models because of conservation laws, a critical exponents subset displays mean-field values ($\nu=1/2$ and $\gamma = 1$) in any dimensions. We treat bulk and boundary dissipation and introduce a new critical exponent relating dissipation and finite size effects. We present numerical simulations that confirm our results. In the case of the forest-fire model, our approach can distinguish between different regimes (SOC-FF and deterministic FF) studied in the literature and determine the full spectrum of critical exponents.

Abstract:
We present a unified dynamical mean-field theory for stochastic self-organized critical models. We use a single site approximation and we include the details of different models by using effective parameters and constraints. We identify the order parameter and the relevant scaling fields in order to describe the critical behavior in terms of usual concepts of non equilibrium lattice models with steady-states. We point out the inconsistencies of previous mean-field approaches, which lead to different predictions. Numerical simulations confirm the validity of our results beyond mean-field theory.

Abstract:
Measurements and data analysis have proved very effective in the study of the Internet's physical fabric and have shown heterogeneities and statistical fluctuations extending over several orders of magnitude. Here we analyze performance measurements obtained by the PingER monitoring infrastructure. We focus on the relationship between the Round-Trip-Time (RTT) and the geographical distance. We define dimensionless variables that contain information on the quality of Internet connections finding that their probability distributions are characterized by a slow power-law decay signalling the presence of scale-free features. These results point out the extreme heterogeneity of the Internet since the transmission speed between different points of the network exhibits very large fluctuations. The associated scaling exponents appear to have fairly stable values in different data sets and thus define an invariant characteristic of the Internet that might be used in the future as a benchmark of the overall state of ``health'' of the Internet. The observed scale-free character should be incorporated in models and analysis of Internet performance.

Abstract:
The spatial structure of populations is a key element in the understanding of the large scale spreading of epidemics. Motivated by the recent empirical evidence on the heterogeneous properties of transportation and commuting patterns among urban areas, we present a thorough analysis of the behavior of infectious diseases in metapopulation models characterized by heterogeneous connectivity and mobility patterns. We derive the basic reaction-diffusion equation describing the metapopulation system at the mechanistic level and derive an early stage dynamics approximation for the subpopulation invasion dynamics. The analytical description uses degree block variables that allows us to take into account arbitrary degree distribution of the metapopulation network. We show that along with the usual single population epidemic threshold the metapopulation network exhibits a global threshold for the subpopulation invasion. We find an explicit analytic expression for the invasion threshold that determines the minimum number of individuals traveling among subpopulations in order to have the infection of a macroscopic number of subpopulations. The invasion threshold is a function of factors such as the basic reproductive number, the infectious period and the mobility process and it is found to decrease for increasing network heterogeneity. We provide extensive mechanistic numerical Monte Carlo simulations that recover the analytical finding in a wide range of metapopulation network connectivity patterns. The results can be useful in the understanding of recent data driven computational approaches to disease spreading in large transportation networks and the effect of containment measures such as travel restrictions.

Abstract:
We study the dynamics of epidemic and reaction-diffusion processes in metapopulation models with heterogeneous connectivity pattern. In SIR-like processes, along with the standard local epidemic threshold, the system exhibits a global invasion threshold. We provide an explicit expression of the threshold that sets a critical value of the diffusion/mobility rate below which the epidemic is not able to spread to a macroscopic fraction of subpopulations. The invasion threshold is found to be affected by the topological fluctuations of the metapopulation network. The presented results provide a general framework for the understanding of the effect of travel restrictions in epidemic containment.

Abstract:
Human mobility and activity patterns mediate contagion on many levels, including the spatial spread of infectious diseases, diffusion of rumors, and emergence of consensus. These patterns however are often dominated by specific locations and recurrent flows and poorly modeled by the random diffusive dynamics generally used to study them. Here we develop a theoretical framework to analyze contagion within a network of locations where individuals recall their geographic origins. We find a phase transition between a regime in which the contagion affects a large fraction of the system and one in which only a small fraction is affected. This transition cannot be uncovered by continuous deterministic models due to the stochastic features of the contagion process and defines an invasion threshold that depends on mobility parameters, providing guidance for controlling contagion spread by constraining mobility processes. We recover the threshold behavior by analyzing diffusion processes mediated by real human commuting data.

Abstract:
We study two driven dynamical systems with conserved energy. The two automata contain the basic dynamical rules of the Bak, Tang and Wiesenfeld sandpile model. In addition a global constraint on the energy contained in the lattice is imposed. In the limit of an infinitely slow driving of the system, the conserved energy $E$ becomes the only parameter governing the dynamical behavior of the system. Both models show scale free behavior at a critical value $E_c$ of the fixed energy. The scaling with respect to the relevant scaling field points out that the developing of critical correlations is in a different universality class than self-organized critical sandpiles. Despite this difference, the activity (avalanche) probability distributions appear to coincide with the one of the standard self-organized critical sandpile.

Abstract:
We present large scale simulations of a stochastic sandpile model in two dimensions. We use moments analysis to evaluate critical exponents and finite size scaling method to consistently test the obtained results. The general picture resulting from our analysis allows us to characterize the large scale behavior of the present model with great accuracy.

Abstract:
Microblogging and mobile devices appear to augment human social capabilities, which raises the question whether they remove cognitive or biological constraints on human communication. In this paper we analyze a dataset of Twitter conversations collected across six months involving 1.7 million individuals and test the theoretical cognitive limit on the number of stable social relationships known as Dunbar's number. We find that the data are in agreement with Dunbar's result; users can entertain a maximum of 100–200 stable relationships. Thus, the ‘economy of attention’ is limited in the online world by cognitive and biological constraints as predicted by Dunbar's theory. We propose a simple model for users' behavior that includes finite priority queuing and time resources that reproduces the observed social behavior.

Abstract:
We present a detailed discussion of a novel dynamical renormalization group scheme: the Dynamically Driven Renormalization Group (DDRG). This is a general renormalization method developed for dynamical systems with non-equilibrium critical steady-state. The method is based on a real space renormalization scheme driven by a dynamical steady-state condition which acts as a feedback on the transformation equations. This approach has been applied to open non-linear systems such as self-organized critical phenomena, and it allows the analytical evaluation of scaling dimensions and critical exponents. Equilibrium models at the critical point can also be considered. The explicit application to some models and the corresponding results are discussed.