Random walk on temporal networks with lasting edges

We consider random walks on dynamical networks where edges appear and disappear during finite time intervals. The process is grounded on three independent stochastic processes determining the walker's waiting-time, the up-time and down-time of edges activation. We first propose a comprehensive analytical and numerical treatment on directed acyclic graphs. Once cycles are allowed in the network, non-Markovian trajectories may emerge, remarkably even if the walker and the evolution of the network edges are governed by memoryless Poisson processes. We then introduce a general analytical framework to characterize such non-Markovian walks and validate our findings with numerical simulations.Comment: 18 pages, 18 figure

A framework for epidemic spreading in multiplex networks of metapopulations

We propose a theoretical framework for the study of epidemics in structured metapopulations, with heterogeneous agents, subjected to recurrent mobility patterns. We propose to represent the heterogeneity in the composition of the metapopulations as layers in a multiplex network, where nodes would correspond to geographical areas and layers account for the mobility patterns of agents of the same class. We analyze both the classical Susceptible-Infected-Susceptible and the Susceptible-Infected-Removed epidemic models within this framework, and compare macroscopic and microscopic indicators of the spreading process with extensive Monte Carlo simulations. Our results are in excellent agreement with the simulations. We also derive an exact expression of the epidemic threshold on this general framework revealing a non-trivial dependence on the mobility parameter. Finally, we use this new formalism to address the spread of diseases in real cities, specifically in the city of Medellin, Colombia, whose population is divided into six socio-economic classes, each one identified with a layer in this multiplex formalism.Comment: 13 pages, 11 figure

Criticality and oscillatory behavior in non-Markovian Contact Process

A Non-Markovian generalization of one-dimensional Contact Process (CP) is being introduced in which every particle has an age and will be annihilated at its maximum age $\tau$. There is an absorbing state phase transition which is controlled by this parameter. The model can demonstrate oscillatory behavior in its approach to the stationary state. These oscillations are also present in the mean-field approximation which is a first-order differential equation with time-delay. Studying dynamical critical exponents suggests that the model belongs to the DP universlity class.Comment: 4 pages, 5 figures, to be published in Phys. Rev.

A simple analytical description of the non-stationary dynamics in Ising spin systems

The analytical description of the dynamics in models with discrete variables (e.g. Isingspins) is a notoriously difficult problem, that can be tackled only undersome approximation.Recently a novel variational approach to solve the stationary dynamical regime has beenintroduced by Pelizzola [Eur. Phys. J. B, 86 (2013) 120], where simpleclosed equations arederived under mean-field approximations based on the cluster variational method. Here wepropose to use the same approximation based on the cluster variational method also for thenon-stationary regime, which has not been considered up to now within this framework. Wecheck the validity of this approximation in describing the non-stationary dynamical regime ofseveral Ising models defined on Erdos-R ́enyi random graphs: westudy ferromagnetic modelswith symmetric and partially asymmetric couplings, models with randomfields and also spinglass models. A comparison with the actual Glauber dynamics, solvednumerically, showsthat one of the two studied approximations (the so-called ‘diamond’approximation) providesvery accurate results in all the systems studied. Only for the spin glass models we find somesmall discrepancies in the very low temperature phase, probably due to the existence of alarge number of metastable states. Given the simplicity of the equations to be solved, webelieve the diamond approximation should be considered as the ‘minimalstandard’ in thedescription of the non-stationary regime of Ising-like models: any new method pretending toprovide a better approximate description to the dynamics of Ising-like models should performat least as good as the diamond approximation

The Cavity Approach to Parallel Dynamics of Ising Spins on a Graph

We use the cavity method to study parallel dynamics of disordered Ising models on a graph. In particular, we derive a set of recursive equations in single site probabilities of paths propagating along the edges of the graph. These equations are analogous to the cavity equations for equilibrium models and are exact on a tree. On graphs with exclusively directed edges we find an exact expression for the stationary distribution of the spins. We present the phase diagrams for an Ising model on an asymmetric Bethe lattice and for a neural network with Hebbian interactions on an asymmetric scale-free graph. For graphs with a nonzero fraction of symmetric edges the equations can be solved for a finite number of time steps. Theoretical predictions are confirmed by simulation results. Using a heuristic method, the cavity equations are extended to a set of equations that determine the marginals of the stationary distribution of Ising models on graphs with a nonzero fraction of symmetric edges. The results of this method are discussed and compared with simulations

Dynamic message-passing approach for kinetic spin models with reversible dynamics

A method to approximately close the dynamic cavity equations for synchronous reversible dynamics on a locally tree-like topology is presented. The method builds on $(a)$ a graph expansion to eliminate loops from the normalizations of each step in the dynamics, and $(b)$ an assumption that a set of auxilary probability distributions on histories of pairs of spins mainly have dependencies that are local in time. The closure is then effectuated by projecting these probability distributions on $n$-step Markov processes. The method is shown in detail on the level of ordinary Markov processes ($n=1$), and outlined for higher-order approximations ($n>1$). Numerical validations of the technique are provided for the reconstruction of the transient and equilibrium dynamics of the kinetic Ising model on a random graph with arbitrary connectivity symmetry.Comment: 6 pages, 4 figure