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Dynamic processes of interacting units on a network are out of equilibrium in general. In the case of a directed tree, the dynamic cavity method provides an efficient tool that characterises the dynamic trajectory of the process for the linear threshold model. However, because of the computational complexity of the method, the analysis has been limited to systems where the largest number of neighbours is small. We devise an efficient implementation of the dynamic cavity method which substantially reduces the computational complexity of the method for systems with discrete couplings. Our approach opens up the possibility to investigate the dynamic properties of networks with fat-tailed degree distribution. We exploit this new implementation to study properties of the non-equilibrium steady-state. We extend the dynamic cavity approach to calculate the pairwise correlations induced by different motifs in the network. Our results suggest that just two basic motifs of the network are able to accurately describe the entire statistics of observed correlations. Finally, we investigate models defined on networks containing bi-directional interactions. We observe that the stationary state associated with networks with symmetric or anti-symmetric interactions is biased towards the active or inactive state respectively, even if independent interaction entries are drawn from a symmetric distribution. This phenomenon, which can be regarded as a form of spontaneous symmetry-breaking, is peculiar to systems formulated in terms of Boolean variables, as opposed to Ising spins.

We analyse biased ensembles of trajectories for the random-field Ising model on a fully-connected lattice, which is described exactly by mean-field theory. By coupling the activity of the system to a dynamical biasing field, we find a range of dynamical phase transitions, including spontaneous symmetry breaking into ordered states. For weak bias, the phase behaviour is controlled by extrema of the free energy, which may be local minima or saddle points. For large bias, the system tends to states of extremal activity, which may differ strongly from free energy minima. We discuss connections of these results to random first-order transition theory of glasses, which motivates an extension of the analysis to random-field Ising models where the dynamical activity is not symmetric under magnetisation reversal., 26 pages, 10 figures

The equilibrium and nonequilibrium properties of ferromagnetic systems may be affected by the long-range nature of the coupling interaction. Here we study the phase separation process of a one-dimensional Ising model in the presence of a power-law decaying coupling, $J(r)=1/r^{1+\sigma}$ with $\sigma >0$, and we focus on the two-time autocorrelation function $C(t,t_w)=\langle s_i(t) s_i(t_w)\rangle$. We find that it obeys the scaling form $C(t,t_w)=f(L(t_w)/L(t))$, where $L(t)$ is the typical domain size at time $t$, and where $f(x)$ can only be of two types. For $\sigma>1$, when domain walls diffuse freely, $f(x)$ falls in the nearest-neighbour (nn) universality class. Conversely, for $\sigma \le 1$, when domain walls dynamics is driven, $f(x)$ displays a new universal behavior. In particular, the so-called Fisher-Huse exponent, which characterizes the asymptotic behavior of $f(x)\simeq x^{-\lambda}$ for $x\gg 1$, is $\lambda=1$ in the nn universality class ($\sigma > 1$) and $\lambda=1/2$ for $\sigma \le 1$., Comment: Accepted for publication in JSTAT, Special issue: New Trends in Nonequilibrium Statistical Mechanics: Classical and Quantum Systems (nesmcq18) (minor changes)

We consider the most general single-spin-flip dynamics for the ferromagnetic Ising chain with nearest-neighbour influence and spin reversal symmetry. This dynamics is a two-parameter extension of Glauber dynamics corresponding respectively to non-linearity and irreversibility. The associated stationary state measure is given by the usual Boltzmann-Gibbs distribution for the ferromagnetic Hamiltonian of the chain. We study the properties of this dynamics both at infinite and at finite temperature, all over its parameter space, with particular emphasis on special lines and points., Comment: 31 pages, 18 figures

We propose a new algorithm for inferring the state of hidden spins and reconstructing the connections in a synchronous kinetic Ising model, given the observed history. Focusing on the case in which the hidden spins are conditionally independent of each other given the state of observable spins, we show that calculating the likelihood of the data can be simplified by introducing a set of replicated auxiliary spins. Belief propagation (BP) and susceptibility propagation (SusP) can then be used to infer the states of hidden variables and to learn the couplings. We study the convergence and performance of this algorithm for networks with both Gaussian-distributed and binary bonds. We also study how the algorithm behaves as the fraction of hidden nodes and the amount of data are changed, showing that it outperforms the Thouless-Anderson-Palmer (TAP) equations for reconstructing the connections. QC 20150625