Phase transitions in an Ising model on a Euclidean network

Scaling laws of failure dynamics on complex networks

The topology of the network of load transmitting connections plays an essential role in the cascading failure dynamics of complex systems driven by the redistribution of load after local breakdown events. In particular, as the network structure is gradually tuned from regular to completely random a transition occurs from the localized to mean field behavior of failure spreading. Based on finite size scaling in the fiber bundle model of failure phenomena, here we demonstrate that outside the localized regime, the load bearing capacity and damage tolerance on the macro-scale, and the statistics of clusters of failed nodes on the micro-scale obey scaling laws with exponents which depend on the topology of the load transmission network and on the degree of disorder of the strength of nodes. Most notably, we show that the spatial structure of damage governs the emergence of the localized to mean field transition: as the network gets gradually randomized failed clusters formed on locally regular patches merge through long range links generating a percolation like transition which reduces the load concentration on the network. The results may help to design network structures with an improved robustness against cascading failure.

An empirical analysis of the Ebola outbreak in West Africa

The data for the Ebola outbreak that occurred in 2014–2016 in three countries of West Africa are analysed within a common framework. The analysis is made using the results of an agent based Susceptible-Infected-Removed (SIR) model on a Euclidean network, where nodes at a distance l are connected with probability P(l) ∝ l^−δ, δ determining the range of the interaction, in addition to nearest neighbors. The cumulative (total) density of infected population here has the form , where the parameters depend on δ and the infection probability q. This form is seen to fit well with the data. Using the best fitting parameters, the time at which the peak is reached is estimated and is shown to be consistent with the data. We also show that in the Euclidean model, one can choose δ and q values which reproduce the data for the three countries qualitatively. These choices are correlated with population density, control schemes and other factors. Comparing the real data and the results from the model one can also estimate the size of the actual population susceptible to the disease. Rescaling the real data a reasonably good quantitative agreement with the simulation results is obtained.

Effect of the nature of randomness on quenching dynamics of the Ising model on complex networks

Randomness is known to affect the dynamical behavior of many systems to a large extent. In this paper we investigate how the nature of randomness affects the dynamics in a zero-temperature quench of the Ising model on two types of random networks. In both networks, which are embedded in a one-dimensional space, the first-neighbor connections exist and the average degree is 4 per node. In random model A the second-neighbor connections are rewired with a probability p, while in random model B additional connections between neighbors at a Euclidean distance l(l > 1) are introduced with a probability P(l) proportionally l(-α). We find that for both models, the dynamics leads to freezing such that the system gets locked in a disordered state. The point at which the disorder of the nonequilibrium steady state is maximum is located. The behavior of dynamical quantities such as residual energy, order parameter, and persistence are discussed and compared. Overall, the behavior of physical quantities are similar, although subtle differences are observed due to the difference in the nature of randomness.

Critical fluctuations in spatial complex networks

An anomalous mean-field solution is known to capture the nontrivial phase diagram of the Ising model in annealed complex networks. Nevertheless, the critical fluctuations in random complex networks remain mean field. Here we show that a breakdown of this scenario can be obtained when complex networks are embedded in geometrical spaces. Through the analysis of the Ising model on annealed spatial networks, we reveal, in particular, the spectral properties of networks responsible for critical fluctuations and we generalize the Ginsburg criterion to complex topologies.

Stripe domain coarsening in geographical small-world networks on a Euclidean lattice

We study phase ordering dynamics of spatially periodic striped patterns on the small-world network that is derived from a two-dimensional regular lattice with distance-dependent random connections. It is demonstrated numerically that addition of spatial disorder in the form of shortcuts makes the growth of domains much slower or even frozen at late times.

Order-disorder phase transition in a chaotic system

For soft-mode turbulence, which is essentially the spatiotemporal chaos caused by the nonlinear interaction between convective modes and Goldstone modes in electroconvection of homeotropic nematics, a type of order-disorder phase transition was revealed, in which a new order parameter was introduced as pattern ordering. We calculated the spatial correlation function and the anisotropy of the convective patterns as a 2D XY system because the convective wave vector could freely rotate in the homeotropic system. We found the hidden order in the chaotic patterns observed beyond the Lifshitz frequency f(L), and a transition from a disordered to a hidden ordered state occurred at the f(L) with the increase of the frequency of the applied voltages.

Phase transition of a one-dimensional Ising model with distance-dependent connections

The critical behavior of the Ising model on a one-dimensional network, which has long-range connections at distances l>1 with the probability theta(l) approximately l(-m), is studied by using Monte Carlo simulations. Through analyzing the Ising model on networks with different m values, this paper discusses the impact of the global correlation, which decays with the increase of m, on the phase transition of the Ising model. Adding the analysis of the finite-size scaling of the order parameter [M], it is observed that in the whole range of 0<m<2, a finite-temperature transition exists, and the critical exponents show consistence with mean-field values, which indicates a mean-field nature of the phase transition.