Special attention network

et al. 2007

Scale-free networks are a recently developed approach to model the interactions found in complex natural and man-made systems. Such networks exhibit a power-law distribution of node link (degree) frequencies n(k) in which a small number of highly connected nodes predominate over a much greater number of sparsely connected ones. In contrast, in an Erdös-Rényi network each of N sites is connected to every site with a low probability p (of the order of 1/N). Then the number k of neighbors will fluctuate according to a Poisson distribution. One can instead assume that each site selects exactly k neighbors among the other sites. Here we compare in both cases the usual network with the directed network, when site A selects site B as a neighbor, and then B influences A but A does not influence B. As we change from undirected to directed scale-free networks, the spontaneous magnetization vanishes after an equilibration time following an Arrhenius law, while the directed ER networks have a positive Curie temperature.

Section: Introductioncontrasting

confidence: 57%

Comparison of Ising Magnet on Directed Versus Undirected Erdös–rényi and Scale-Free Networks

Sumour

El-Astal

et al. 2007

“…Sumour and Shabat [7,8] investigated Ising models on directed Barabási-Albert networks [9] with the usual Glauber dynamics. No spontaneous magnetisation was found, in contrast to the case of undirected Barabási-Albert networks [10,11,12] where a spontaneous magnetisation was found lower a critical temperature which increases logarithmically with system size. More recently, Lima and Stauffer [13] simulated directed square, cubic and hypercubic lattices in two to five dimensions with heat bath dynamics in order to separate the network effects from the effects of directedness.…”

Section: Introductioncontrasting

confidence: 56%

Majority-Vote on Directed Barabási–albert Networks

2006

On directed Barabási-Albert networks with two and seven neighbours selected by each added site, the Ising model was seen not to show a spontaneous magnetisation. Instead, the decay time for flipping of the magnetisation followed an Arrhenius law for Metropolis and Glauber algorithms, but for Wolff cluster flipping the magnetisation decayed exponentially with time. On these networks the Majority-vote model with noise is now studied through Monte Carlo simulations. However, in this model, the order-disorder phase transition of the order parameter is well defined in this system. We calculate the value of the critical noise parameter q c for several values of connectivity z of the directed Barabási-Albert network. The critical exponentes β/ν, γ/ν and 1/ν were calculated for several values of z.Keywords:Monte Carlo simulation,vote , networks, nonequilibrium. IntroductionIt has been argued that nonequilibrium stochastic spin systems on regular square lattice with up-down symmetry fall in the universality class of the equilibrium Ising model [1]. This conjecture was found in several models that do not obey detailed balance [2,3,4]. Campos et al.[5] investigated the majority-vote model on small-world network by rewiring the two dimensional square lattice. These small-world networks, aside from presenting quenched disorder, also posses long-range interactions. They found that the critical exponents γ/ν and β/ν are different from the Ising model and depend on the rewiring probability. However, it was not evident that the exponent change was due to the disordered nature of the network or due to the presence of long-range interactions. Lima et al.[6] studied the majority-vote model on Voronoi-Delaunay random lattices with periodic boundary conditions. These lattices posses natural quenched disorder in their conecctions. They showed that presence of quenched connectivity disorder is enough to alter the exponents β/ν and γ/ν from the pure model and therefore that is a relevant term to such non-equilibrium phase-transition. Sumour and Shabat [7,8]

“…No spontaneous magnetisation was found, in contrast to the case of undirected Barabási-Albert networks [4,5,6] where a spontaneous magnetisation was found lower a critical temperature which increases logarithmically with system size. Lima and Stauffer [7] simulated directed square, cubic and hypercubic lattices in two to five dimensions with heat bath dynamics in order to separate the network effects form the effects of directedness.…”

Section: Introductionmentioning

confidence: 69%

Majority-Vote on Directed Small-World Networks

Luz

2007

On directed Small-World networks the Majority-vote model with noise is now studied through Monte Carlo simulations. In this model, the orderdisorder phase transition of the order parameter is well defined in this system. We calculate the value of the critical noise parameter q c for several values of rewiring probability p of the directed Small-World network. The critical exponentes β/ν, γ/ν and 1/ν were calculated for several values of p.