Ising model spin S=1 on directed Barabasi-Albert networks

On directed Barabasi-Albert networks with two and seven neighbours selected by each added site, the Ising model with spin S=1/2 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 Ising model spin S=1 is now studied through Monte Carlo simulations. However, in this model, the order-disorder phase transition is well defined in this system. We have obtained a first-order phase transition for values of connectivity m=2 and m=7 of the directed Barabasi-Albert network.Comment: 8 pages for Int. J. Mod. Phys. C; e-mail: [email protected]

Majority-vote on undirected Barabasi-Albert networks

On Barabasi-Albert networks with z neighbours selected by each added site, the Ising model was seen to show a spontaneous magnetisation. This spontaneous magnetisation was found below a critical temperature which increases logarithmically with system size. 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 and this wasn't found to increase logarithmically with system size. We calculate the value of the critical noise parameter q_c for several values of connectivity $z$ of the undirected Barabasi-Albert network. The critical exponentes beta/nu, gamma/nu and 1/nu were calculated for several values of z.Comment: 15 pages with numerous figure

Self-organized criticality in a model of collective bank bankruptcies

The question we address here is of whether phenomena of collective bankruptcies are related to self-organized criticality. In order to answer it we propose a simple model of banking networks based on the random directed percolation. We study effects of one bank failure on the nucleation of contagion phase in a financial market. We recognize the power law distribution of contagion sizes in 3d- and 4d-networks as an indicator of SOC behavior. The SOC dynamics was not detected in 2d-lattices. The difference between 2d- and 3d- or 4d-systems is explained due to the percolation theory.Comment: For Int. J. Mod. Phys. C 13, No. 3, six pages including four figure

Voter model on Sierpinski fractals

We investigate the ordering of voter model on fractal lattices: Sierpinski Carpets and Sierpinski Gasket. We obtain a power law ordering, similar to the behavior of one-dimensional system, regardless of fractal ramification.Comment: 7 pages, 5 EPS figures, 1 table, uses elsart.cl

Majority-vote on directed Small-World networks

On directed Small-World networks the Majority-vote model with noise is now studied through Monte Carlo simulations. 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 rewiring probability p of the directed Small-World network. The critical exponentes beta/nu, gamma/nu and 1/nu were calculated for several values of p.Comment: 16 pages including 9 figures, for Int. J. Mod. Phys.

Comparison of Ising magnet on directed versus undirected Erdos-Renyi and scale-free network

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 Erdos-Renyi network each of N sites is connected to every site with a low probability p (of the orde r 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.Comment: 10 pages including all figures, for Int. J, Mod. Phys. C 1

Collective firm bankruptcies and phase transition in rating dynamics

We present a simple model of firm rating evolution. We consider two sources of defaults: individual dynamics of economic development and Potts-like interactions between firms. We show that such a defined model leads to phase transition, which results in collective defaults. The existence of the collective phase depends on the mean interaction strength. For small interaction strength parameters, there are many independent bankruptcies of individual companies. For large parameters, there are giant collective defaults of firm clusters. In the case when the individual firm dynamics favors dumping of rating changes, there is an optimal strength of the firm's interactions from the systemic risk point of view