Abstract:The existence of spontaneous magnetization of Ising spins on directed Barabasi-Albert networks is investigated with seven neighbors, by using Monte Carlo simulations. In large systems we see the magnetization for different temperatures T to decay after a characteristic time τ (T ), which is extrapolated to diverge at zero temperature.Keywords: Monte Carlo simulations, Directed Barabasi-Albert networks, Magnetization, Fortran program Introduction:The Ising magnet is since decades a standard tool of computational physics [1]. We apply it here to scale-free networks [2], where previous simulations [3] indicated a Curie temperature increasing logarithmically with increasing system size N. In contrast to that work we use here directed [4] as opposed to undirected networks and then apply the standard Glauber kinetic Ising model [1] to the fixed network.

Chen and Dohm predicted theoretically in 2004 that the widely believed universality principle is violated in the Ising model on the simple cubic lattice with more than only six nearest neighbours. Schulte and Drope by Monte Carlo simulations found such violation, but not in the predicted direction. Selke and Shchur tested the square lattice. Here we check only this universality for the susceptibility ratio near the critical point. For this purpose we study first the standard Ising model on a simple cubic lattice with six nearest neighbours, then with six nearest and twelve next-nearest neighbours, and compare the results with the Chen-Dohm lattice of six nearest neighbours and only half of the twelve next-nearest neighbours. We do not confirm the violation of universality found by Schulte and Drope in the susceptibility ratio.

Mortality, birth rates and retirement play a major role in demographic changes. In most cases, mortality rates decreased in the past century without noticeable decrease in fertility rates, this leads to a significant increase in population growth. In many poor countries like Palestinian territories the number of births has fallen and the life expectancy increased.In this article we concentrate on measuring, analyzing and extrapolating the age structure in Palestine a few decades ago into future. A Fortran program has been designed and used for the simulation and analysis of our statistical data. This study of demographic change in Palestine has shown that Palestinians will have in future problems as the strongest age cohorts are the above-60-year olds. We therefore recommend the increase of both the retirement age and women employment.

In usual scale-free networks of Barabási-Albert type, a newly added node selects randomly m neighbors from the already existing network nodes, proportionally to the number of links these had before. Then the number N (k) of nodes with k links each decays as 1/k γ where γ = 3 is universal, i.e. independent of m. Now we use a limited directedness in the construction of the network, as a result of which the exponent γ decreases from 3 to 2 for increasing m.

In this paper we study urban segregation of two different communities A and B, rich and poor, distributed randomly on finite samples, to check cheap and expensive residences. For this purpose we avoid the complications of the Schelling model which are not necessary and instead we use the Ising model on 500 × 500 square lattices, which gives similar results, with random magnetic field at lower and higher temperatures (k B T /J = 2.0, 99.0) in finite times equal to 40, 400, 4000 and 40 000. This random-field Ising magnet is a suitable model, where each site of the square lattice carries a magnetic field ±h which is randomly up (expensive) or down (cheap). The resulting addition to the energy prefers up-spins on the expensive and down-spins on the cheap sites. Our simulations were carried out using a 50-line FORTRAN program. We present at a lower temperature (2.0) a time series of pictures, separating growing from non-growing domains. A small random field (h = ±0.1) allows for large domains, while a large random field (h = ±0.9) allows only small clusters. At higher temperature (99.0) we could not obtain growing domains.

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.

We check the existence of a spontaneous magnetisation of Ising and Potts spins on semi-directed Barabasi-Albert networks by Monte Carlo simulations. We verified that the magnetisation for different temperatures T decays after a characteristic time τ (T ), which we extrapolate to diverge at positive temperatures T c (N ) by a Vogel-Fulcher law, with T c (N ) increasing logarithmically with network size N .

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