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

Abstract

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.

Clusterization, Frustration and Collectivity in Random Networks

“…As we see, the obtained Curie temperature increases with k . Also, the mean-fieldcharacter of the system is confirmed, as it was concluded in [22], because the curves M 2 (T ) are approximately linear. For k = 2, the value of the Curie temperature, T c /J ≈ 1.7 agrees with the one obtained in [22].…”

“…The Curie temperature for the random network was obtained already by Sumour et al [22]. However, up to our knowledge the clusterization coefficient has not been varied in these networks.…”

“…One timestep is equivalent to N attempts to flip a spin. For the comparison with the results of [22], the calculations for k = 4 are repeated for k = 3 and k = 2, N = 10 4 . The obtained curves for the squared magnetization are presented in Fig.…”

Phase Transition in the Ising Model on Local-World Evolving Networks

“…10 Recently, the Ising model has been applied to networks with a more complicated connectivity. [11][12][13][14][15][16][17][18][19][20][21][22] Ising spins put onto network vertices are found to experience an effective phase transition from ferromagnetism to paramagnetism upon heating. Theoretical and numerical studies have demonstrated various universal critical behaviors that are determined by the structure of these networks.…”

“…However, as far as we know, most studies considered dynamic systems only on small-world networks [11][12][13][14][15] and scale-free networks. [16][17][18][19][20][21][22] Here, we investigate the phase transition of the Ising model on local-world evolving networks. Monte Carlo simulations with the standard Metropolis local update algorithms are performed on these networks, and various thermodynamic quantities are measured.…”