Ferromagnetic phase transition in Barabási–Albert networks

Abstract

Ising spins put onto a Barabási-Albert scale-free network show an effective phase transition from ferromagnetism to paramagnetism upon heating, with an effective critical temperature increasing as the logarithm of the system size. Starting with all spins up and upon equilibration pinning the few most-connected spins down nucleates the phase with most of the spins down.

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

“…Sumour and Shabat [5,6] investigated Ising models with spin S = 1/2 on directed BA networks [4] with the usual Glauber dynamics. No spontaneous magnetisation was found, in contrast to the case of undirected BA networks [7,8,9] where a spontaneous magnetisation was found below a critical temperature which increases logarithmically with system size. In S = 1/2 systems on undirected, scale-free hierarchical-lattice small-world networks [10], conventional and algebraic (Berezinskii-Kosterlitz-Thouless) ordering, with finite transition temperatures, have been found.…”

Scale-Free Network Theory in Studying the Structure of the Road Network in Poland

“…Since the explosion of the complex network science, many real-world networks have been examined. Examples include the structure of the Internet [1], criminal groups [2,3], cyber-terrorism [1], social groups [4][5][6], trade [7][8][9], railroads and gas/oil pipeline systems [10], energy networks [11][12], scientific cooperation networks [13], citation networks [14] and many others.…”

Special attention network

“…Recently Aleksiejuk et al [1] proposed a social network model for opinion formation by introducing ferromagnetically coupled Ising spins on a Barabási-Albert network [2]. The strength of the couplings between linked nodes, J, was taken to be uniform, independent of the number of connections to a node.…”

“…Simulations of this model indicated the existence of a critical "temperature" below which opinion formation is possible (two-phase coexistence) and above which common opinion is unstable (disordered phase). A peculiar feature of this model is that simulations indicate that the critical temperature depends on the number of nodes, or system size, N in the manner T c ∝ log(N), so that in the thermodynamic limit an initially imposed common opinion always persists (in the absence of an external opposing "field"), no matter how weak the uniform coupling J between each pair of connected persons, or "partners" [1]. Incidentally, this peculiar divergence of T c with N is missed in the simplest mean-field approximation to the model [1], but is captured correctly in Email address: joseph.indekeu@fys.kuleuven.ac.be (J. O. Indekeu).…”