Ising model on a small world network

Pękalski, Andrzej

doi:10.1103/physreve.64.057104

Cited by 79 publications

(69 citation statements)

References 13 publications

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“…We shall see below that the smallworld structure at larger p translates into a distinctive critical behavior for the Ising model on this network. (7) and (8). For other p, we have calculated ℓn numerically, with an accuracy of ±0.3%.…”

Section: Clustering Coefficient CMmentioning

confidence: 99%

“…Since the pioneering network models of Watts-Strogatz [4], which exemplified the first two properties, and Barabási-Albert [5], which showed how the third could arise from particular mechanisms of network growth, significant advances have taken place in understanding how these properties affect statistical systems. The Ising model has been studied on small-world networks [6,7,8,9,10], along with the XY model [11], and on Barabási-Albert scale-free networks [12,13]. On random graphs with arbitrary degree distributions, the Ising model shows a range of possible critical behaviors depending on the moments of the distribution (or in the specific case of scale-free distributions, the exponent describing the power-law tail) [14,15], a fact which is accounted for by a phenomenological theory of critical phenomena on these types of networks [16].…”

Section: Introductionmentioning

confidence: 99%

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Inverted Berezinskii-Kosterlitz-Thouless singularity and high-temperature algebraic order in an Ising model on a scale-free hierarchical-lattice small-world network

2006

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We have obtained exact results for the Ising model on a hierarchical lattice incorporating three key features characterizing many real-world networks-a scale-free degree distribution, a high clustering coefficient, and the small-world effect. By varying the probability p of long-range bonds, the entire spectrum from an unclustered, non-small-world network to a highly-clustered, small-world system is studied. Using the self-similar structure of the network, we obtain analytic expressions for the degree distribution P (k) and clustering coefficient C for all p, as well as the average path length ℓ for p = 0 and 1. The ferromagnetic Ising model on this network is studied through an exact renormalization-group transformation of the quenched bond probability distribution, using up to 562,500 renormalized probability bins to represent the distribution. For p < 0.494, we find powerlaw critical behavior of the magnetization and susceptibility, with critical exponents continuously varying with p, and exponential decay of correlations away from Tc. For p ≥ 0.494, in fact where the network exhibits small-world character, the critical behavior radically changes: We find a highly unusual phase transition, namely an inverted Berezinskii-Kosterlitz-Thouless singularity, between a low-temperature phase with non-zero magnetization and finite correlation length and a hightemperature phase with zero magnetization and infinite correlation length, with power-law decay of correlations throughout the phase. Approaching Tc from below, the magnetization and the susceptibility respectively exhibit the singularities of exp(−C/ √ Tc − T ) and exp(D/ √ Tc − T ), with C and D positive constants. With long-range bond strengths decaying with distance, we see a phase transition with power-law critical singularities for all p, and evaluate an unusually narrow critical region and important corrections to power-law behavior that depend on the exponent characterizing the decay of long-range interactions.

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Section: Clustering Coefficient CMmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Inverted Berezinskii-Kosterlitz-Thouless singularity and high-temperature algebraic order in an Ising model on a scale-free hierarchical-lattice small-world network

2006

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“…In recent years, a large amount of work has been devoted to the study of small-world networks, mainly numerical [1] with emphasis e.g. on biophysical networks [2]- [4] or social networks [5] and, to a lesser extent, analytically [6,7].…”

Section: Introductionmentioning

confidence: 99%

Thermodynamics of spin systems on small-world hypergraphs

2006

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We study the thermodynamic properties of spin systems on small-world hypergraphs, obtained by superimposing sparse Poisson random graphs with p-spin interactions onto a one-dimensional Ising chain with nearest-neighbor interactions. We use replica-symmetric transfer-matrix techniques to derive a set of fixed-point equations describing the relevant order parameters and free energy, and solve them employing population dynamics. In the special case where the number of connections per site is of the order of the system size we are able to solve the model analytically. In the more general case where the number of connections is finite we determine the static and dynamic ferromagneticparamagnetic transitions using population dynamics. The results are tested against Monte-Carlo simulations.

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“…For a chain the neighbors of i are thus i − 1, i + 1, i + R where R is a random distance; these neighbors can be compared with those of the honeycomb lattice: i ± 1, i ± L. Therefore it cannot be excluded that ordering is found also in one dimension. This simplified model is closer to small-world networks [2,3,4].…”

Section: Introductionmentioning

confidence: 99%