Small-world phenomena in physics: the Ising model

Gitterman, Moshe

doi:10.1088/0305-4470/33/47/304

Cited by 116 publications

(154 citation statements)

References 17 publications

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“…Such graphs have been used, for example, to improve scalability of parallel computer algorithms [6][7][8][9][10]. Furthermore, the critical behavior of models of materials, such as the Ising model, Heisenberg model, and random-walker models have been studied on SW graphs [11][12][13][14][15][16][17][18][19][20][21]. The result of these studies is that models on SW graphs exhibit mean-field scaling, namely they have an effective dimension at or above the upper critical dimension of the model (which for the Ising model without disorder is d=4).…”

Section: Introductionmentioning

confidence: 99%

Critical behavior of ising models with random long-range (small-world) interactions

Zhang¹,

Novotny²

2006

Braz. J. Phys.

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The critical scaling behavior of Ising models with long range interactions is studied. These long-range interactions, when imposed in addition to interactions on a regular lattice, lead to small world graphs. Large-scale Monte Carlo simulations, together with finite-size scaling, is used to obtain the critical behavior of a number of different models. These include the z-model introduced by Scalettar, standard small-world bonds superimposed on a square lattice, and physical small-world bonds superimposed on a square lattice. These scaling results provide further evidence to support the existence of physical (quasi-) small-world nanomaterials.

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Section: Introductionmentioning

confidence: 99%

Critical behavior of ising models with random long-range (small-world) interactions

Zhang¹,

Novotny²

2006

Braz. J. Phys.

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“…Quite naturally, various spin models of statistical mechanics have been studied on an underlying complex network [2,3,4]. These studies serve a twofold purposes: Firstly, they aid studies of the static network structure.…”

Section: Introductionmentioning

confidence: 99%

Dynamic critical behavior of theXYmodel in small-world networks

et al. 2003

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The critical behavior of the XY model on small-world network is investigated by means of dynamic Monte Carlo simulations. We use the short-time relaxation scheme, i.e., the critical behavior is studied from the nonequilibrium relaxation to equilibrium. Static and dynamic critical exponents are extracted through the use of the dynamic finite-size scaling analysis. It is concluded that the dynamic universality class at the transition is of the mean-field nature. We also confirm numerically that the value of dynamic critical exponent is independent of the rewiring probability P for P > ∼ 0.03.

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“…The suppression of critical fluctuations of the virtual time horizon is also closely related to the emergence of mean-field-like phase transitions and phase ordering in non-frustrated interacting systems [1,[64][65][66][67][68][69][70].…”

Section: Discussionmentioning

confidence: 99%

Small-World Synchronized Computing Networks for Scalable Parallel Discrete-Event Simulations

et al. 2004

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Abstract. We study the scalability of parallel discrete-event simulations for arbitrary short-range interacting systems with asynchronous dynamics. When the synchronization topology mimics that of the short-range interacting underlying system, the virtual time horizon (corresponding to the progress of the processing elements) exhibits Kardar-Parisi-Zhang-like kinetic roughening. Although the virtual times, on average, progress at a nonzero rate, their statistical spread diverges with the number of processing elements, hindering efficient data collection. We show that when the synchronization topology is extended to include quenched random communication links between the processing elements, they make a close-to-uniform progress with a nonzero rate, without global synchronization. We discuss in detail a coarse-grained description for the small-world synchronized virtual time horizon and compare the findings to those obtained by "simulating the simulations" based on the exact algorithmic rules.

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Small-world phenomena in physics: the Ising model

Cited by 116 publications

References 17 publications

Critical behavior of ising models with random long-range (small-world) interactions

Critical behavior of ising models with random long-range (small-world) interactions

Dynamic critical behavior of theXYmodel in small-world networks

Small-World Synchronized Computing Networks for Scalable Parallel Discrete-Event Simulations

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