Ising model in small-world networks

Herrero, Carlos P.

doi:10.1103/physreve.65.066110

Cited by 144 publications

(116 citation statements)

References 24 publications

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“…The primary use of the small-world model has been as a substrate for the investigation of various processes taking place on graphs, such as percolation [294,325,326,360], coloring [388,406], coupled oscillators [37,201,416], iterated games [1,135,231,416], diffusion processes [150,173,216,258,259,289,329], epidemic processes [28,235,255,293,427,428], and spin models [40,191,202,256,337,429]. Some of this work is discussed further in Section VIII.…”

Section: Average Path Lengthmentioning

confidence: 99%

The Structure and Function of Complex Networks

Newman¹

2003

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Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

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Section: Average Path Lengthmentioning

confidence: 99%

The Structure and Function of Complex Networks

Newman¹

2003

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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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“…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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Ising model in small-world networks

Cited by 144 publications

References 24 publications

The Structure and Function of Complex Networks

The Structure and Function of Complex Networks

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

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

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