# XY model in small-world networks.

@article{Kim2001XYMI, title={XY model in small-world networks.}, author={B. J. Kim and Hyunsuk Hong and Petter Holme and Gun Sang Jeon and Petter Minnhagen and M.Y. Choi}, journal={Physical review. E, Statistical, nonlinear, and soft matter physics}, year={2001}, volume={64 5 Pt 2}, pages={ 056135 } }

The phase transition in the XY model on one-dimensional small-world networks is investigated by means of Monte Carlo simulations. It is found that long-range order is present at finite temperatures, even for very small values of the rewiring probability, suggesting a finite-temperature transition for any nonzero rewiring probability. Nature of the phase transition is discussed in comparison with the globally coupled XY model.

## 77 Citations

### Dynamic critical behavior of the XY model in small-world networks.

- PhysicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2003

The critical behavior of the XY model on small-world network is investigated by means of dynamic Monte Carlo simulations and it is concluded that the dynamic universality class at the transition is of the mean-field nature.

### In an Attempt to Introduce Long-range Interactions into Small-world Networks

- Physics
- 2009

Distinguishing the long-range bonds with the regular ones, the critical temperature of the spin-lattice Guassian model built on two typical Small-world Networks (SWNs) is studied. The results show…

### Emergence of Long Range Order in the XY Model on Diluted Small World Networks

- Physics, Computer Science
- 2013

The XY model on diluted Small World networks is studied and the presence of a second order phase transition appearing even for very low p and γ, when the topology is still near the regular lattice one is highlighted and a dependence of the critical energy ϵ c on the rewiring probability p is observed.

### Potts model with q=3 and 4 states on directed small-world network

- PhysicsComput. Phys. Commun.
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### Finite-size scaling in complex networks.

- PhysicsPhysical review letters
- 2007

A finite-size-scaling (FSS) theory is proposed for various models in complex networks and based on the droplet-excitation (hyperscaling) argument, the values of the FSS exponents for the Ising model, the susceptible-infected-susceptible model, and the contact process are conjecture.

### Non-equilibrium Phase Transitions in 2D Small-World Networks: Competing Dynamics

- PhysicsOpen Physics
- 2019

Abstract This article offers a detailed analysis of the Ising model in 2D small-world networks with competing Glauber and Kawasaki dynamics. The non-equilibrium stationary state phase transitions are…

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- 2018

The critical properties of the Potts model with $$q=3$$q=3 and 8 states in one-dimension on directed small-world networks are investigated. This disordered system is simulated by updating it with the…

### Irreversible growth of binary mixtures on small-world networks.

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- 2006

The small-world networks are shown to trigger criticality, a phenomenon analogous to similar observations reported recently in the investigation of equilibrium systems.

### Stochastic Growth in a Small World

- Physics
- 2003

We considered the Edwards-Wilkinson model on a small-world network. We studied the finite-size behavior of the surface width by performing exact numerical diagonalization for the underlying coupling…

### Phase transitions in an Ising model on a Euclidean network.

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- 2006

This work investigates the critical behavior of the Ising model on a one-dimensional network where spins interact with extra neighbors apart from their nearest neighbors for 0<or=delta<2, and observes that there is a finite temperature phase transition in the entire range.

## References

SHOWING 1-10 OF 39 REFERENCES

### First-order transition in small-world networks

- Geology
- 2000

The small-world transition is a first-order transition at zero density p of shortcuts, whereby the normalised shortest-path distance = /L undergoes a discontinuity in the thermodynamic limit. On…

### Renormalization Group Analysis of the Small-World Network Model

- Physics, Computer Science
- 1999

### Small-world phenomena in physics: the Ising model

- Physics
- 2000

The Ising system with a small fraction of random long-range interactions is the simplest example of small-world phenomena in physics. Considering the latter both in an annealed and in a quenched…

### Small world effect in an epidemiological model.

- Computer SciencePhysical review letters
- 2001

A model for the spread of an infection is analyzed for different population structures and finds a transition to self-sustained oscillations in the size of the infected subpopulation at a finite value of the disorder of the network.

### On the properties of small-world network models

- Physics
- 1999

Abstract:We study the small-world networks recently introduced by Watts and Strogatz [Nature 393, 440 (1998)], using analytical as well as numerical tools. We characterize the geometrical properties…

### Scaling and percolation in the small-world network model.

- MathematicsPhysical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
- 1999

There is one nontrivial length-scale in the small-world network model of Watts and Strogatz, analogous to the correlation length in other systems, which is well-defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero, giving a normal critical point in this limit.

### Incipient spanning cluster on small-world networks

- Physics
- 2001

We analyze the scaling properties of the largest cluster size for the site percolation problem on small-world graphs. It is shown how the presence of the extra length-scale, the small-world crossover…

### Exact solution of site and bond percolation on small-world networks.

- MathematicsPhysical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
- 2000

An exact solution of the model for both site and bond percolation is given, including the position of thePercolation transition at which epidemic behavior sets in, the values of the critical exponents governing this transition, the mean and variance of the distribution of cluster sizes below the transition, and the size of the giant component (epidemic) above the transition.

### Ordering, metastability and phase transitions in two-dimensional systems

- Physics
- 1973

A new definition of order called topological order is proposed for two-dimensional systems in which no long-range order of the conventional type exists. The possibility of a phase transition…

### Emergence of scaling in random networks

- Computer ScienceScience
- 1999

A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.