### Abstract

We introduce a simple algorithm that constructs scale-free random graphs efficiently: each vertex i has a prescribed weight P_{i} ∝^{i-μ} (0 < μ < 1) and an edge can connect vertices i and j with rate P_{i}P_{j}. Corresponding equilibrium ensemble is identified and the problem is solved by the q → 1 limit of the q-state Potts model with inhomogeneous interactions for all pairs of spins. The number of loops as well as the giant cluster size and the mean cluster size are obtained in the thermodynamic limit as a function of the edge density. Various critical exponents associated with the percolation transition are also obtained together with finite-size scaling forms. The process of forming the giant cluster is qualitatively different between the cases of λ > 3 and 2 < λ < 3, where λ = 1 + μ^{-1} is the degree distribution exponent. While for the former, the giant cluster forms abruptly at the percolation transition, for the latter, however, the formation of the giant cluster is gradual and the mean cluster size for finite N shows double peaks.

Original language | English |
---|---|

Pages (from-to) | 1149-1159 |

Number of pages | 11 |

Journal | Pramana - Journal of Physics |

Volume | 64 |

Issue number | 6 SPEC. ISS. |

Publication status | Published - 2005 Jun 1 |

Externally published | Yes |

### Fingerprint

### Keywords

- Percolation transition
- Potts model
- Scale-free random graph

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Pramana - Journal of Physics*,

*64*(6 SPEC. ISS.), 1149-1159.

**Scale-free random graphs and Potts model.** / Lee, D. S.; Goh, Kwang-Il; Kahng, B.; Kim, D.

Research output: Contribution to journal › Article

*Pramana - Journal of Physics*, vol. 64, no. 6 SPEC. ISS., pp. 1149-1159.

}

TY - JOUR

T1 - Scale-free random graphs and Potts model

AU - Lee, D. S.

AU - Goh, Kwang-Il

AU - Kahng, B.

AU - Kim, D.

PY - 2005/6/1

Y1 - 2005/6/1

N2 - We introduce a simple algorithm that constructs scale-free random graphs efficiently: each vertex i has a prescribed weight Pi ∝i-μ (0 < μ < 1) and an edge can connect vertices i and j with rate PiPj. Corresponding equilibrium ensemble is identified and the problem is solved by the q → 1 limit of the q-state Potts model with inhomogeneous interactions for all pairs of spins. The number of loops as well as the giant cluster size and the mean cluster size are obtained in the thermodynamic limit as a function of the edge density. Various critical exponents associated with the percolation transition are also obtained together with finite-size scaling forms. The process of forming the giant cluster is qualitatively different between the cases of λ > 3 and 2 < λ < 3, where λ = 1 + μ-1 is the degree distribution exponent. While for the former, the giant cluster forms abruptly at the percolation transition, for the latter, however, the formation of the giant cluster is gradual and the mean cluster size for finite N shows double peaks.

AB - We introduce a simple algorithm that constructs scale-free random graphs efficiently: each vertex i has a prescribed weight Pi ∝i-μ (0 < μ < 1) and an edge can connect vertices i and j with rate PiPj. Corresponding equilibrium ensemble is identified and the problem is solved by the q → 1 limit of the q-state Potts model with inhomogeneous interactions for all pairs of spins. The number of loops as well as the giant cluster size and the mean cluster size are obtained in the thermodynamic limit as a function of the edge density. Various critical exponents associated with the percolation transition are also obtained together with finite-size scaling forms. The process of forming the giant cluster is qualitatively different between the cases of λ > 3 and 2 < λ < 3, where λ = 1 + μ-1 is the degree distribution exponent. While for the former, the giant cluster forms abruptly at the percolation transition, for the latter, however, the formation of the giant cluster is gradual and the mean cluster size for finite N shows double peaks.

KW - Percolation transition

KW - Potts model

KW - Scale-free random graph

UR - http://www.scopus.com/inward/record.url?scp=21544433271&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21544433271&partnerID=8YFLogxK

M3 - Article

VL - 64

SP - 1149

EP - 1159

JO - Pramana - Journal of Physics

JF - Pramana - Journal of Physics

SN - 0304-4289

IS - 6 SPEC. ISS.

ER -