### Abstract

We study the bond percolation problem in random graphs of N weighted vertices, where each vertex i has a prescribed weight P_{i} and an edge can connect vertices i and j with rate P_{i}P_{j}. The problem is solved by the q→1 limit of the q-state Potts model with inhomogeneous interactions for all pairs of spins. We apply this approach to the static model having P_{i}∝i^{-μ} (0<μ<1) so that the resulting graph is scale-free with the degree exponent λ=1+1/μ. 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, and their associated critical exponents are also obtained. Finite-size scaling behaviors are derived using the largest cluster size in the critical regime, which is calculated from the cluster size distribution, and checked against numerical simulation results. We find that the process of forming the giant cluster is qualitatively different between the cases of λ>3 and 2<λ<3. 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) | 351-380 |

Number of pages | 30 |

Journal | Nuclear Physics B |

Volume | 696 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2004 Sep 27 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Nuclear and High Energy Physics

### Cite this

*Nuclear Physics B*,

*696*(3), 351-380. https://doi.org/10.1016/j.nuclphysb.2004.06.029

**Evolution of scale-free random graphs : Potts model formulation.** / Lee, D. S.; Goh, Kwang-Il; Kahng, B.; Kim, D.

Research output: Contribution to journal › Article

*Nuclear Physics B*, vol. 696, no. 3, pp. 351-380. https://doi.org/10.1016/j.nuclphysb.2004.06.029

}

TY - JOUR

T1 - Evolution of scale-free random graphs

T2 - Potts model formulation

AU - Lee, D. S.

AU - Goh, Kwang-Il

AU - Kahng, B.

AU - Kim, D.

PY - 2004/9/27

Y1 - 2004/9/27

N2 - We study the bond percolation problem in random graphs of N weighted vertices, where each vertex i has a prescribed weight Pi and an edge can connect vertices i and j with rate PiPj. The problem is solved by the q→1 limit of the q-state Potts model with inhomogeneous interactions for all pairs of spins. We apply this approach to the static model having Pi∝i-μ (0<μ<1) so that the resulting graph is scale-free with the degree exponent λ=1+1/μ. 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, and their associated critical exponents are also obtained. Finite-size scaling behaviors are derived using the largest cluster size in the critical regime, which is calculated from the cluster size distribution, and checked against numerical simulation results. We find that the process of forming the giant cluster is qualitatively different between the cases of λ>3 and 2<λ<3. 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 study the bond percolation problem in random graphs of N weighted vertices, where each vertex i has a prescribed weight Pi and an edge can connect vertices i and j with rate PiPj. The problem is solved by the q→1 limit of the q-state Potts model with inhomogeneous interactions for all pairs of spins. We apply this approach to the static model having Pi∝i-μ (0<μ<1) so that the resulting graph is scale-free with the degree exponent λ=1+1/μ. 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, and their associated critical exponents are also obtained. Finite-size scaling behaviors are derived using the largest cluster size in the critical regime, which is calculated from the cluster size distribution, and checked against numerical simulation results. We find that the process of forming the giant cluster is qualitatively different between the cases of λ>3 and 2<λ<3. 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.

UR - http://www.scopus.com/inward/record.url?scp=4444258698&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4444258698&partnerID=8YFLogxK

U2 - 10.1016/j.nuclphysb.2004.06.029

DO - 10.1016/j.nuclphysb.2004.06.029

M3 - Article

VL - 696

SP - 351

EP - 380

JO - Nuclear Physics B

JF - Nuclear Physics B

SN - 0550-3213

IS - 3

ER -