Evolution of scale-free random graphs: Potts model formulation

被引:62
作者
Lee, DS
Goh, KI
Kahng, B
Kim, D
机构
[1] Seoul Natl Univ, Sch Phys, Seoul 151747, South Korea
[2] Seoul Natl Univ, Ctr Theoret Phys, Seoul 151747, South Korea
关键词
D O I
10.1016/j.nuclphysb.2004.06.029
中图分类号
O412 [相对论、场论]; O572.2 [粒子物理学];
学科分类号
摘要
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 --> I 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 proportional to i(-mu) (0 < mu < 1) so that the resulting graph is scale-free with the degree exponent. lambda = 1 + 1/mu. 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 lambda > 3 and 2 < lambda < 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. (C) 2004 Elsevier B.V. All rights reserved.
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页码:351 / 380
页数:30
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