Potts model on random trees

We study the Potts model on locally tree-like random graphs of arbitrary degree distribution. Using a population dynamics algorithm we numerically solve the problem exactly. We confirm our results with simulations. Comparisons with a previous approach are made, showing where its assumption of uniform local fields breaks down for networks with nodes of low degree.

[1]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[2]  R. Baxter.  Exactly solved models in statistical mechanics , 1982 .

[3]  M. Newman,et al.  Random graphs with arbitrary degree distributions and their applications. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Cohen,et al.  Resilience of the internet to random breakdowns , 2000, Physical review letters.

[5]  S. Bornholdt,et al.  Coevolutionary games on networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Alessandro Vespignani,et al.  Epidemic spreading in scale-free networks. , 2000, Physical review letters.

[7]  Sergey N. Dorogovtsev,et al.  Ising Model on Networks with an Arbitrary Distribution of Connections , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Ginestra Bianconi,et al.  Clogging and self-organized criticality in complex networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  F. Ball,et al.  Epidemics with two levels of mixing , 1997 .

[10]  R. Baxter.  Potts model at the critical temperature , 1973 .

[11]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[12]  S. N. Dorogovtsev,et al.  Potts model on complex networks , 2004 .

[13]  H. Herrmann,et al.  Self-organized criticality on small world networks , 2001, cond-mat/0110239.

[14]  R. Zecchina,et al.  Ferromagnetic ordering in graphs with arbitrary degree distribution , 2002, cond-mat/0203416.

[15]  S. N. Dorogovtsev,et al.  Evolution of networks , 2001, cond-mat/0106144.

[16]  D S Callaway,et al.  Network robustness and fragility: percolation on random graphs. , 2000, Physical review letters.

[17]  Per Bak,et al.  Small Worlds: The Dynamics of Networks between Order and Randomness, by Duncan J. Watts , 2000 .

[18]  Derek de Solla Price,et al.  A general theory of bibliometric and other cumulative advantage processes , 1976, J. Am. Soc. Inf. Sci..

[19]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[20]  D. Watts,et al.  Small Worlds: The Dynamics of Networks between Order and Randomness , 2001 .

[21]  Wang,et al.  Nonuniversal critical dynamics in Monte Carlo simulations. , 1987, Physical review letters.

[22]  S. Havlin,et al.  Scale-free networks are ultrasmall. , 2002, Physical review letters.

[23]  M. Kuperman,et al.  Social games in a social network. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  M. Newman,et al.  Scaling and percolation in the small-world network model. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[25]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..