Dynamic critical behavior of the [Formula presented] model in small-world networks

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 67, Issue 3, 2003, Pages 4-

  Medvedyeva, Kateryna ^a   Holme, Petter ^a   Minnhagen, Petter ^a,b   Kim, Beom Jun ^c  

^a UMEÅ UNIVERSITY   (Sweden)

^b NORDITA   (Denmark)

^c Ajou University   (South Korea)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85037232477     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.67.036118     Document Type: Article

References (36)

1. S.H. Strogatz, Science 284, 79 (1999)
2. S.N. Dorogovtsev, J.F.F. Mendes, Adv. Phys. 51, 1079 (2002)
3. R. Albert and A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002).
4. A. Barrat and M. Weigt, Eur. Phys. J. B 13, 547 (2000).
5. B.J. Kim, H. Hong, P. Holme, G.S. Jeon, P. Minnhagen, and M.Y. Choi, Phys. Rev. E 64, 056135 (2001).
6. M. Gitterman, J. Phys. A 33, 8373 (2000)
7. P. Svenson, Phys. Rev. E 64, 036122 (2001)
8. Phys. Rev. EA. Pȩkalski, 64, 057104 (2001)
9. Phys. Rev. EH. Hong, B.J. Kim, and M.Y. Choi, 66, 018101 (2002)
10. Phys. Rev. EC.P. Herrero, 65, 066110 (2002)
11. A. Aleksiejuk, J.A. Holyst, and D. Stauffer, Physica A 310, 260 (2002)
12. G. Bianconi, e-print cond-mat/0204455;
13. A. Aleksiejuk-Fronczak, e-print cond-mat/0206027;
14. D. Boyer and O. Miramontes, e-print cond-mat/0210352.
15. P. Holme, e-print cond-mat/030113.
16. D.J. Watts and S.H. Strogatz, Nature (London) 393, 440 (1998).
17. J. Toner and Y. Tu, Phys. Rev. Lett. 75, 4326 (1995).
18. Z.B. Li, L. Schülke, and B. Zheng, Phys. Rev. Lett. 74, 3396 (1995).
19. M. Silvèro Soares, J. Kamphorst Leal da Silva, and F.C. Sá Barreto, Phys. Rev. B 55, 1021 (1997).
20. H.J. Luo and B. Zheng, Mod. Phys. Lett. B 11, 15412 (2000).
21. B.J. Kim, P. Minnhagen, S.K. Oh, and J.S. Chung, Phys. Rev. B 64, 024406 (2001).
22. N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller, J. Chem. Phys. 21, 1087 (1953).
23. B.J. Kim, Phys. Rev. B 63, 024503 (2001)
24. K. Medvedyeva, B.J. Kim, and P. Minnhagen, Physica C 355, 6 (2001).
25. L.M. Jensen, B.J. Kim, and P. Minnhagen, Phys. Rev. B 61, 15 412 (2000).
26. S. Wansleben and D.P. Landau, Phys. Rev. B 43, 6006 (1991).
27. In d dimensional regular lattices, we have (Formula presented) and (Formula presented) with the linear size of the system L, the correlation length (Formula presented) and the correlation volume (Formula presented) Accordingly, at (Formula presented) we have (Formula presented) Since (Formula presented) diverges as (Formula presented) we also have (Formula presented) For systems where the correlation length is not well defined, but the correlation volume is well defined (as in the present work), (Formula presented) and (Formula presented) need to be measured to detect the dynamic and static universality class of the system. In the mean-field systems corresponding to (Formula presented) (Formula presented) and (Formula presented) are expected.
28. M.E.J. Newman and D.J. Watts, Phys. Rev. E 60, 7332 (1999).
29. M. Barthelemy and L.A.N. Amaral, Phys. Rev. Lett. 82, 3180 (1999)
30. Phys. Rev. Lett.M. BarthelemyL.A.N. Amaral82, 5180 (2000)
31. M. Ozana, Europhys. Lett. 55, 762 (2001).
32. If we assume a logarithmic correction to the characteristic length scale (Formula presented) as it was suggested by A.J. Bray, A.J. Briant, and D.K. Jervis, Phys. Rev. Lett. 84, 1503 (2000), we obtain a value (Formula presented) closer to 0.5.
33. To arrive at the result (Formula presented) we set the threshold for breakdown of scaling (i.e., where we would have needed yet larger system sizes) to (Formula presented) and average all points for larger P values.
34. H.E. Stanley, S.V. Buldyrev, A.L. Goldberger, S. Havlin, C.-K. Peng, and M. Simons, Physica A 200, 4 (1993).
35. W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge University Press, Cambridge, 1992).
36. B. Efron, The Jackknife, the Bootstrap, and Other Resampling Plans (SIAM, Philadelphia, 1982).