Uncorrelated random networks

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 67, Issue 4 2, 2003, Pages 461181-461187

  Burda, Z ^a,b   Krzywicki, A ^c  

^a BIELEFELD UNIVERSITY   (Germany)

^b JAGIELLONIAN UNIVERSITY   (Poland)

^c UNIVERSITÉ PARIS SUD   (France)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; COMPUTER SIMULATION; CORRELATION METHODS; TREES (MATHEMATICS);

RANDOM NETWORKS;

STATISTICAL MECHANICS;

EID: 0037865862     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (26)

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2. S. Dorogovtsev, J. F. F. Mendes, and A. N. Samukhin, preprint cond-mat/0204111.
3. S. Dorogovtsev, J. F. F. Mendes, and A. N. Samukhin, preprint cond-mat/0206131.
4. J. Berg and M. Lässig, preprint cond-mat/0205589.
5. M. Bauer and D. Bernard, preprint cond-mat/0206150.
6. M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Phys. Rev. E 64, 026118 (2001).
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8. A.-L. Barabási and R. Albert, Rev. Mod. Phys. 74, 47 (2002).
9. S. N. Dorogovtsev and J. F. F. Mendes, Adv. Phys. 51, 1079 (2002).
10. See any field theory textbook. A short introduction can be found in D. Bessis, C. Itzykson, and J. B. Zuber, Adv. Appl. Math. 1, 109 (1980).
11. M. Molloy and B. Reed, Random Struct. Algorithms 6, 161 (1995); Combinatorics, Probab. Comput. 7, 295 (1998).
12. M. Molloy and B. Reed, Random Struct. Algorithms 6, 161 (1995); Combinatorics, Probab. Comput. 7, 295 (1998).
13. The factor 1/2 reflects the symmetry between the two orientations of a link forming a tadpole and the factor 1/m! reflects the symmetry of link permutations in an m-tuple connection between nodes.
14. N. Metropolis et al., J. Chem. Phys. 21, 1087 (1953); K. Binder, Monte Carlo Methods in Statistical Physics (Springer, Berlin, 1986); a pedagogical introduction can be found in M. Creutz, L. Jacobs, and C. Rebbi, Phys. Rep. 95, 201 (1983).
15. N. Metropolis et al., J. Chem. Phys. 21, 1087 (1953); K. Binder, Monte Carlo Methods in Statistical Physics (Springer, Berlin, 1986); a pedagogical introduction can be found in M. Creutz, L. Jacobs, and C. Rebbi, Phys. Rep. 95, 201 (1983).
16. N. Metropolis et al., J. Chem. Phys. 21, 1087 (1953); K. Binder, Monte Carlo Methods in Statistical Physics (Springer, Berlin, 1986); a pedagogical introduction can be found in M. Creutz, L. Jacobs, and C. Rebbi, Phys. Rep. 95, 201 (1983).
17. P. Bialas, Z. Burda, and D. Johnston, Nucl. Phys. B 493, 505 (1997); 542, 413 (1999).
18. P. Bialas, Z. Burda, and D. Johnston, Nucl. Phys. B 493, 505 (1997); 542, 413 (1999).
19. j eliminates the bias. One can drop this factor in the Metropolis test by selecting the link to be rewired as follows: select at random, in the register of nodes, the node j. Then, pick at random one of the links converging to j. With this method the mapping on the balls-in-boxes model is even more obvious, since one samples directly the nodes to be updated. In the model of Ref. [14] the node orders are the only degrees of freedom.
20. S. N. Dorogovtsev, J. F. F. Mendes, and A. N. Samukhin, Phys. Rev. E 63, 062101 (2001).
21. P. L. Krapivsky and S. Redner, Phys. Rev. E 63, 066123 (2001).
22. n This does not affect the scaling properties of the cut-off.
23. r = 20.4(2), 166(6), 1273(15), and 9374(71). This behavior can be inferred from the analytic argument of Sec. III. We do not count redundant links there, but we estimate the weight of degenerate graphs, which involves the same information.
24. A. Krzywicki, preprint cond-mat/0110574.
25. A.-L. Barabási and R. Albert, Science 286, 509 (1999).
26. This construction is described in detail in Ref. [11], but not as a method of constructing nondegenerate graphs, the actual object of the study of the authors. Molloy and Reed find it useful to map the ensemble of graphs they are interested in on the less constrained ensemble of "random constructions," i.e., degenerate graphs (this approach was earlier used by other authors; see Ref. [7]). We are indebted to M. Bauer and D. Bernard for calling our attention to this point.