Reciprocal relation between the fractal and the small-world properties of complex networks

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

Volumn 82, Issue 3, 2010, Pages

  Kawasaki, F ^a   Yakubo, K ^a  

^a HOKKAIDO UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

CHARACTERISTIC LENGTH; COMPLEX NETWORKS; CROSSOVER BEHAVIOR; FRACTALITY; LENGTH SCALE; PERCOLATION TRANSITION; RECIPROCAL RELATIONS; SMALL WORLDS;

SOLVENTS;

FRACTAL DIMENSION;

EID: 78651253818     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.82.036113     Document Type: Article

References (44)

1. D. R. Albert and A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002). 10.1103/RevModPhys.74.47
2. S. N. Dorogovtsev and J. F. F. Mendes, Evolution of Networks: From Biological Nets to the Internet and WWW (Oxford University Press, Oxford, 2003).
3. Handbook of Graphs and Networks: From the Genome to the Internet, edited by, S. Bornholdt, and, H. G. Schuster, (WILEY-VCH, Weinheim, 2003).
4. D. J. Watts and S. H. Strogatz, Nature (London) 393, 440 (1998). 10.1038/30918
5. A.-L. Barabási and R. Albert, Science 286, 509 (1999). 10.1126/science.286.5439.509
6. C. Song, S. Havlin, and H. A. Makse, Nature (London) 433, 392 (2005). 10.1038/nature03248
7. G. Csányi and B. Szendröi, Phys. Rev. E 70, 016122 (2004). 10.1103/PhysRevE.70.016122
8. C. Song, S. Havlin, and H. A. Makse, Nat. Phys. 2, 275 (2006). 10.1038/nphys266
9. L. Barrière, F. Comellas, and C. Dalfó, J. Phys. A 39, 11739 (2006). 10.1088/0305-4470/39/38/003
10. Z. Z. Zhang, S. G. Zhou, L. C. Chen, and J. Guan, Eur. Phys. J. B 64, 277 (2008). 10.1140/epjb/e2008-00299-1
11. M. Guida and F. Maria, Chaos, Solitons Fractals 31, 527 (2007). 10.1016/j.chaos.2006.02.007
12. J. S. Kim, K.-I. Goh, G. Salvi, E. Oh, B. Kahng, and D. Kim, Phys. Rev. E 75, 016110 (2007). 10.1103/PhysRevE.75.016110
13. J. S. Kim, K.-I. Goh, B. Kahng, and D. Kim, New J. Phys. 9, 177 (2007). 10.1088/1367-2630/9/6/177
14. J. S. Kim, K.-I. Goh, B. Kahng, and D. Kim, Chaos 17, 026116 (2007). 10.1063/1.2737827
15. L. H. Zou, W. J. Pei, T. Lia, Z. Hea, and Y. Cheung, Physica A 380, 592 (2007). 10.1016/j.physa.2007.02.060
16. L. K. Gallos, C. M. Song, and H. A. Makse, Physica A 386, 686 (2007). 10.1016/j.physa.2007.07.069
17. S. Carmi, P. L. Krapivsky, and D. ben-Avraham, Phys. Rev. E 78, 066111 (2008). 10.1103/PhysRevE.78.066111
18. R. F. S. Andrade, J. G. V. Miranda, S. T. R. Pinho, and T. P. Lobao, Eur. Phys. J. B 61, 247 (2008). 10.1140/epjb/e2008-00049-5
19. M. Kitsak, S. Havlin, G. Paul, M. Riccaboni, F. Pammolli, and H. E. Stanley, Phys. Rev. E 75, 056115 (2007). 10.1103/PhysRevE.75.056115
20. F. Radicchi, J. J. Ramasco, A. Barrat, and S. Fortunato, Phys. Rev. Lett. 101, 148701 (2008). 10.1103/PhysRevLett.101.148701
21. F. Radicchi, A. Barrat, S. Fortunato, and J. J. Ramasco, Phys. Rev. E 79, 026104 (2009). 10.1103/PhysRevE.79.026104
22. H. D. Rozenfeld, C. Song, and H. A. Makse, Phys. Rev. Lett. 104, 025701 (2010). 10.1103/PhysRevLett.104.025701
23. R. Cohen, K. Erez, D. ben-Avraham, and S. Havlin, Phys. Rev. Lett. 86, 3682 (2001). 10.1103/PhysRevLett.86.3682
24. R. Cohen and S. Havlin, Physica A 336, 6 (2004). 10.1016/j.physa.2004.01. 005
25. S. N. Dorogovtsev and A. V. Goltsev, Rev. Mod. Phys. 80, 1275 (2008). 10.1103/RevModPhys.80.1275
26. R. Albert, H. Jeong, and A.-L. Barabási, Nature (London) 401, 130 (1999). 10.1038/43601
27. G. Concas, M. F. Locci, M. Marchesi, S. Pinna, and I. Turnu, Europhys. Lett. 76, 1221 (2006). 10.1209/epl/i2006-10384-1
28. C. R. Myers, Phys. Rev. E 68, 046116 (2003). 10.1103/PhysRevE.68.046116
29. Center for Complex Network Research (CCNR), http://www.nd.edu/networks/ resources.htm
30. Center for Cancer Systems Biology (CCSB), http://ccsb.dfci.harvard.edu/ web/www/ccsb/
31. We should note that the network data set for the PIN of H. sapiens is different from that used in Ref.. Therefore, the fractal dimension of this network differs from that reported in Ref..
32. C. Song, L. K. Gallos, S. Havlin, and H. A. Makse, J. Stat. Mech.: Theory Exp. (2007), P03006. 10.1088/1742-5468/2007/03/P03006
33. Although we first cover high degree hubs for reducing the computing time, the fractal property obtained by this method characterizes the entire structure of the network without removing hubs.
34. The SHM model has actually four parameter, m, n, a, and g. Meanings of these parameters are the ratio of the number of nodes in the renormalized graph to that in the prerenormalized one (m), the ratio of the degree of the renormalized node to that of the prerenormalized node (n), the ratio of the diameter of the renormalized network to that of the prerenormalized network (a), and the generation (g). We choose m = 2, n = 6, a = 2, and g = 6 for Figs. and m = 2, n = 6, a = 2, and g = 7 for Fig..
35. P. Erdos and A. Rényi, Publ. Math. (Debrecen) 6, 290 (1959).
36. G. Caldarelli, A. Capocci, P. De Los Rios, and M. A. Munõz, Phys. Rev. Lett. 89, 258702 (2002). 10.1103/PhysRevLett.89.258702
37. D. Garlaschelli, A. Capocci, and G. Caldarelli, Nat. Phys. 3, 813 (2007). 10.1038/nphys729
38. M. Mitobe and K. Yakubo, J. Phys. Soc. Jpn. 78, 124002 (2009). 10.1143/JPSJ.78.124002
39. M. Molloy and B. Reed, Random Struct. Algorithms 6, 161 (1995). 10.1002/rsa.3240060204
40. M. Molloy and B. Reed, Combinatorics, Probab. Comput. 7, 295 (1998). 10.1017/S0963548398003526
41. R. Cohen, K. Erez, D. ben-Avraham, and S. Havlin, Phys. Rev. Lett. 85, 4626 (2000). 10.1103/PhysRevLett.85.4626
42. D. S. Callaway, M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Phys. Rev. Lett. 85, 5468 (2000). 10.1103/PhysRevLett.85.5468
43. M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Phys. Rev. E 64, 026118 (2001). 10.1103/PhysRevE.64.026118
44. A. Bunde and S. Havlin, in Fractals and Disordered Systems, edited by, A. Bunde, and, S. Havlin, (Springer, New York, 1996).