Abstract.
We make a mapping from Sierpinski fractals to a new class of networks, the incompatibility networks, which are scale-free, small-world, disassortative, and maximal planar graphs. Some relevant characteristics of the networks such as degree distribution, clustering coefficient, average path length, and degree correlations are computed analytically and found to be peculiarly rich. The method of network representation can be applied to some real-life systems making it possible to study the complexity of real networked systems within the framework of complex network theory.
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References
R. Albert, A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002)
S.N. Dorogovtsev, J.F.F. Mendes, Adv. Phys. 51, 1079 (2002)
M.E.J. Newman, SIAM Review 45, 167 (2003)
S. Boccaletti, V. Latora, Y. Moreno, M. Chavezf, D.-U. Hwanga, Phy. Rep. 424, 175 (2006)
L.da.F. Costa, F.A. Rodrigues, G. Travieso, P.R.V. Boas, Adv. Phys. 56, 167 (2007)
A.-L. Barabási, R. Albert, Science 286, 509 (1999)
D.J. Watts, H. Strogatz, Nature (London) 393, 440 (1998)
R. Pastor-Satorras, A. Vespignani, Phys. Rev. Lett. 86, 3200 (2001)
D.S. Callaway, M.E.J. Newman, S.H. Strogatz, D.J. Watts, Phys. Rev. Lett. 85, 5468 (2000)
R. Cohen, K. Erez, D. ben-Avraham, S. Havlin, Phys. Rev. Lett. 86, 3682 (2001)
M. Barahona, L.M. Pecora, Phys. Rev. Lett. 89, 054101 (2002)
G. Szabó, G. Fáth, Phy. Rep. 446, 97 (2007)
S.N. Dorogovtsev, J.F.F. Mendes, A.N. Samukhin, Phys. Rev. Lett. 85, 4633 (2000)
R. Albert, A.-L. Barabási, Phys. Rev. Lett. 85, 5234 (2000)
G. Bianconi, A.-L. Barabási, Europhys. Lett. 54, 436 (2001)
Z.Z. Zhang, L.L. Rong, B. Wang, S.G. Zhou, J.H. Guan, Physica A 380, 639 (2007)
A. Barrat, M. Barthélemy, A. Vespignani, Phys. Rev. Lett. 92, 228701 (2004)
W.X. Wang, B.H. Wang, B. Hu, G. Yan, Q. Ou, Phys. Rev. Lett. 94, 188702 (2005)
Z.Z. Zhang, S.G. Zhou, L.J. Fang, J.H. Guan, Y.C. Zhang, Europhys. Lett. 79, 38007 (2007)
Z.Z. Zhang, S.G. Zhou, L.C. Chen, J.H. Guan, L.J. Fang, Y.C. Zhang, Eur. Phys. J. B 59 99 (2007)
M. Hinczewski, Phys. Rev. E 75, 061104 (2007)
Z.Z. Zhang, S.G. Zhou, Z.Y. Wang, Z. Shen, J. Phys. A 40, 11863 (2007)
W. Sierpinski, Comptes Rendus (Paris) 160, 302 (1915)
C.A. Reiter, Comput. Graphics 18, 885 (1994)
S. Hutchinson, Indiana Univ. Math. J. 30, 713 (1981)
B.M. Hambly, Probab. Theory Related Fields 94, 1 (1992)
M. Rosvall, A. Trusina, P. Minnhagen, K. Sneppen, Phys. Rev. Lett. 94, 028701 (2005)
C. Haslinger, P.F. Stadler, Bull. Math. Biol. 61, 437 (1999)
A. Kabakçıoğlu, A.L. Stella, Phys. Rev. E 70, 011802 (2004)
M. Vendruscolo, B. Subramanian, I. Kanter, E. Domany, J. Lebowitz, Phys. Rev. E 59, 977 (1999)
A. Kabakçıoğlu, A.L. Stella, Phys. Rev. E 72, 055102(R) (2005)
J.L. Jacobsen, N. Read, H. Saleur, Phys. Rev. Lett. 93, 038701 (2004)
D.B. West, Introduction to Graph Theory (Prentice-Hall, Upper Saddle River, NJ, 2001)
J.S. Andrade Jr., H.J. Herrmann, R.F.S. Andrade, L.R.da Silva, Phys. Rev. Lett. 94, 018702 (2005)
Z.Z. Zhang, F. Comellas, G. Fertin, L.L. Rong, J. Phys. A 39, 1811 (2006)
T. Zhou, G. Yan, B.H. Wang, Phys. Rev. E 71, 046141 (2005)
Z.Z. Zhang, L.L. Rong, F. Comellas, Physica A 364, 610 (2006)
Z.Z. Zhang, L.L. Rong, S.G. Zhou, Phys. Rev. E 74, 046105 (2006)
S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, Phys. Rev. E 65, 066122 (2002)
F. Comellas, G. Fertin, A. Raspaud, Phys. Rev. E 69, 037104 (2004)
Z.Z. Zhang, S.G. Zhou, L.C. Chen, Eur. Phys. J. B 58, 337 (2007)
E. Ravasz, A.-L. Barabási, Phys. Rev. E 67, 026112 (2003)
M. Hinczewski, A.N. Berker, Phys. Rev. E 73, 066126 (2006)
Z.Z. Zhang, L.C. Chen, S.G. Zhou, L.J. Fang, J.H. Guan, Y.C. Zhang, e-print arXiv:0606.3491
S. Maslov, K. Sneppen, Science 296, 910 (2002)
R. Pastor-Satorras, A. Vázquez, A. Vespignani, Phys. Rev. Lett. 87, 258701 (2001)
A. Vázquez, R. Pastor-Satorras, A. Vespignani, Phys. Rev. E 65, 066130 (2002)
M.E.J. Newman, Phys. Rev. Lett. 89, 208701 (2002)
M.E.J. Newman, Phys. Rev. E 67, 026126 (2003)
Z.Z. Zhang, S.G. Zhou, Physica A, 380, 621 (2007)
J.P.K. Doye, C.P. Massen, Phys. Rev. E 71, 016128 (2005)
Z.Z. Zhang, L.L. Rong, S.G. Zhou, Physica A 377, 329 (2007)
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Zhang, Z., Zhou, S., Zou, T. et al. Incompatibility networks as models of scale-free small-world graphs. Eur. Phys. J. B 60, 259–264 (2007). https://doi.org/10.1140/epjb/e2007-00344-7
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DOI: https://doi.org/10.1140/epjb/e2007-00344-7