Abstract
We study the percolation transition in evolving scale-free networks. A new node is added at each step and is connected to a random number of old nodes according to the preferential attachment mechanism. We give the critical value of the emergence of the giant component and prove that the transition is of infinite order. We also obtain asymptotic expressions for the cluster size distribution in the subcritical, critical, and supercritical regimes.
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References
A. L. Barabási and R. Alber, Science, 286, 509–512 (1999); arXiv:cond-mat/9910332v1 (1999).
M. E. J. Newman, A. L. Barabasi, and D. J. Watts, The Structure and Dynamics of Networks, Princeton Univ. Press, Princeton, N. J. (2006).
L. F. Costa, O. N. Oliveira Jr, G. Travieso, F. A. Rodrigues, P. R. Villas Boas, L. Antiqueira, M. P. Viana, and L. E. Correa Rocha, Adv. Phys., 60, 329–412 (2011); arXiv:0711.3199v3 [physicssoc-ph] (2007).
A. L. Barabási, R. Albert, and H. Jeong, Phys. A, 272, 173–187 (1999); arXiv:cond-mat/9907068v1 [cond-mat. dis-nn] (1999).
P. L. Krapivsky, S. Redner, and F. Leyvraz, Phys. Rev. Lett., 85, 4629–4632 (2000); arXiv:cond-mat/0005139v2 (2000).
S. N. Dorogovtsev and J. F. F. Mendes, Phys. Rev. E, 63, 025101 (2001); arXiv:cond-mat/0009065v1 (2000).
A. Barrat, M. Barthélemy, and A. Vespignani, Phys. Rev. Lett., 92, 228701 (2004); arXiv:cond-mat/0401057v2 (2004).
R. Albert and A. L. Barabási, Phys. Rev. Lett., 85, 5234–5237 (2000); arXiv:cond-mat/0005085v1 (2000).
R. Albert and A. L. Barabási, Rev. Modern Phys., 74, 47–97 (2002); arXiv:cond-mat/0106096v1 (2001).
R. Cohen, K. Erez, D. Ben-Avraham, and S. Havlin, Phys. Rev. Lett., 85, 4626–4628 (2000); arXiv:cond-mat/0007048v2 (2000).
D. S. Callaway, M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Phys. Rev. Lett., 85, 5468–5471 (2000); arXiv:cond-mat/0007300v2 (2000).
S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, Rev. Modern Phys., 80, 1275–1335 (2008); arXiv: 0705.0010v6 [cond-matstat-mech] (2007).
Z. Xu, Y. Higuchi, and C. Hu, Theor. Math. Phys., 172, 901–910 (2012).
P. Erdôs and A. Rényi, Publ. Math., 6, 290–297 (1959).
P. Erdôs and A. Rényi, Pub. Math. Inst. Hung. Acad. Sci. Ser. A, 5, 17–61 (1960).
D. S. Callaway, J. E. Hopcroft, J. M. Kleinberg, M. E. J. Newman, and S. H. Strogatz, Phys. Rev. E, 64, 041902 (2001); arXiv:cond-mat/0104546v2 (2001).
S. N. Dorogovtsev, J. F. F. Mendes, and A. N. Samukhin, Phys. Rev. E, 64, 066110 (2001); arXiv:cond-mat/0106141v2 (2001).
J. Kim, P. L. Krapivsky, B. Kahng, and S. Redner, Phys. Rev. E, 66, 055101 (2002); arXiv:cond-mat/0203167v3 (2002).
S. Coulomb and M. Bauer, Eur. Phys. J. B, 35, 377–389 (2003); arXiv:cond-mat/0212371v1 (2002).
P. L. Krapivsky and B. Derrida, Phys. A, 340, 714–724 (2004); arXiv:cond-mat/0408161v1 (2004).
A. D. Polyanin and V. F. Zaitsev, Handbook on Ordinary Differential Equations [in Russian], Fizmatlit, Moscow (2001); English transl.: Handbook of Exact Solutions for Ordinary Differential Equations, Chapman and Hall/CRC Press, Boca Raton (2002).
G. Polya and G. Latta, Complex Variables, Wiley, New York (1974).
D. Lancaster, J. Phys. A, 35, 1179–1194 (2002).
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This research was supported by the National Natural Science Foundation of China (Grant No. 11401368).
Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 186, No. 3, pp. 496–507, March, 2016.
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Li, Y., Sun, Y. Emergence of the giant component in preferential-attachment growing networks. Theor Math Phys 186, 430–439 (2016). https://doi.org/10.1134/S0040577916030107
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DOI: https://doi.org/10.1134/S0040577916030107