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Exact calculations of first-passage properties on the pseudofractal scale-free web. (English) Zbl 1374.82014
Summary: In this paper, we consider discrete time random walks on the pseudofractal scale-free web (PSFW) and we study analytically the related first passage properties. First, we classify the nodes of the PSFW into different levels and propose a method to derive the generation function of the first passage probability from an arbitrary starting node to the absorbing domain, which is located at one or more nodes of low-level (i.e., nodes with large degree). Then, we calculate exactly the first passage probability, the survival probability, the mean, and the variance of first passage time by using the generating functions as a tool. Finally, for some illustrative examples corresponding to given choices of starting node and absorbing domain, we derive exact and explicit results for such first passage properties. The method we propose can as well address the cases where the absorbing domain is located at one or more nodes of high-level on the PSFW, and it can also be used to calculate the first passage properties on other networks with self-similar structure, such as (u, v) flowers and recursive scale-free trees.
©2015 American Institute of Physics

MSC:
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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[1] Lovász, L., Combinatorics: Paul erdös is Eighty, 2, 1-46, (1993), Keszthely: Keszthely, Hungary
[2] Weiss, G., Aspects and Applications of the Random Walk, (1994), Amsterdam, North-Holland: Amsterdam, North-Holland, Netherlands · Zbl 0925.60079
[3] Havlin, S.; Ben-Avraham, D., Adv. Phys., 36, 695, (1987)
[4] Ben-Avraham, D.; Havlin, S., Diffusion and Reactions in Fractals and Disordered Systems, (2004), Cambridge University Press: Cambridge University Press, Cambridge, UK
[5] Metzler, R.; Klafter, J., Phys. Rep., 339, 1, (2000) · Zbl 0984.82032
[6] Chepizhko, O.; Peruani, F., Phys. Rev. Lett., 111, 160604, (2013)
[7] Redner, S., A Guide to First-Passage Processes, (2007), Cambridge University Press: Cambridge University Press, Cambridge, UK · Zbl 1128.60002
[8] Meyer, B.; Chevalier, C.; Voituriez, R.; Bénichou, O. O., Phys. Rev. E, 83, 051116, (2011)
[9] Condamin, S.; Bénichou, O.; Moreau, M., Phys. Rev. Lett., 95, 260601, (2005)
[10] Heijs, D. J.; Malyshev, V. A.; Knoester, J., J. Chem. Phys., 121, 4884, (2004)
[11] Kim, S. K., J. Chem. Phys., 28, 1057, (1958)
[12] Zhang, Z. Z.; Sheng, Y. B.; Hu, Z. Y.; Chen, G. R., Chaos, 22, 043129, (2012) · Zbl 1319.05120
[13] Condamin, S.; Bénichou, O.; Tejedor, V.; Voituriez, R.; Klafter, J., Nature, 450, 77, (2007)
[14] Kozak, J. J.; Balakrishnan, V., Phys. Rev. E, 65, 021105, (2002)
[15] Bentz, J. L.; Turner, J. W.; Kozak, J. J., Phys. Rev. E, 82, 011137, (2010)
[16] Zhang, Z. Z.; Guan, J. H.; Xie, W. L.; Qi, Y.; Zhou, S. G., Europhys. Lett., 86, 10006, (2009)
[17] Zhang, Z. Z.; Gao, S. Y.; Xie, W. L., Chaos, 20, 043112, (2010) · Zbl 1311.05175
[18] Comellas, F.; Miralles, A., Phys. Rev. E, 81, 061103, (2010)
[19] Zhang, Z. Z.; Qi, Y.; Zhou, S. G.; Gao, S. Y.; Guan, J. H., Phys. Rev. E, 81, 016114, (2010)
[20] Lin, Y.; Wu, B.; Zhang, Z. Z., Phys. Rev. E, 82, 031140, (2010)
[21] Zhang, Z. Z.; Lin, Y.; Ma, Y. J., J. Phys. A: Math. Theor., 44, 075102, (2011) · Zbl 1210.05160
[22] Zhang, Z. Z.; Wu, B.; Zhang, H. J.; Zhou, S. G.; Guan, J. H.; Wang, Z. G., Phys. Rev. E, 81, 031118, (2010)
[23] Zhang, Z. Z.; Li, X. T.; Lin, Y.; Chen, G. R., J. Stat. Mech.: Theory Exp., 2011, P08013
[24] Agliari, E., Phys. Rev. E, 77, 011128, (2008)
[25] Zhang, Z. Z.; Lin, Y.; Zhou, S. G.; Wu, B.; Guan, J. H., New J. Phys., 11, 103043, (2009)
[26] Wu, B.; Lin, Y.; Zhang, Z. Z.; Chen, G. R., J. Chem. Phys., 137, 044903, (2012)
[27] Agliari, E.; Sartori, F.; Cattivelli, L.; Cassi, D., Phys. Rev. E, 91, 052132, (2015)
[28] Agliari, E.; Burioni, R., Phys. Rev. E, 80, 031125, (2009)
[29] Agliari, E.; Burioni, R.; Manzotti, A., Phys. Rev. E, 82, 011118, (2010)
[30] Peng, J. H.; Xu, G., J. Chem. Phys., 40, 134102, (2014)
[31] Peng, J. H.; Xiong, J.; Xu, G., Physica A, 407, 231, (2014) · Zbl 1395.60112
[32] Bénichou, O.; Chevalier, C.; Klafter, J.; Meyer, B.; Voituriez, R., Nat. Chem., 2, 472, (2010)
[33] Bénichou, O.; Voituriez, R., Phys. Rep., 539, 225, (2014) · Zbl 1358.60003
[34] Dorogovtsev, S. N.; Goltsev, A. V.; Mendes, J. F. F., Phys. Rev. E, 65, 066122, (2002)
[35] Zhang, Z. Z.; Rong, L. L.; Zhou, S. G., Physica A, 377, 329, (2007)
[36] Zhang, Z. Z.; Zhou, S. G.; Chen, L. C., Eur. Phys. J. B, 58, 337, (2007)
[37] Zhang, Z. Z.; Liu, H. X.; Wu, B.; Zhou, S. G., Europhys. Lett., 90, 68002, (2010)
[38] Bollt, E. M.; ben-Avraham, D., New J. Phys., 7, 26, (2005)
[39] Zhang, Z. Z.; Qi, Y.; Zhou, S. G.; Xie, W. L.; Guan, J. H., Phys. Rev. E, 79, 021127, (2009)
[40] Kahng, B.; Redner, S., J. Phys. A, 22, 887, (1989)
[41] Rudnick, J.; Gaspari, G., Elements of the Random Walk: An introduction for Advanced Students and Researchers, (2004), Cambridge University Press: Cambridge University Press, Cambridge, UK · Zbl 1086.60003
[42] Meyer, B.; Agliari, E.; Bénichou, O.; Voituriez, R., Phys. Rev. E, 85, 026113, (2012)
[43] Rozenfeld, H. D.; Havlin, S.; ben-Avraham, D., New J. Phys., 9, 175, (2007)
[44] Actually, the PSFW is self-similar in a weak sense: it contains subgraphs that resemble the whole, but lacks the affine transformation of scale associated with self-similarity in fractals. In fact, its dimension is infinite and this is why the name pseudofractal.
[45] When k = 0 the starting nodes are two and topologically equivalent; therefore, the average over starting nodes simply returns the mean time obtained for any of the two starting nodes.
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