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As a basic dynamical process, random walk on networks is fundamental to many branches of science, and has attracted much attention. A difficult problem in the study of random walk is how to obtain the exact solution for the mean trapping time (MTT) of this process. The MTT is defined as the mean time for the walker staring from any node in the network to first reach the trap node. In this paper, we study random walk on the Koch network with a trap located at the highest degree node and calculate the solution for MTT. The accurate expression for the MTT is obtained through the recurrence relation and the structure properties of the Koch network. We confirm the correctness of the MTT result by direct numerical calculations based on the Laplacian matrix of Koch network. It can be seen from the obtained results that in the large limit of network size, the MTT increases linearly with the size of network increasing. Comparison between the MTT result of the Koch network with that of the other networks, such as complete graph, regular lattices, Sierpinski fractals, and T-graph, shows that the Koch has a high transmission efficiency.
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Keywords:
- complex networks /
- Koch network /
- random walk /
- mean trapping time
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[2] Cami A, Deo N 2008 Networks 51 211
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[13] Fouss F, Pirotte A, Renders J, Saerens M 2007 IEEE T. Knowl. Data En. 19 355
[14] Berkhin P 2005 Internet Mathematics 2 73
[15] Zhang Z Z, Li X T, Lin Y, Chen G R 2011 J. Stat. Mech. 2011 08013
[16] Bénichou O, Coppey M, Moreau M, Suet P H, Voituriez R 2005 Phys. Rev. Lett. 94 198101
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[19] Kozak J J, Balakrishnan V 2002 Phys. Rev. E 65 021105
[20] Kozak J J, Balakrishnan V 2002 Int. J. Bifurcation Chaos Appl. Sci. Eng. 12 2379
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[22] Zhang Z Z, Qi Y, Zhou S G, Xie W L, Guan J H 2009 Phys. Rev. E 79 021127
[23] Wu S Q, Zhang Z Z, Chen G R 2011 Eur. Phys. J. B 82 91
[24] Zhang Z Z, Guan J H, Xie W L, Qi Y, Zhou S G 2009 Europhys. Lett. 86 10006
[25] Zhang Z Z, Zhou S G, Xie W L, Chen L C, Lin Y, Guan J H 2009 Phys. Rev. E 79 061113
[26] Liu J X, Kong X M 2010 Acta Phys. Sin. 59 2244 (in Chinese) [刘甲雪, 孔祥木 2010 物理学报 59 2244]
[27] Zhang Z Z, Gao S Y, Chen L C, Zhou S G, Zhang H J, Gan J H 2010 J. Phys. A: Math. Theor. 43 395101
[28] Zhang Z Z, Gao S Y, Xie W L 2010 Chaos 20 043112
[29] Zhang Z Z, Gao S Y 2011 Euro. Phys. J. B 80 209
[30] Wu B, Zhang Z Z, Chen G R 2012 J. Phys. A: Math. Theor. 45 025102
[31] Newman M E J 2002 Phys. Rev. Lett. 89 208701
[32] Kemeny J G, Snell J L 1976 Finite Markov Chains (lst Ed.) (New York: Springer) p210
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[1] Albert R, Jeong H, Barabasi A L 1999 Nature 401 130
[2] Cami A, Deo N 2008 Networks 51 211
[3] Faloutsos M, Faloutsos P, Faloutsos C 1999 Comput. Commun. Rev. 29 251
[4] Xu T, Chen R, He Y 2004 Int. J. Mod. Phys. B 18 2599
[5] Guimerá R, Amaral L A N 2004 Eur. Phys. J. B 38 381
[6] Dorogovtsev S N, Goltsev A V, Mendes J F F 2008 Rev. Mod. Phys. 80 1275
[7] Spitzer F 1964 Principles of Random Walk (1st Ed.) (Princeton, N. J.: van Nostrand) p402
[8] Lloyd A L, May R M 2001 Science 292 1316
[9] Shlesinger M F 2006 Nature 443 281
[10] Pandit S A, Amritkar R E 2001 Phys. Rev. E 63 041104
[11] Noh J D, Rieger H 2004 Phys. Rev. Lett. 92 118701
[12] Lee S M, Yook S H, Kim Y 2008 Physica A 387 3033
[13] Fouss F, Pirotte A, Renders J, Saerens M 2007 IEEE T. Knowl. Data En. 19 355
[14] Berkhin P 2005 Internet Mathematics 2 73
[15] Zhang Z Z, Li X T, Lin Y, Chen G R 2011 J. Stat. Mech. 2011 08013
[16] Bénichou O, Coppey M, Moreau M, Suet P H, Voituriez R 2005 Phys. Rev. Lett. 94 198101
[17] Loverdo C, Bénichou O, Moreau M, Voituriez R 2008 Nat. Phys. 4 134
[18] Montroll E W 1969 J. Math. Phys. 10 753
[19] Kozak J J, Balakrishnan V 2002 Phys. Rev. E 65 021105
[20] Kozak J J, Balakrishnan V 2002 Int. J. Bifurcation Chaos Appl. Sci. Eng. 12 2379
[21] Agliari E 2008 Phys. Rev. E 77 011128
[22] Zhang Z Z, Qi Y, Zhou S G, Xie W L, Guan J H 2009 Phys. Rev. E 79 021127
[23] Wu S Q, Zhang Z Z, Chen G R 2011 Eur. Phys. J. B 82 91
[24] Zhang Z Z, Guan J H, Xie W L, Qi Y, Zhou S G 2009 Europhys. Lett. 86 10006
[25] Zhang Z Z, Zhou S G, Xie W L, Chen L C, Lin Y, Guan J H 2009 Phys. Rev. E 79 061113
[26] Liu J X, Kong X M 2010 Acta Phys. Sin. 59 2244 (in Chinese) [刘甲雪, 孔祥木 2010 物理学报 59 2244]
[27] Zhang Z Z, Gao S Y, Chen L C, Zhou S G, Zhang H J, Gan J H 2010 J. Phys. A: Math. Theor. 43 395101
[28] Zhang Z Z, Gao S Y, Xie W L 2010 Chaos 20 043112
[29] Zhang Z Z, Gao S Y 2011 Euro. Phys. J. B 80 209
[30] Wu B, Zhang Z Z, Chen G R 2012 J. Phys. A: Math. Theor. 45 025102
[31] Newman M E J 2002 Phys. Rev. Lett. 89 208701
[32] Kemeny J G, Snell J L 1976 Finite Markov Chains (lst Ed.) (New York: Springer) p210
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