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In this paper, we construct a cost constrained spatial network by adding long-range connections to the one-dimensional circle. The probability for a long-range connection between nodes i and j is pij dij- ( 0), where dij is the lattice distance and the total length of the long-range connections is set to be =cN(c 0), where c is a positive constant and N is the network size. According to the simulation and numeric results, we find an optimal power-law exponent 0 such that the mean first-passage time is shortest. Furthermore, the shortest mean first-passage time has the power law relationships with the network size N. With the increase of network size N and the total cost , the optimal power-law exponent 0 increases monotonically and approaches 1.5.
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Keywords:
- spatial networks /
- random walks /
- total energy constraint /
- mean first-passage time
[1] Xu D, Li X, Wang X F 2007 Acta Phys. Sin. 56 1313 (in Chinese) [许丹, 李翔, 汪小帆 2007 56 1313]
[2] Boccalettia S, Latorab V, Morenod Y, Chavez M, Hwang D U 2006 Physics Reports 424 175
[3] Weng W G, Ni S J, Shen S F, Yuan H Y 2007 Acta Phys. Sin. 56 1938 (in Chinese) [翁文国, 倪顺江, 申世飞, 袁宏永 2007 56 1938]
[4] Noh JD, Rieger H 2004 Phys. Rev. Lett. 92 118701
[5] Fronczak A, Fronczak P 2009 Phys. Rev. E 80 016107
[6] Herrero C P 2005 Phys. Rev. E 71 016103
[7] Campos P R A, Moreira F G B 2005 Phys. Rev. E 71 061921
[8] Wang S P, Pei W J 2009 Physica A 388 514
[9] Masuda N, Konno N 2004 Phys. Rev. E 69 066113
[10] Lee S, Yook S H, Kima Y 2008 Physica A 387 3033
[11] Aldous D 1990 SIAM Journal on Discrete Mathematics 3 450
[12] Tadi?B, Rodgers G J, Thurner S 2007 International Journal of Bifurcation and Chaos 17 2363
[13] Kleinberg J M 2000 Nature 406 845
[14] Cajueiro D O 2009 Phys. Rev. E 79 046103
[15] Kim B J, Yoon C N, Han S K, Jeong H 2002 Phys. Rev. E 65 027103
[16] Zhou J, Liu Z H 2009 Physica A 388 1228
[17] Jiang Z H, Wang H, Gao C 2011 Acta Phys. Sin. 60 058903 (in Chinese) [姜志宏, 王晖, 高超 2011 60 058903]
[18] Zhang Z Z, Qi Y, Zhou S G, Gao S Y, Guan J H 2010 Phys. Rev. E 81 016114
[19] Zhang Z Z, Lin Y, Gao S Y, Zhou S G, Guan J H, Li M 2009 Phys. Rev. E 80 051120
[20] Zhang Z Z, Wu B, Zhang H J, Zhou S G, Guan J H, Wang Z G 2010 Phys. Rev. E 81 031118
[21] Haynes C P, Roberts A P 2008 Phys. Rev. E 78 041111
[22] Tejedor V, Bénichou O, Voituriez R 2009 Phys. Rev. E 80 065104
[23] Agliari E 2008 Phys. Rev. E 77 011128
[24] Zhang Z Z, Qi Y, Zhou S G, Xie W L, Guan J H 2009 Phys. Rev. E 79 021127
[25] Zeng A, Zhou D, Hu Y Q, Fan Y, Di Z R 2011 Physica A 390 3962
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[1] Xu D, Li X, Wang X F 2007 Acta Phys. Sin. 56 1313 (in Chinese) [许丹, 李翔, 汪小帆 2007 56 1313]
[2] Boccalettia S, Latorab V, Morenod Y, Chavez M, Hwang D U 2006 Physics Reports 424 175
[3] Weng W G, Ni S J, Shen S F, Yuan H Y 2007 Acta Phys. Sin. 56 1938 (in Chinese) [翁文国, 倪顺江, 申世飞, 袁宏永 2007 56 1938]
[4] Noh JD, Rieger H 2004 Phys. Rev. Lett. 92 118701
[5] Fronczak A, Fronczak P 2009 Phys. Rev. E 80 016107
[6] Herrero C P 2005 Phys. Rev. E 71 016103
[7] Campos P R A, Moreira F G B 2005 Phys. Rev. E 71 061921
[8] Wang S P, Pei W J 2009 Physica A 388 514
[9] Masuda N, Konno N 2004 Phys. Rev. E 69 066113
[10] Lee S, Yook S H, Kima Y 2008 Physica A 387 3033
[11] Aldous D 1990 SIAM Journal on Discrete Mathematics 3 450
[12] Tadi?B, Rodgers G J, Thurner S 2007 International Journal of Bifurcation and Chaos 17 2363
[13] Kleinberg J M 2000 Nature 406 845
[14] Cajueiro D O 2009 Phys. Rev. E 79 046103
[15] Kim B J, Yoon C N, Han S K, Jeong H 2002 Phys. Rev. E 65 027103
[16] Zhou J, Liu Z H 2009 Physica A 388 1228
[17] Jiang Z H, Wang H, Gao C 2011 Acta Phys. Sin. 60 058903 (in Chinese) [姜志宏, 王晖, 高超 2011 60 058903]
[18] Zhang Z Z, Qi Y, Zhou S G, Gao S Y, Guan J H 2010 Phys. Rev. E 81 016114
[19] Zhang Z Z, Lin Y, Gao S Y, Zhou S G, Guan J H, Li M 2009 Phys. Rev. E 80 051120
[20] Zhang Z Z, Wu B, Zhang H J, Zhou S G, Guan J H, Wang Z G 2010 Phys. Rev. E 81 031118
[21] Haynes C P, Roberts A P 2008 Phys. Rev. E 78 041111
[22] Tejedor V, Bénichou O, Voituriez R 2009 Phys. Rev. E 80 065104
[23] Agliari E 2008 Phys. Rev. E 77 011128
[24] Zhang Z Z, Qi Y, Zhou S G, Xie W L, Guan J H 2009 Phys. Rev. E 79 021127
[25] Zeng A, Zhou D, Hu Y Q, Fan Y, Di Z R 2011 Physica A 390 3962
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