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The couplings among different networks facilitate their communications, while at the same time they also bring the risk of enhancing the wide spread of cascading failures to the coupled networks. Given that there is usually the time-delay during the spread of failures and more than one coupling link a node might possess, a cascading failure model for scale-free multi-coupling-link coupled networks is built in this paper, based on time-delay coupled map lattices (CML) model, which may be wider representative than previous models. Our research shows that in BA (Barabási-Albert) scale-free coupled networks, there is a threshold hT ≈ 3: when the coupling strength is bellow this threshold, the stronger coupling strength corresponds to a lower invulnerability; and vice versa, the stronger coupling strength would bring a higher invulnerability. In addition, our studies show that the presence of time-delay not only prolongs the failure spreading time during which measures can be taken to suppress cascading failures, but also has a significant influence on the eventual cascading size, for detail, if intra-layer time-delay τ1 and inter-layer time-delay τ2 can have any values, then the multiples of the two numbers will cause larger cascading size. We hope our research can provide a reference for building high-invulnerable coupled networks or the increase of the invulnerability of the coupled networks.
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Keywords:
- coupled networks /
- coupled map lattices /
- coupling strength /
- time-delay
[1] Watts D J, Strogatz S H 1998 Nature 393 440
[2] Barabási A L, Albert R 1999 Science 286 509
[3] Liu G, Li Y S 2012 Acta Phys. Sin. 61 108901 (in Chinese)[刘刚, 李永树2012 61 108901]
[4] Boccaletti S, Latora V, Moreno Y, Chavez M, Hwanga D U 2006 Phys. Rep. 424 175
[5] Zhao L, Park K, Lai Y C 2004 Phys. Rev. E 70 035101 (R)
[6] Motter A E, Lai Y C 2002 Phys. Rev. E 66 065102
[7] Crucitti P, Latora V, Marchiori M 2004 Phys. Rev. E 69 045104(R)
[8] Wang X F, Xu J 2004 Phys. Rev. E 70 056113
[9] Wang J W 2012 Physica A 391 4004
[10] Wang J W, Rong L L, Zhang L, Zhang Z Z 2008 Physica A 387 6671
[11] Ash J, Newth D 2007 Physica A 380 673
[12] Motter A E 2004 Phys. Rev. Lett. 93 098701
[13] Dou B L, Wang X G, Zhang S Y 2010 Physica A 389 2310
[14] Hu K, Hu T, Tang Y 2010 Chin. Phys. B 19 080206
[15] Buldyrev S V, Parshani R, Paul G, Stanley H E, Havlin S 2010 Nature 464 1025
[16] Shao J, Buldyrev S V, Havlin S, Stanley H E 2011 Phys. Rev. E 83 036116
[17] Gao J X, Buldyrev S V, Stanley H E, Havlin S 2012 Nature Physics. 8 40
[18] Li W, Bashan A, Buldyrev S V, Stanley H E, Havlin S 2012 Phys. Rev. Lett. 108 228702
[19] Huang X Q, Shao S, Wang H J, Buldyrev S V, Stanley H E, Havlin S 2013 EPL 101 18002
[20] Brummitt C D, D Souza R M, Leicht E A 2012 PNAS 109 E680
[21] Tan F, Xia Y X, Zhang W P, Jin X Y 2013 EPL 102 28009
[22] Qiu Y Z 2013 Physica A 392 1920
[23] Xu J, Wang X F 2005 Physica A 349 685
[24] Cui D, Gao Z Y, Zhao X M 2008 Chin. Phys. B 17 1703
[25] Cui D, Gao Z Y, Zheng J F 2009 Chin. Phys. B 18 992
[26] Holme P, Kim B J, Yoon C N, Han S K 2002 Phys. Rev. E 65 056109
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[1] Watts D J, Strogatz S H 1998 Nature 393 440
[2] Barabási A L, Albert R 1999 Science 286 509
[3] Liu G, Li Y S 2012 Acta Phys. Sin. 61 108901 (in Chinese)[刘刚, 李永树2012 61 108901]
[4] Boccaletti S, Latora V, Moreno Y, Chavez M, Hwanga D U 2006 Phys. Rep. 424 175
[5] Zhao L, Park K, Lai Y C 2004 Phys. Rev. E 70 035101 (R)
[6] Motter A E, Lai Y C 2002 Phys. Rev. E 66 065102
[7] Crucitti P, Latora V, Marchiori M 2004 Phys. Rev. E 69 045104(R)
[8] Wang X F, Xu J 2004 Phys. Rev. E 70 056113
[9] Wang J W 2012 Physica A 391 4004
[10] Wang J W, Rong L L, Zhang L, Zhang Z Z 2008 Physica A 387 6671
[11] Ash J, Newth D 2007 Physica A 380 673
[12] Motter A E 2004 Phys. Rev. Lett. 93 098701
[13] Dou B L, Wang X G, Zhang S Y 2010 Physica A 389 2310
[14] Hu K, Hu T, Tang Y 2010 Chin. Phys. B 19 080206
[15] Buldyrev S V, Parshani R, Paul G, Stanley H E, Havlin S 2010 Nature 464 1025
[16] Shao J, Buldyrev S V, Havlin S, Stanley H E 2011 Phys. Rev. E 83 036116
[17] Gao J X, Buldyrev S V, Stanley H E, Havlin S 2012 Nature Physics. 8 40
[18] Li W, Bashan A, Buldyrev S V, Stanley H E, Havlin S 2012 Phys. Rev. Lett. 108 228702
[19] Huang X Q, Shao S, Wang H J, Buldyrev S V, Stanley H E, Havlin S 2013 EPL 101 18002
[20] Brummitt C D, D Souza R M, Leicht E A 2012 PNAS 109 E680
[21] Tan F, Xia Y X, Zhang W P, Jin X Y 2013 EPL 102 28009
[22] Qiu Y Z 2013 Physica A 392 1920
[23] Xu J, Wang X F 2005 Physica A 349 685
[24] Cui D, Gao Z Y, Zhao X M 2008 Chin. Phys. B 17 1703
[25] Cui D, Gao Z Y, Zheng J F 2009 Chin. Phys. B 18 992
[26] Holme P, Kim B J, Yoon C N, Han S K 2002 Phys. Rev. E 65 056109
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