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Interdependent networks are extremely fragile because a very small node failure in one network would trigger a cascade of failures in the entire system. Therefore, the resilience of interdependent networks is always a critical issue studied by researchers in different fields. Existing studies mainly focused on protecting several influential nodes for enhancing robustness of interdependent networks before the networks suffer random failures. In reality, it is necessary to repair a failing interdependent network in time to prevent it from suffering total breakdown. Recent investigations introduce a failure-recovery model for studying the concurrent failure process and recovery process of interdependent networks based on a random recovery strategy. This stochastic strategy covers repairing a small fraction of mutual boundary nodes which are the failed neighbors of the giant connected component of each network, with a random probability of recovery . Obviously, the random recovery is simple and straightforward. Here, we analyze the recovery process of interdependent networks with two types of connectivity links, i.e., the first-type connectivity links and the second-type connectivity links, which represent the mutual boundary nodes(being also failed nodes) linked to survival nodes in current giant connected component, and linked to failed nodes out of current giant connected component in networks, respectively. We find that when mutual boundary nodes have more first-type connectivity links, the current giant connected component has higher average degree and immediately makes better interdependent network resilience, on the other hand, more second-type connectivity links generate more candidates during the recovery procedure, and indirectly make better system resilience. In short, two types of connectivity links of mutual boundary nodes both have great effects on the resilience of interdependent networks during the recovery. In this paper, we propose a new recovery strategy (preferential recovery based on connectivity link, or PRCL) to identify the mutual boundary node recovery influence in interdependent networks, based on the failure-recovery model. By defining two indexes that represent the numbers of first-type and links second-type connectivity links, respectively, we calculate the boundary influence with one parameter f by combining together with two indexes. After calculating all boundary nodes in the current process, we obtain a boundary importance index which is more accurate to indicate recovery influence of boundary node for each boundary node in interdependent networks. Our strategy is applied to interdependent networks constructed by ER random network or/and scale-free network with the same average degree. And a dynamical model of random failure based on percolation theory is used to make a comparison of performance between PRCL and other recovery strategies(including random recovery, preferential recovery based on degree, preferential recovery based on local centrality) in terms of four quantitative indices, i.e., probability of existence of the giant connected component, number of iteration steps, recovery robustness and average degree of the steady state of the giant connected component. Experiments on different interdependent networks (ER-ER/SF-SF/ER-SF/SF-ER) demonstrate that with a very small number of mutual boundary node recoveries by PRCL strategy, the resilience and robustness of entire system under the recovery process can be greatly enhanced. Finally, the only parameter f in PRCL strategy is also discussed. Our strategy is meaningful in practice as it can largely enhance interdependent network resilience and contribute to the decrease of system breakdown risk.
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Keywords:
- interdependent networks /
- cascading failures /
- failures-recovery model /
- preferential recovery strategy
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[2] Buldyrev S V, Parshani R, Paul G, Stanley H E, Havlin S 2010 Nature 464 1025
[3] Gao J X, Buldyrev S V, Stanley H E, Havlin S 2012 Nat. Phys. 8 40
[4] Chen S M, L H, Xu Q G, Xu Y F, Lai Q 2015 Acta Phys. Sin. 64 048902 (in Chinese)[陈世明, 吕辉, 徐青刚, 许云飞, 赖强 2015 物理学报 64 048902]
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[11] Albert R, Albert I, Nakarado G L 2004 Phys. Rev. E 69 025103
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[13] Zhang Z K, Liu C, Zhan X X, Lu X, Zhang C X, Zhang Y C 2016 Phys. Rep. 65 1
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[21] Schneider C M, Moreira A A, Andrade J S, Havlin S, Herrmann H J 2011 Proc. Natl. Acad. Sci. USA 108 3838
[22] Huang X Q, Gao J X, Buldyrev S V, Havlin S, Stanley H E 2011 Phys. Rev. E 83 065101
[23] Hu F Y, Yeung C H, Yang S N, Wang W P, Zeng A 2016 Sci. Rep. 6 24522
[24] Majdandzic A, Podobnki B, Buldyrev S V, Kenett D Y, Havlin S, Stanley H E 2013 Nat. Phys. 10 34
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[26] Weng J S, Lim E P, Jiang J, He Q 2010 Proceedings of the Third ACM International Conference on Web Search and Data Mining (New York:ACM Press) pp261-270
[27] Liu C, Zhang Z K 2014 Commun. Nonlinear. Sci. 19 896
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[31] Wang X Y, Cao J Y, Qin X M 2016 Plos One 11 e0160545
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[33] Valdez L D, Macri P A, Braunstein L A 2014 J. Phys. A:Math. Theor. 47 055002
[34] Freeman L C 1979 Social Networks 1 215
[35] Chen D B, L L Y, Shang M S, Zhang Y C, Zhou T 2012 Physica A 391 1777
[36] Erds P, Rnyi A 1959 Publ. Math. Debrecen 6 290
[37] Newman M E 2003 SIAM Rev. 45 167
[38] Radicchi F 2015 Nat. Phys. 11 7
[39] Liu R R, Jia C X, Zhang J L, Wang B H 2012 J. Univ. Shanghai Sci. Technol. 34 235 (in Chinese)[刘润然, 贾春晓, 章剑林, 汪秉宏 2012 上海理工大学学报 34 235]
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[1] Vespignani A 2010 Nature 464 984
[2] Buldyrev S V, Parshani R, Paul G, Stanley H E, Havlin S 2010 Nature 464 1025
[3] Gao J X, Buldyrev S V, Stanley H E, Havlin S 2012 Nat. Phys. 8 40
[4] Chen S M, L H, Xu Q G, Xu Y F, Lai Q 2015 Acta Phys. Sin. 64 048902 (in Chinese)[陈世明, 吕辉, 徐青刚, 许云飞, 赖强 2015 物理学报 64 048902]
[5] Rinaldi S M, Peerenboom J P, Kelly T K 2001 IEEE Contr. Syst. 21 11
[6] Morris R G, Barthelemy M 2013 Sci. Rep. 3 2764
[7] Liu L J, Yin Y F, Zhang Z H, Malaiya Y K 2016 Plos One 10 e0164777
[8] Korkali M, Veneman J G, Tivnan B F, Bagrow J P, Hines P D H 2017 Sci. Rep. 7 44499
[9] Wang X F, Li X, Chen G R 2012 Network Science:An Introduction (Beijing:Higher Education Press) (in Chinese)[汪小帆, 李翔, 陈关荣 2012 网络科学导论(北京:高等教育出版社)]
[10] Cohen R, Erez K, Ben-Avraham D, Havlin S 2001 Phys. Rev. Lett. 86 3682
[11] Albert R, Albert I, Nakarado G L 2004 Phys. Rev. E 69 025103
[12] Gong K, Tang M, Hui P M, Zhang H F, Younghae D, Lai Y C 2013 Plos One 8 83489
[13] Zhang Z K, Liu C, Zhan X X, Lu X, Zhang C X, Zhang Y C 2016 Phys. Rep. 65 1
[14] Schneider C M, Yazdani N, Arajo N A M, Havlin S, Herrmann H 2013 Sci. Rep. 3 1969
[15] Du R J, Dong G G, Tian L X, Liu R R 2016 Physica A 450 687
[16] Gong M G, Ma L J, Cai Q, Jiao L C 2015 Sci. Rep. 5 8439
[17] Wang J D, Lao S Y, Ruan Y R, Bai L, Hou L L 2017 Appl. Sci. 7 597
[18] Shang Y L 2016 Sci. Rep. 6 30521
[19] Shekhtman L M, Danziger M M, Havlin S 2016 Chaos Solition. Fract. 90 28
[20] Muro M A D, Rocca C E L, Stanley H E, Havlin S, Braunstein L A 2016 Sci. Rep. 6 22834
[21] Schneider C M, Moreira A A, Andrade J S, Havlin S, Herrmann H J 2011 Proc. Natl. Acad. Sci. USA 108 3838
[22] Huang X Q, Gao J X, Buldyrev S V, Havlin S, Stanley H E 2011 Phys. Rev. E 83 065101
[23] Hu F Y, Yeung C H, Yang S N, Wang W P, Zeng A 2016 Sci. Rep. 6 24522
[24] Majdandzic A, Podobnki B, Buldyrev S V, Kenett D Y, Havlin S, Stanley H E 2013 Nat. Phys. 10 34
[25] Liu J G, Lin J H, Guo Q, Zhou T 2016 Sci. Rep. 6 21380
[26] Weng J S, Lim E P, Jiang J, He Q 2010 Proceedings of the Third ACM International Conference on Web Search and Data Mining (New York:ACM Press) pp261-270
[27] Liu C, Zhang Z K 2014 Commun. Nonlinear. Sci. 19 896
[28] Ren X L, L L Y 2014 Chin. Sci. Bull. 13 1175 (in Chinese)[任晓龙, 吕琳媛 2014 科学通报 13 1175]
[29] Liu R R, Li M, Jia C X, Wang B H 2016 Sci. Rep. 6 25294
[30] Sun S W, Wu Y F, Ma Y L, Wang L, Gao Z K, Xia C Y 2016 Sci. Rep. 6 32983
[31] Wang X Y, Cao J Y, Qin X M 2016 Plos One 11 e0160545
[32] Boccaletti S, Bianconi G, Criado R, del Genio C I, Gmez-Gardees J, Romance M, Sendia-Nadal I, Wang Z, Zanin M 2014 Phys. Rep. 544 1
[33] Valdez L D, Macri P A, Braunstein L A 2014 J. Phys. A:Math. Theor. 47 055002
[34] Freeman L C 1979 Social Networks 1 215
[35] Chen D B, L L Y, Shang M S, Zhang Y C, Zhou T 2012 Physica A 391 1777
[36] Erds P, Rnyi A 1959 Publ. Math. Debrecen 6 290
[37] Newman M E 2003 SIAM Rev. 45 167
[38] Radicchi F 2015 Nat. Phys. 11 7
[39] Liu R R, Jia C X, Zhang J L, Wang B H 2012 J. Univ. Shanghai Sci. Technol. 34 235 (in Chinese)[刘润然, 贾春晓, 章剑林, 汪秉宏 2012 上海理工大学学报 34 235]
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