- LI Ming ,
- WANG Binghong

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- 1. School of Engineering Science, University of Science and Technology of China, Hefei 230026, China;

2. Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China

**Received date: **2017-03-30

**Revised date: **2017-06-16

**Online published: **2017-07-29

The percolation model describes the emergence of a giant component in a system, of which the sites are connected randomly with some mechanisms. Although the rule is very simple, the percolation model involves many concepts of statistical physics and nonlinear physics, and has also been applied to a large variety of natural, technological and social systems, most of which can be viewed as networks. In recent years, a new kind of networks, the multilayer networks, have been proposed to study the complex and interacting systems. Based on the percolation in these networks, a number of natural, technological and social problems have been solved. At the same time, the percolation transition itself also has some new properties, such as the discontinuous transition. This paper analyzes the mechanism of the percolation on multilayer networks first, and then briefly discusses the discontinuous transition in this model. Furthermore, the theoretical methods are also be reviewed.

**Key words：**
percolation; multilayer networks; phase transition

LI Ming
,
WANG Binghong
. Percolation transition on multilayer networks[J]. *Science & Technology Review*, 2017
, 35(14)
: 50
-55
.
DOI: 10.3981/j.issn.1000-7857.2017.14.006

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[27] Cellai D, López E, Zhou J, et al. Percolation in multiplex networks with overlap[J]. Physical Review E, 2013, 88(5):052811.

[28] Hu Y, Zhou D, Zhang R, et al. Percolation of interdependent networks with intersimilarity[J]. Physical Review E, 2013, 88(5):052805.

[29] Parshani R, Rozenblat C, Ietri D, et al. Inter-similarity between cou pled networks[J]. Europhysics Letters, 2011, 92(6):68002.

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[31] Bianconi G, Dorogovtsev S N. Multiple percolation transitions in a con figuration model of a network of networks[J]. Physical Review E, 2014, 89(6):062814.

[32] Radicchi F. Percolation in real interdependent networks[J]. Nature Physics, 2015, 11(7):597-602.

[33] Bianconi G, Radicchi F. Percolation in real multiplex networks[J]. Physical Review E, 2016, 94(6):060301.

[34] Watanabe S, Kabashima Y. Cavity-based robustness analysis of inter dependent networks:Influences of intranetwork and internetwork de gree-degree correlations[J]. Physical Review E, 2014, 89(1):012808.

[35] Cellai D, Dorogovtsev S N, Bianconi G. Message passing theory for percolation models on multiplex networks with link overlap[J]. Physi cal Review E, 2016, 94(3):032301.

[36] Hackett A, Cellai D, Gómez S, et al. Bond percolation on multiplex networks[J]. Physical Review X, 2016, 6(2):021002.

[37] Bianconi G. Epidemic spreading and bond percolation in multilayer networks[J]. Journal of Statistical Mechanics:Theory and Experiment, 3, 2017, 3(3):034001.

[38] Li M, Liu R R, Peng D, et al. Channel is more important than effec tiveness in spreading dynamics on multiplex networks[J]. arXiv pre print arXiv, 2016:1604.05209.

[39] Parshani R, Buldyrev S V, Havlin S. Critical effect of dependency groups on the function of networks[J]. Proceedings of the National Academy of Sciences of the United States of America, 2011, 108(3):1007-1010.

[40] Gao J, Buldyrev S V, Havlin S, et al. Robustness of a network of net works[J]. Physical Review Letters, 2011, 107(19):195701.

[41] Gao J, Buldyrev S V, Stanley H E, et al. Percolation of a general net work of networks[J]. Physical Review E, 2013, 88(6):062816.

[42] Gao J, Buldyrev S V, Stanley H E, et al. Networks formed from inter dependent networks[J]. Nature physics, 2012, 8(1):40-48.

[43] Azimi-Tafreshi N, Gómez-Gardenes J, Dorogovtsev S N. K-core per colation on multiplex networks[J]. Physical Review E, 2014, 90(3):032816.

[44] Brummitt C D, Lee K M, Goh K I. Multiplexity-facilitated cascades in networks[J]. Physical Review E, 2012, 85(4):045102.

[45] Gao J, Li D, Havlin S. From a single network to a network of networks[J]. National Science Review, 2014, 1(3):346-356.

[46] Kivelä M, Arenas A, Barthelemy M, et al. Multilayer networks[J]. Jour nal of complex networks, 2014, 2(3):203-271.

[47] Danziger M M, Bashan A, Berezin Y, et al. An introduction to interde pendent networks[J]. Nonlinear Dynamics of Electronic Systems, 2014, 438:189-202.

[2] Christensen K, Moloney N R. Complexity and criticality[M]. London:Im perial College Press, 2005.

[3] Ziff R M, Scullard C R. Exact bond percolation thresholds in two di mensions[J]. Journal of Physics A:Mathematical and General, 2006, 39(49):15083.

[4] Newman M E J, Ziff R M. Efficient Monte Carlo algorithm and highprecision results for percolation[J]. Physical Review Letters, 2000, 85(19):4104.

[5] Deng Y, Blöte H W J. Monte Carlo study of the site-percolation model in two and three dimensions[J]. Physical Review E, 2005, 72(1):016126.

[6] Newman M E J. Spread of epidemic disease on networks[J]. Physical re view E, 2002, 66(1):016128.

[7] Pastor-Satorras R, Castellano C, Van Mieghem P, et al. Epidemic pro cesses in complex networks[J]. Reviews of Modern Physics, 2015, 87(3):925.

[8] Cohen R, Erez K, Ben-Avraham D, et al. Resilience of the internet to random breakdowns[J]. Physical Review Letters, 2000, 85(21):4626.

[9] Cohen R, Erez K, Ben-Avraham D, et al. Breakdown of the internet un der intentional attack[J]. Physical Review Letters, 2001, 86(16):3682.

[10] Cohen R, Havlin S. Complex networks:Structure, robustness and func tion[M]. Cambridge:Cambridge University Press, 2010.

[11] Boccaletti S, Bianconi G, Criado R, et al. The structure and dynamics of multilayer networks[J]. Physics Reports, 2014, 544(1):1-122.

[12] Li W, Bashan A, Buldyrev S V, et al. Cascading failures in interdepen dent lattice networks:The critical role of the length of dependency links[J]. Physical Review Letters, 2012, 108(22):228702.

[13] Lowinger S, Cwilich G A, Buldyrev S V. Interdependent lattice net works in high dimensions[J]. Physical Review E, 2016, 94(5):052306.

[14] Son S W, Grassberger P, Paczuski M. Percolation transitions are not al ways sharpened by making networks interdependent[J]. Physical Re view Letters, 2011, 107(19):195702.

[15] Liu X W, Deng Y, Jacobsen J L. Recursive percolation[J]. Physical Re view E, 2015, 92(1):010103.

[16] Buldyrev S V, Parshani R, Paul G, et al. Catastrophic cascade of fail ures in interdependent networks[J]. Nature, 2010, 464(7291):1025-1028.

[17] Bashan A, Berezin Y, Buldyrev S V, et al. The extreme vulnerability of interdependent spatially embedded networks[J]. Nature Physics, 2013, 9(10):667-672.

[18] Grassberger P. Percolation transitions in the survival of interdependent agents on multiplex networks, catastrophic cascades, and solid-on-sol id surface growth[J]. Physical Review E, 2015, 91(6):062806.

[19] Son S W, Bizhani G, Christensen C, et al. Percolation theory on inter dependent networks based on epidemic spreading[J]. Europhysics Let ters, 2012, 97(1):16006.

[20] Feng L, Monterola C P, Hu Y. The simplified self-consistent probabili ties method for percolation and its application to interdependent net works[J]. New Journal of Physics, 2015, 17(6):063025.

[21] 李明. 网络逾渗与级联故障[D]. 合肥:中国科学技术大学近代物理系, 2014. Li Ming. Network percolation and cascading failures[D]. Hefei:Depart ment of Modern Physics, University of Science and Technology of Chi na, 2014.

[22] Li M, Wang B H. Percolation on networks with dependence links[J]. Chinese Physics B, 2014, 23(7):076402.

[23] Baxter G J, Dorogovtsev S N, Goltsev A V, et al. Avalanche collapse of interdependent networks[J]. Physical Review Letters, 2012, 109(24):248701.

[24] Zhou D, Bashan A, Cohen R, et al. Simultaneous first-and second-or der percolation transitions in interdependent networks[J]. Physical Re view E, 2014, 90(1):012803.

[25] Lee D, Choi S, Stippinger M, et al. Hybrid phase transition into an ab sorbing state:Percolation and avalanches[J]. Physical Review E, 2016, 93(4):042109.

[26] Parshani R, Buldyrev S V, Havlin S. Interdependent networks:Reduc ing the coupling strength leads to a change from a first to second or der percolation transition[J]. Physical Review Letters, 2010, 105(4):048701.

[27] Cellai D, López E, Zhou J, et al. Percolation in multiplex networks with overlap[J]. Physical Review E, 2013, 88(5):052811.

[28] Hu Y, Zhou D, Zhang R, et al. Percolation of interdependent networks with intersimilarity[J]. Physical Review E, 2013, 88(5):052805.

[29] Parshani R, Rozenblat C, Ietri D, et al. Inter-similarity between cou pled networks[J]. Europhysics Letters, 2011, 92(6):68002.

[30] Valdez L D, Macri P A, Stanley H E, et al. Triple point in correlated interdependent networks[J]. Physical Review E, 2013, 88(5):050803.

[31] Bianconi G, Dorogovtsev S N. Multiple percolation transitions in a con figuration model of a network of networks[J]. Physical Review E, 2014, 89(6):062814.

[32] Radicchi F. Percolation in real interdependent networks[J]. Nature Physics, 2015, 11(7):597-602.

[33] Bianconi G, Radicchi F. Percolation in real multiplex networks[J]. Physical Review E, 2016, 94(6):060301.

[34] Watanabe S, Kabashima Y. Cavity-based robustness analysis of inter dependent networks:Influences of intranetwork and internetwork de gree-degree correlations[J]. Physical Review E, 2014, 89(1):012808.

[35] Cellai D, Dorogovtsev S N, Bianconi G. Message passing theory for percolation models on multiplex networks with link overlap[J]. Physi cal Review E, 2016, 94(3):032301.

[36] Hackett A, Cellai D, Gómez S, et al. Bond percolation on multiplex networks[J]. Physical Review X, 2016, 6(2):021002.

[37] Bianconi G. Epidemic spreading and bond percolation in multilayer networks[J]. Journal of Statistical Mechanics:Theory and Experiment, 3, 2017, 3(3):034001.

[38] Li M, Liu R R, Peng D, et al. Channel is more important than effec tiveness in spreading dynamics on multiplex networks[J]. arXiv pre print arXiv, 2016:1604.05209.

[39] Parshani R, Buldyrev S V, Havlin S. Critical effect of dependency groups on the function of networks[J]. Proceedings of the National Academy of Sciences of the United States of America, 2011, 108(3):1007-1010.

[40] Gao J, Buldyrev S V, Havlin S, et al. Robustness of a network of net works[J]. Physical Review Letters, 2011, 107(19):195701.

[41] Gao J, Buldyrev S V, Stanley H E, et al. Percolation of a general net work of networks[J]. Physical Review E, 2013, 88(6):062816.

[42] Gao J, Buldyrev S V, Stanley H E, et al. Networks formed from inter dependent networks[J]. Nature physics, 2012, 8(1):40-48.

[43] Azimi-Tafreshi N, Gómez-Gardenes J, Dorogovtsev S N. K-core per colation on multiplex networks[J]. Physical Review E, 2014, 90(3):032816.

[44] Brummitt C D, Lee K M, Goh K I. Multiplexity-facilitated cascades in networks[J]. Physical Review E, 2012, 85(4):045102.

[45] Gao J, Li D, Havlin S. From a single network to a network of networks[J]. National Science Review, 2014, 1(3):346-356.

[46] Kivelä M, Arenas A, Barthelemy M, et al. Multilayer networks[J]. Jour nal of complex networks, 2014, 2(3):203-271.

[47] Danziger M M, Bashan A, Berezin Y, et al. An introduction to interde pendent networks[J]. Nonlinear Dynamics of Electronic Systems, 2014, 438:189-202.

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