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Core-periphery structure in heterogeneous adaptive network and its inhibiting effect on epidemic spreading

Yang Hui Tang Ming Cai Shi-Min Zhou Tao

Core-periphery structure in heterogeneous adaptive network and its inhibiting effect on epidemic spreading

Yang Hui, Tang Ming, Cai Shi-Min, Zhou Tao
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  • The study of epidemic spreading in node-property heterogeneous adaptive network shows that node-property heterogeneity can greatly increase the epidemic threshold, and the initial network can adaptively self-organize into a more robust degree heterogeneous network structure. The difference in epidemic spreading between homogeneous and heterogeneous node-property adaptive networks is of great importance for understanding the threshold increasing in the heterogeneous node-property adaptive network. In this paper, we study the transient spreading process in the heterogeneous node-property adaptive network. In order to capture the core-periphery structure in the network, we define two hierarchical structure indicators. When both indicators are of large values in the network, not only is the network of strong core-periphery property, but also less susceptible nodes are more likely to be in the core area of the network. By combining them with various network structure properties, such as the average degree ratio and static threshold of transient network, we analyze the evolution of network structure and show the self-organizing formation process of robust degree heterogeneous structure by numerical simulations. We find that the threshold increase is basically due to the formation of core-periphery structure, where the less susceptible nodes are more likely to be reallocated to the core area of the network by rewiring. In view of this, we propose a new preference rewiring strategy. The results show that the new strategy can increase the epidemic threshold by faciliating the formation of core-periphery structure, which verifies the correctness of the transient network structure analysis. It not only helps to deeply understand the epidemic spreading process in the node-property heterogeneous adaptive network, but also provides new ideas for putting forward the strategy of controlling epidemics.
      Corresponding author: Tang Ming, tangminghan007@gmail.com
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11105025, 11575041, 61433014) and the Program of Outstanding Ph. D. Candidate in Academic Research by UESTC (Grant No. YBXSZC20131036).
    [1]

    Erdös P,Rényi A 1960 Publ.Math.Inst.Hungar.Acad.Sci. 5 17

    [2]

    Watts D J, Strogatz S H 1998 Nature 393 440

    [3]

    Barabási A L, Albert R 1999 Science 286 509

    [4]

    Gross T, Blasius B 2008 J. R. Soc. Interface 5 259

    [5]

    Holme P, Saramäki J 2012 Phys. Rep. 519 97

    [6]

    De Domenico M, Solé-Ribalta A, Cozzo E, Kivelä M, Moreno Y, Porter M A, Gómez S, Arenas A 2013 Phys.Rev.X 3 041022

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    Boccaletti S, Bianconi G, Criado R, Del Genio C I, Gómez-Gardeñes J, Romance M, Sendiña-Nadal I, Wang Z, Zanin M 2014 Phys.Rep. 544 1

    [8]

    Kivelä M, Arenas A, Barthelemy M, Gleeson J P, Moreno Y, Porter M A 2014 J. Complex Networks 2 203

    [9]

    De Domenico M, Nicosia V, Arenas A, Latora V 2015 Nature Commun. 6 6864

    [10]

    Zhao Y, Zheng M, Liu Z 2014 Chaos: An Interdisciplinary Journal of Nonlinear Science 24 043129

    [11]

    Szell M, Lambiotte R, Thurner S 2010 Proc.Natl.Acad.Sci. 107 13636

    [12]

    Palla G, Derényi I, Farkas I, Vicsek T 2005 Nature 435 814

    [13]

    Li R Q, Tang M, Xu B M 2013 Acta Phys.Sin. 62 168903 (in Chinese) [李睿琪, 唐明, 许伯铭 2013 物理学报 62 168903]

    [14]

    Galvani A P, May R M 2005 Nature 438 293

    [15]

    Lloyd-Smith J O, Schreiber S J, Kopp P E, Getz W M 2005 Nature 438 355

    [16]

    Lipsitch M, Cohen T, Cooper B, Robins J M, Ma S, James L, Gopalakrishma G, Chew S K, Tan C C, Samore M H, Fisman D, Murray M 2003 Science 300 1966

    [17]

    Yang H, Tang M, Hui P M 2012 Complex Systems and Complexity Science 9 63 (in Chinese) [杨慧,唐明,许伯铭 2012 复杂系统与复杂科学 9 63]

    [18]

    Gross T, D'Lima C J D, Blasius B 2006 Phys.Rev.Lett. 96 208701

    [19]

    Marceau V, Noël P A, Hébert-Dufresne L, Allard A, Dubé L J 2010 Phys.Rev.E 82 036116

    [20]

    Shaw L B, Schwartz I B 2008 Phys.Rev.E 77 066101

    [21]

    Zanette D H, Risau-Gusmán S 2008 J. Biol. Phys. 34 135

    [22]

    Risau-Gusmán S, Zanette D H 2009 J. Theor. Biol. 257 52

    [23]

    Gross T, Kevrekidis I G 2008 Europhys. Lett. 82 38004

    [24]

    Shaw L B, Schwartz I B 2010 Phys.Rev.E 81 046120

    [25]

    Yang H, Tang M, Zhang H F 2012 New J. Phys. 14 123017

    [26]

    Liu H K,Yang H,Tang M, Zhou T 2014 Sci. Sin.: Phys. Mech. Astron. 44 32 (in Chinese) [刘宏鲲, 杨慧, 唐明, 周涛 2014 中国科学: 物理学 力学 天文学 44 32]

    [27]

    Van Segbroeck S, Santos F C, Pacheco J M 2010 PLoS Comput. Biol. 6 e1000895

    [28]

    Zhu G, Chen G, Xu X J, Fu X C 2013 J. Theor. Biol. 317 133

    [29]

    Miller J C 2007 Phys.Rev.E 76 010101

    [30]

    Miller J C 2008 J.Appl.Prob. 45 498

    [31]

    Neri F M, Pérez-Reche F J, Taraskin S N, Gilligan C A 2011 J.R.Soc.Interface 8 201

    [32]

    Neri F M, Bates A, Fuchtbauer W S, Pérez-Reche F J, Taraskin S N, Otten W, Bailey D J, Gilligan C A 2011 PLoS Comput. Biol. 7 e1002174

    [33]

    Smilkov D, Hidalgo C A, Kocarev L 2014 Sci.Rep. 4 4795

    [34]

    Yan G, Zhou T, Wang J, Fu Z Q, Wang B H 2005 Chin. Phys. Lett. 22 510

    [35]

    Yang Z M, Zhou T 2012 Phys. Rev. E 85 056106

    [36]

    Wang W, Tang M, Zhang H F, Gao H, Do Y, Liu Z H 2014 Phys.Rev.E 90 042803

    [37]

    Li X, Cao L, Cao G F 2010 The Eur. Phys. J. B 75 319

    [38]

    Yang H, Tang M, Gross T 2015 Sci.Rep. 5 13122

    [39]

    Pastor-Satorras R, Vespignani A 2001 Phys.Rev.Lett. 86 3200

    [40]

    Kitsak M, Gallos L K, Havlin S, Liljeros F, Muchnik L, Stanley H E, Makse H A 2010 Nature Phys. 6 888

    [41]

    Liu Y, Tang M, Zhou T, Do Y 2015 Sci. Rep. 5 9602

    [42]

    Liu Y, Tang M, Zhou T, Do Y 2015 Sci. Rep. 5 13172

    [43]

    Wang Y,Chakrabarti D,Wang C,Faloutsos C 2003 in Proceedings of the 22nd Symposium on Reliable Distributed Systems Florence, Italy 6-8 October, 2003 (IEEE) pp 25-34

  • [1]

    Erdös P,Rényi A 1960 Publ.Math.Inst.Hungar.Acad.Sci. 5 17

    [2]

    Watts D J, Strogatz S H 1998 Nature 393 440

    [3]

    Barabási A L, Albert R 1999 Science 286 509

    [4]

    Gross T, Blasius B 2008 J. R. Soc. Interface 5 259

    [5]

    Holme P, Saramäki J 2012 Phys. Rep. 519 97

    [6]

    De Domenico M, Solé-Ribalta A, Cozzo E, Kivelä M, Moreno Y, Porter M A, Gómez S, Arenas A 2013 Phys.Rev.X 3 041022

    [7]

    Boccaletti S, Bianconi G, Criado R, Del Genio C I, Gómez-Gardeñes J, Romance M, Sendiña-Nadal I, Wang Z, Zanin M 2014 Phys.Rep. 544 1

    [8]

    Kivelä M, Arenas A, Barthelemy M, Gleeson J P, Moreno Y, Porter M A 2014 J. Complex Networks 2 203

    [9]

    De Domenico M, Nicosia V, Arenas A, Latora V 2015 Nature Commun. 6 6864

    [10]

    Zhao Y, Zheng M, Liu Z 2014 Chaos: An Interdisciplinary Journal of Nonlinear Science 24 043129

    [11]

    Szell M, Lambiotte R, Thurner S 2010 Proc.Natl.Acad.Sci. 107 13636

    [12]

    Palla G, Derényi I, Farkas I, Vicsek T 2005 Nature 435 814

    [13]

    Li R Q, Tang M, Xu B M 2013 Acta Phys.Sin. 62 168903 (in Chinese) [李睿琪, 唐明, 许伯铭 2013 物理学报 62 168903]

    [14]

    Galvani A P, May R M 2005 Nature 438 293

    [15]

    Lloyd-Smith J O, Schreiber S J, Kopp P E, Getz W M 2005 Nature 438 355

    [16]

    Lipsitch M, Cohen T, Cooper B, Robins J M, Ma S, James L, Gopalakrishma G, Chew S K, Tan C C, Samore M H, Fisman D, Murray M 2003 Science 300 1966

    [17]

    Yang H, Tang M, Hui P M 2012 Complex Systems and Complexity Science 9 63 (in Chinese) [杨慧,唐明,许伯铭 2012 复杂系统与复杂科学 9 63]

    [18]

    Gross T, D'Lima C J D, Blasius B 2006 Phys.Rev.Lett. 96 208701

    [19]

    Marceau V, Noël P A, Hébert-Dufresne L, Allard A, Dubé L J 2010 Phys.Rev.E 82 036116

    [20]

    Shaw L B, Schwartz I B 2008 Phys.Rev.E 77 066101

    [21]

    Zanette D H, Risau-Gusmán S 2008 J. Biol. Phys. 34 135

    [22]

    Risau-Gusmán S, Zanette D H 2009 J. Theor. Biol. 257 52

    [23]

    Gross T, Kevrekidis I G 2008 Europhys. Lett. 82 38004

    [24]

    Shaw L B, Schwartz I B 2010 Phys.Rev.E 81 046120

    [25]

    Yang H, Tang M, Zhang H F 2012 New J. Phys. 14 123017

    [26]

    Liu H K,Yang H,Tang M, Zhou T 2014 Sci. Sin.: Phys. Mech. Astron. 44 32 (in Chinese) [刘宏鲲, 杨慧, 唐明, 周涛 2014 中国科学: 物理学 力学 天文学 44 32]

    [27]

    Van Segbroeck S, Santos F C, Pacheco J M 2010 PLoS Comput. Biol. 6 e1000895

    [28]

    Zhu G, Chen G, Xu X J, Fu X C 2013 J. Theor. Biol. 317 133

    [29]

    Miller J C 2007 Phys.Rev.E 76 010101

    [30]

    Miller J C 2008 J.Appl.Prob. 45 498

    [31]

    Neri F M, Pérez-Reche F J, Taraskin S N, Gilligan C A 2011 J.R.Soc.Interface 8 201

    [32]

    Neri F M, Bates A, Fuchtbauer W S, Pérez-Reche F J, Taraskin S N, Otten W, Bailey D J, Gilligan C A 2011 PLoS Comput. Biol. 7 e1002174

    [33]

    Smilkov D, Hidalgo C A, Kocarev L 2014 Sci.Rep. 4 4795

    [34]

    Yan G, Zhou T, Wang J, Fu Z Q, Wang B H 2005 Chin. Phys. Lett. 22 510

    [35]

    Yang Z M, Zhou T 2012 Phys. Rev. E 85 056106

    [36]

    Wang W, Tang M, Zhang H F, Gao H, Do Y, Liu Z H 2014 Phys.Rev.E 90 042803

    [37]

    Li X, Cao L, Cao G F 2010 The Eur. Phys. J. B 75 319

    [38]

    Yang H, Tang M, Gross T 2015 Sci.Rep. 5 13122

    [39]

    Pastor-Satorras R, Vespignani A 2001 Phys.Rev.Lett. 86 3200

    [40]

    Kitsak M, Gallos L K, Havlin S, Liljeros F, Muchnik L, Stanley H E, Makse H A 2010 Nature Phys. 6 888

    [41]

    Liu Y, Tang M, Zhou T, Do Y 2015 Sci. Rep. 5 9602

    [42]

    Liu Y, Tang M, Zhou T, Do Y 2015 Sci. Rep. 5 13172

    [43]

    Wang Y,Chakrabarti D,Wang C,Faloutsos C 2003 in Proceedings of the 22nd Symposium on Reliable Distributed Systems Florence, Italy 6-8 October, 2003 (IEEE) pp 25-34

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  • Received Date:  30 September 2015
  • Accepted Date:  04 December 2015
  • Published Online:  05 March 2016

Core-periphery structure in heterogeneous adaptive network and its inhibiting effect on epidemic spreading

    Corresponding author: Tang Ming, tangminghan007@gmail.com
  • 1. Web Science Center, University of Electronic Science and Technology of China, Chengdu 611731, China;
  • 2. Big Data Research Center, University of Electronic Science and Technology of China, Chengdu 611731, China
Fund Project:  Project supported by the National Natural Science Foundation of China (Grant Nos. 11105025, 11575041, 61433014) and the Program of Outstanding Ph. D. Candidate in Academic Research by UESTC (Grant No. YBXSZC20131036).

Abstract: The study of epidemic spreading in node-property heterogeneous adaptive network shows that node-property heterogeneity can greatly increase the epidemic threshold, and the initial network can adaptively self-organize into a more robust degree heterogeneous network structure. The difference in epidemic spreading between homogeneous and heterogeneous node-property adaptive networks is of great importance for understanding the threshold increasing in the heterogeneous node-property adaptive network. In this paper, we study the transient spreading process in the heterogeneous node-property adaptive network. In order to capture the core-periphery structure in the network, we define two hierarchical structure indicators. When both indicators are of large values in the network, not only is the network of strong core-periphery property, but also less susceptible nodes are more likely to be in the core area of the network. By combining them with various network structure properties, such as the average degree ratio and static threshold of transient network, we analyze the evolution of network structure and show the self-organizing formation process of robust degree heterogeneous structure by numerical simulations. We find that the threshold increase is basically due to the formation of core-periphery structure, where the less susceptible nodes are more likely to be reallocated to the core area of the network by rewiring. In view of this, we propose a new preference rewiring strategy. The results show that the new strategy can increase the epidemic threshold by faciliating the formation of core-periphery structure, which verifies the correctness of the transient network structure analysis. It not only helps to deeply understand the epidemic spreading process in the node-property heterogeneous adaptive network, but also provides new ideas for putting forward the strategy of controlling epidemics.

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