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The centrality reflects the importance of a node in a complex network, which plays an important role in the propagation dynamics. Many researches in the field of node ranking estimation have revealed the characteristics of higher centrality in the structural dynamics and propagation dynamics. However, there are few reports about the effect of nodes with a relatively lower centrality on propagation process. In this paper, we focus on the effect of heterogeneous structural characteristics on propagation dynamics. First, we select four centrality measurements (i.e., degree, coreness, betweenness, and eigenvector) and initialize source nodes with the maximum and minimum centralities respectively. Then, based on the email propagation model and the SI model, the massive numbers of elaborate simulations are implemented in twelve scale-free networks. These networks include three networks generated by the Barabsi-Albert model, four synthetic networks compiled by the GLP (generalized linear preference) algorithm, and five benchmark networks. The simulation results contain two parts: one is the crossover phenomenon of two propagation processes, and the other is the correlation between the crossover point and the proportion of the initial source nodes. We present the crossover of two propagations by calculating the total infected nodes, the incremental infected nodes, and the average degree of the incremental infected nodes. The average degrees of the incremental infected nodes in both synthetic networks and benchmark networks show that there exist two kinds of diffusion modes (i.e., fan-shaped type and single-strand type). With the increase of the initial source nodes, the interaction between two modes results in the different dynamic changes of two propagations with respect to propagation speed, which may lead to the crossover of two propagations in terms of propagation scale in the propagation process. Specifically, the increase of the initial source nodes would suppress the propagation process in which nodes with the maximum centralities are portrayed as propagating sources. However, such an effect is not observed in the propagation process in which nodes with the minimum centralities are portrayed as propagating sources. Our further simulation indicates that the crossover points appear earlier as the proportion of the initial source nodes increases. And by employing the discrete-time method, we find that such a phenomenon can be triggered exactly by increasing the initial source nodes. This work reveals that the influence of the nodes with the minimum centralities should be taken into consideration because the initial infected nodes with a lower centrality will lead to a larger propagation scale if the initial proportion is high.
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Keywords:
- complex networks /
- heterogeneous structure /
- centrality measures /
- propagation speed
[1] Zhang H F, Zhang J, Zhou C S, Small M, Wang B H 2010 New J. Phys. 12 023015
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[10] Liu Y Y, Slotine J J, Barabsi A L 2011 Nature 473 167
[11] Borgatti S P 2005 Soc. Networks 27 55
[12] Gao C, Liu J M, Zhong N 2011 Knowl. Inf. Syst. 27 253
[13] Jiang J J, Wen S, Yu S, Xiang Y, Zhou W L 2016 IEEE Trans. Depend. Secure. pp 1
[14] Zhang X Z, Zhang Y B, L T Y, Yin Y 2014 Physica A 442 100
[15] Han X, Shen Z S, Wang W X, Di Z R 2015 Phys. Rev. Lett. 114 028701
[16] Wang X F, Chen G R 2003 IEEE Circ. Syst. Mag. 3 6
[17] Strogatz S H 2001 Nature 410 268
[18] Watts D J, Strogatz S H 1998 Nature 393 440
[19] Barabsi A L, Albert R 1999 Science 286 509
[20] Wang X F, Li X, Chen G R 2012 Network Science: An Introduction (Beijing: Higher Education Press) pp270-275 (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) 第270-275页]
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[22] Barthlemy M, Barrat A, Pastor-Satorras R, Vespignani A 2004 Phys. Rev. Lett. 92 178701
[23] Albert R, Barabsi A L 2002 Rev. Mod. Phys. 74 47
[24] Goh K I, Kahng B, Kim D 2001 Phys. Rev. Lett. 87 278701
[25] Estrada E, Rodrguez-Velzquez J A 2006 Physica A 364 581
[26] Pastor-Satorras R, Vespignani A 2001 Phys. Rev. E 63 066117
[27] Bu T, Towsley D 2002 Proceedings of the 21st Annual Joint Conference of the IEEE Computer and Communications Societies New York, USA, June 23-27, 2002 p638
[28] Stanford Network Analysis Project, Leskovec J https:// snap.stanford.edu/data/ca-GrQc.html [2017-4-18]
[29] Stanford Network Analysis Project, Leskovec J https://snap.stanford.edu/data/ca-HepTh.html [2017-4-18]
[30] Zou C C, Towsley D, Gong W 2007 IEEE Trans. Depend. Secure. 4 105
[31] Bogu M, Pastor-Satorras R, Vespignani A 2003 Statistical Mechanics of Complex Networks (Berlin: Springer-Verlag) p127
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[1] Zhang H F, Zhang J, Zhou C S, Small M, Wang B H 2010 New J. Phys. 12 023015
[2] Saito K, Kimura M, Ohara K, Motoda H 2016 Inform. Sci. 329 985
[3] Fu C, Min L, Yang J, Xu D L, Liu X Y, Han L S 2015 Proceedings of IEEE International Conference on Computer and Information Technology; Ubiquitous Computing and Communications; Dependable, Autonomic and Secure Computing; Pervasive Intelligence and Computing Liverpool, United Kingdom, October 26-28, 2015 p1725
[4] Ren X L, L L Y 2014 Chin. Sci. Bull. 59 1175 (in Chinese) [任晓龙, 吕琳媛 2014 科学通报 59 1175]
[5] Zhao J, Yu L, Li J R, Zhou P 2015 Chin. Phys. B 24 058904
[6] Song B, Jiang G P, Song Y R, Xia L L 2015 Chin. Phys. B 24 100101
[7] Liu J G, Ren Z M, Guo Q, Wang B H 2013 Acta Phys. Sin. 62 178901 (in Chinese) [刘建国, 任卓明, 郭强, 汪秉宏 2013 物理学报 62 178901]
[8] Freeman L C 1978 Soc. Networks 1 215
[9] Kitsak M, Gallos L K, Havlin S, Liljeros F, Muchnik L, Stanley H E, Makse H A 2010 Nat. Phys. 6 888
[10] Liu Y Y, Slotine J J, Barabsi A L 2011 Nature 473 167
[11] Borgatti S P 2005 Soc. Networks 27 55
[12] Gao C, Liu J M, Zhong N 2011 Knowl. Inf. Syst. 27 253
[13] Jiang J J, Wen S, Yu S, Xiang Y, Zhou W L 2016 IEEE Trans. Depend. Secure. pp 1
[14] Zhang X Z, Zhang Y B, L T Y, Yin Y 2014 Physica A 442 100
[15] Han X, Shen Z S, Wang W X, Di Z R 2015 Phys. Rev. Lett. 114 028701
[16] Wang X F, Chen G R 2003 IEEE Circ. Syst. Mag. 3 6
[17] Strogatz S H 2001 Nature 410 268
[18] Watts D J, Strogatz S H 1998 Nature 393 440
[19] Barabsi A L, Albert R 1999 Science 286 509
[20] Wang X F, Li X, Chen G R 2012 Network Science: An Introduction (Beijing: Higher Education Press) pp270-275 (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) 第270-275页]
[21] Pastor-Satorras R, Vespignani A 2001 Phys. Rev. Lett. 86 3200
[22] Barthlemy M, Barrat A, Pastor-Satorras R, Vespignani A 2004 Phys. Rev. Lett. 92 178701
[23] Albert R, Barabsi A L 2002 Rev. Mod. Phys. 74 47
[24] Goh K I, Kahng B, Kim D 2001 Phys. Rev. Lett. 87 278701
[25] Estrada E, Rodrguez-Velzquez J A 2006 Physica A 364 581
[26] Pastor-Satorras R, Vespignani A 2001 Phys. Rev. E 63 066117
[27] Bu T, Towsley D 2002 Proceedings of the 21st Annual Joint Conference of the IEEE Computer and Communications Societies New York, USA, June 23-27, 2002 p638
[28] Stanford Network Analysis Project, Leskovec J https:// snap.stanford.edu/data/ca-GrQc.html [2017-4-18]
[29] Stanford Network Analysis Project, Leskovec J https://snap.stanford.edu/data/ca-HepTh.html [2017-4-18]
[30] Zou C C, Towsley D, Gong W 2007 IEEE Trans. Depend. Secure. 4 105
[31] Bogu M, Pastor-Satorras R, Vespignani A 2003 Statistical Mechanics of Complex Networks (Berlin: Springer-Verlag) p127
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