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Complex network as a key approach to understanding many complex systems, such as biological, chemical, physical, technological and social systems, is ubiquitous in nature and society. Synchronization of large-scale complex networks is one of the most important issues in network science. In the last two decades, much attention has been paid to the synchronization of complex dynamic networks, especially the meso-scale networks. However, many real networks consist of even hundreds of millions of nodes. Analyzing the synchronization of such large-scale coupled complex dynamic networks often generate a large number of coupled differential equations, which may make many synchronization algorithms inapplicable for meso-scale networks due to the complexities of simulation experiments. Coarse graining method can map the large-scale networks into meso-scale networks while preserving some of topological properties or dynamic charac-teristics of the original network. Especially, the spectral coarse-graining scheme, as a typical coarse graining method, is proposed to reduce the network size while preserving the synchronization capacity of the initial network. Nevertheless, plenty of studies demonstrate that the components of eigenvectors for the eigenvalue of the coupling matrix, which can depict the ability to synchronizing networks, distribute unevenly. Most of the components distribute concentrically and the intervals are small, while some other components distribute dispersedly and the intervals are large, which renders the applications of original spectral coarse graining method unsatisfactory. Inspired by the adaptive clustering, we propose an improved spectral coarse graining algorithm, which clusters the same or the similar nodes in the network according to the distance between the components of eigenvectors for the eigenvalue of network coupling matrices, so that the nodes with the same or the similar dynamic properties can be effectively clustered together. Compared with the original spectral coarse graining algorithm, this method can improve the accuracy of the result of clustering. Meanwhile, our method can greatly reduce algorithm complexity, and obtain better spectral coarse graining result. Finally, numerical simulation experiments are implemented in four typical complex networks: NW network, ER network, BA scale-free network and clustering network. The comparison of results demonstrate that our method outperforms the original spectral coarse graining approach under various criteria, and improves the effect of coarse graining and the ability to synchronize networks.
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Keywords:
- complex network /
- synchronization /
- spectral coarse graining /
- improved algorithm
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[4] Barahona M, Pecora L M 2002 Phys. Rev. Lett. 89 054101
[5] Wang X F, Chen G R 2002 Int. J. Bifurcat. Chaos 12 187
[6] Motter A E, Zhou C S, Kurths J 2005 Phys. Rev. E 71 016116
[7] Nishikawa T, Motter A E 2006 Physica D 224 77
[8] Zhou J, Lu J A, Lu J H 2006 IEEE Trans. Auto. Control 51 652
[9] Arenas A, Daz-Guilera A, Kurths J, Moreno Y, Zhou C S 2008 Phys. Rep. 469 93
[10] Zhu T X, Wu Y, Xiao J H 2012 Acta Phys. Sin. 61 040502 (in Chinese) [朱廷祥, 吴晔, 肖井华 2012 61 040502]
[11] Xu M M, Lu J A, Zhou J 2016 Acta Phys. Sin. 65 028902 (in Chinese) [徐明明, 陆君安, 周进 2016 65 028902]
[12] Kurkcuoglu O, Jernigan R L, Doruker P 2004 Polymer 45 649
[13] Marrink S J, Vries A H D, Mark A E 2004 J. Phys. Chem. B 108 750
[14] Bornholdt S 2005 Science 310 449
[15] Chen J, Lu J A, Lu X F, Wu X Q, Chen G R 2013 Commun. Nonlinear Sci. 18 3036
[16] Zeng A, L L Y 2011 Phys. Rev. E 83 056123
[17] Saunders M G, Voth G A 2013 Annu. Rev. Biophys. 42 73
[18] Kim B J 2004 Phys. Rev. Lett. 93 168701
[19] Chen H S, Hou Z H, Xin H W, Yan Y J 2010 Phys. Rev. E 82 011107
[20] Gfeller D, Rios P D L 2007 Phys. Rev. Lett. 99 038701
[21] Gfeller D, Rios P D L 2008 Phys. Rev. Lett. 100 174104
[22] Chen G R, Wang X F, Li X, L J H 2009 Some Recent Advances in Complex Networks Synchronization (Heidelberg: Springer) pp3-16
[23] Lu J A, Liu H, Chen J 2016 Synchronization in Complex Dynamical Networks (Beijing: Higher Education Press) pp120-125 (in Chinese) [陆君安, 刘慧, 陈娟 2016 复杂动态网络的同步(北京:高等教育出版社) 第120125页]
[24] Kuramoto Y 1975 Lect. Notes Phys. 39 420
[25] Acebrn J A, Bonilla L L, Prez Vicente C J, Ritort F, Spigler R 2005 Rev. Mod. Phys. 77 137
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[1] Pecora L M, Carroll T L 1998 Phys. Rev. Lett. 80 2109
[2] Jost J, Joy M P 2001 Phys. Rev. E 65 016201
[3] Wang X F, Chen G R 2002 IEEE Trans. Circuits-I 49 54
[4] Barahona M, Pecora L M 2002 Phys. Rev. Lett. 89 054101
[5] Wang X F, Chen G R 2002 Int. J. Bifurcat. Chaos 12 187
[6] Motter A E, Zhou C S, Kurths J 2005 Phys. Rev. E 71 016116
[7] Nishikawa T, Motter A E 2006 Physica D 224 77
[8] Zhou J, Lu J A, Lu J H 2006 IEEE Trans. Auto. Control 51 652
[9] Arenas A, Daz-Guilera A, Kurths J, Moreno Y, Zhou C S 2008 Phys. Rep. 469 93
[10] Zhu T X, Wu Y, Xiao J H 2012 Acta Phys. Sin. 61 040502 (in Chinese) [朱廷祥, 吴晔, 肖井华 2012 61 040502]
[11] Xu M M, Lu J A, Zhou J 2016 Acta Phys. Sin. 65 028902 (in Chinese) [徐明明, 陆君安, 周进 2016 65 028902]
[12] Kurkcuoglu O, Jernigan R L, Doruker P 2004 Polymer 45 649
[13] Marrink S J, Vries A H D, Mark A E 2004 J. Phys. Chem. B 108 750
[14] Bornholdt S 2005 Science 310 449
[15] Chen J, Lu J A, Lu X F, Wu X Q, Chen G R 2013 Commun. Nonlinear Sci. 18 3036
[16] Zeng A, L L Y 2011 Phys. Rev. E 83 056123
[17] Saunders M G, Voth G A 2013 Annu. Rev. Biophys. 42 73
[18] Kim B J 2004 Phys. Rev. Lett. 93 168701
[19] Chen H S, Hou Z H, Xin H W, Yan Y J 2010 Phys. Rev. E 82 011107
[20] Gfeller D, Rios P D L 2007 Phys. Rev. Lett. 99 038701
[21] Gfeller D, Rios P D L 2008 Phys. Rev. Lett. 100 174104
[22] Chen G R, Wang X F, Li X, L J H 2009 Some Recent Advances in Complex Networks Synchronization (Heidelberg: Springer) pp3-16
[23] Lu J A, Liu H, Chen J 2016 Synchronization in Complex Dynamical Networks (Beijing: Higher Education Press) pp120-125 (in Chinese) [陆君安, 刘慧, 陈娟 2016 复杂动态网络的同步(北京:高等教育出版社) 第120125页]
[24] Kuramoto Y 1975 Lect. Notes Phys. 39 420
[25] Acebrn J A, Bonilla L L, Prez Vicente C J, Ritort F, Spigler R 2005 Rev. Mod. Phys. 77 137
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