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Laplacian Spectra and Synchronization Processes on Complex Networks

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Handbook of Optimization in Complex Networks

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 57))

Abstract

The spectrum of the Laplacian matrix of a network contains a great deal of information about the network structure and plays a fundamental role in the dynamical behavior of the network. This chapter is to explore and analyze the Laplacian eigenvalue distributions of several typical network models, to study the network dynamics towards synchronization at a mesoscale level of description, and to report the finding of a relation between the spectral information of the Laplacian matrix and the dynamics in the network synchronization process. First, an example of adding long-distance edges is given to show that the network synchronizability may not be directly inferred from statistical properties of the network. Then, the Laplacian eigenvalues of several representative complex networks are shown to possess very different properties, and yet they also share some common features meanwhile. Further, the correlation between the Laplacian spectrum and the node-degree sequence of a network is investigated, revealing that scale-free networks have the highest correlation values, followed by random networks and then by small-world networks. To that end, a simple local prediction–correction algorithm is presented for approximating the eigenvalue λ i+1 from λ i , i=1, 2, ⋯, N, where N is the network size. Finally, it is shown that the processes of synchronization and generalized synchronization (GS) display different patterns, depending intrinsically on the topological structures of the networks. It is found that in the process of synchronization (or GS), roughly speaking, synchronization (or GS) first starts from a small part of hub nodes and then spreads to the other nodes with smaller degrees. It is also demonstrated that, for community networks, a typical synchronization process generally starts from partial synchronization through cluster synchronization to evolve to global complete synchronization.

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Acknowledgements

This work is supported in part by the Chinese National Natural Science Foundation (Grant Nos. 11172215, 60804039 and 60974081), in part by the National Basic Research 973 Program of China under Grant No. 2007CB310805, and in part by the Hong Kong Research Grants Council (Grants NSFC-HK N-CityU107/07 and GRF1117/10E).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei, 430072, China

    Juan Chen & Jun-an Lu

  2. Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China

    Choujun Zhan & Guanrong Chen

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  1. Juan Chen
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  2. Jun-an Lu
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  3. Choujun Zhan
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  4. Guanrong Chen
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Correspondence to Guanrong Chen .

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Editors and Affiliations

  1. Science and Engineering, Department of Computer and Information, University of Florida, Gainesville, 3260, Florida, USA

    My T. Thai

  2. , Department of Industrial & Systems Engin, University of Florida, Weil Hall 401, Gainesville, 32611-6595, Florida, USA

    Panos M. Pardalos

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Chen, J., Lu, Ja., Zhan, C., Chen, G. (2012). Laplacian Spectra and Synchronization Processes on Complex Networks. In: Thai, M., Pardalos, P. (eds) Handbook of Optimization in Complex Networks. Springer Optimization and Its Applications(), vol 57. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-0754-6_4

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