Laplacian Spectra and Synchronization Processes on Complex Networks

Juan Chen (), Jun-an Lu (), Choujun Zhan () and Guanrong Chen ()
Additional contact information
Juan Chen: Wuhan University
Jun-an Lu: Wuhan University
Choujun Zhan: City University of Hong Kong
Guanrong Chen: City University of Hong Kong

Chapter Chapter 4 in Handbook of Optimization in Complex Networks, 2012, pp 81-113 from Springer

Abstract: 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.

Keywords: Random Network; Laplacian Matrix; Degree Sequence; Generalize Synchronization; Synchronization Process (search for similar items in EconPapers)
Date: 2012
References: Add references at CitEc
Citations: View citations in EconPapers (3)

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-1-4614-0754-6_4

Ordering information: This item can be ordered from
http://www.springer.com/9781461407546

DOI: 10.1007/978-1-4614-0754-6_4

Access Statistics for this chapter

More chapters in Springer Optimization and Its Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().