中国物理B ›› 2009, Vol. 18 ›› Issue (8): 3122-3130.doi: 10.1088/1674-1056/18/8/004

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Are networks with more edges easier to synchronize, or not?

王文旭1, 段志生2, 刘超2, 陈关荣3   

  1. (1)Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China; (2)State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, China; (3)State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, China;Department of Electronic Engineering, City University of Hong Kong, Hong Kong, C
  • 收稿日期:2008-09-15 修回日期:2008-11-25 出版日期:2009-08-20 发布日期:2009-08-20
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grant Nos 10832006 and 60674093), the Foundation for Key Program of Educational Ministry, China (Grant No 107110) and the City University of Hong Kong under the Research Enhancement Scheme and SRG (Grant No 9041335).

Are networks with more edges easier to synchronize, or not?

Duan Zhi-Sheng(段志生)a)†, Wang Wen-Xu(王文旭)b), Liu Chao(刘超)a), and Chen Guan-Rong(陈关荣)a)b)   

  1. a State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, China; b Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China
  • Received:2008-09-15 Revised:2008-11-25 Online:2009-08-20 Published:2009-08-20
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grant Nos 10832006 and 60674093), the Foundation for Key Program of Educational Ministry, China (Grant No 107110) and the City University of Hong Kong under the Research Enhancement Scheme and SRG (Grant No 9041335).

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摘要： In this paper, the relationship between network synchronizability and the edge-addition of its associated graph is investigated. First, it is shown that adding one edge to a cycle definitely decreases the network synchronizability. Then, since sometimes the synchronizability can be enhanced by changing the network structure, the question of whether the networks with more edges are easier to synchronize is addressed. Based on a subgraph and complementary graph method, it is shown by examples that the answer is negative even if the network structure is arbitrarily optimized. This reveals that generally there are redundant edges in a network, which not only make no contributions to synchronization but actually may reduce the synchronizability. Moreover, a simple example shows that the node betweenness centrality is not always a good indicator for the network synchronizability. Finally, some more examples are presented to illustrate how the network synchronizability varies following the addition of edges, where all the examples show that the network synchronizability globally increases but locally fluctuates as the number of added edges increases.

Abstract: In this paper, the relationship between network synchronizability and the edge-addition of its associated graph is investigated. First, it is shown that adding one edge to a cycle definitely decreases the network synchronizability. Then, since sometimes the synchronizability can be enhanced by changing the network structure, the question of whether the networks with more edges are easier to synchronize is addressed. Based on a subgraph and complementary graph method, it is shown by examples that the answer is negative even if the network structure is arbitrarily optimized. This reveals that generally there are redundant edges in a network, which not only make no contributions to synchronization but actually may reduce the synchronizability. Moreover, a simple example shows that the node betweenness centrality is not always a good indicator for the network synchronizability. Finally, some more examples are presented to illustrate how the network synchronizability varies following the addition of edges, where all the examples show that the network synchronizability globally increases but locally fluctuates as the number of added edges increases.

Key words: complex network, complementary graph, synchronizability, edge addition

中图分类号:  (Networks and genealogical trees)

  • 89.75.Hc
02.10.Ox (Combinatorics; graph theory) 05.45.Xt (Synchronization; coupled oscillators)