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**The changes on synchronizing ability of coupled networks from ring networks to chain networks.**
*(English)*
Zbl 1169.34327

Summary: In this paper, two different ring networks with unidirectional couplings and with bidirectional couplings were discussed by theoretical analysis. It was found that the effects on synchronizing ability of the two different structures by cutting a link are completely opposite. The synchronizing ability will decrease if the change is from bidirectional ring to bidirectional chain. Moreover, the change on synchronizing ability will be four times if the number of \(N\) is large enough. However, it will increase obviously from unidirectional ring to unidirectional chain. It will be \(N^{2}/(2\pi ^{2})\) times if the number of \(N\) is large enough. The numerical simulations confirm the conclusion in quality. This paper also discusses the effects on synchronization by adding one link with different length d to these two different structures. It can be seen that the effects are different. Theoretical results are accordant to numerical simulations. Synchronization is an essential physics problem. These results proposed in this paper have some important reference meanings on the real world networks, such as the bioecological system networks, the designing of the circuit, etc.

### MSC:

34D05 | Asymptotic properties of solutions to ordinary differential equations |

05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |

05C90 | Applications of graph theory |

15A18 | Eigenvalues, singular values, and eigenvectors |

### Keywords:

complex networks; synchronizing ability; unidirectional couplings; bidirectional couplings; ring; chain; coupling matrix
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\textit{X. Han} and \textit{J. Lu}, Sci. China, Ser. F 50, No. 4, 615--624 (2007; Zbl 1169.34327)

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