### Abstract

Previous work has revealed that the synchronizability of a scale-free network tends to be suppressed when its clustering coefficient is increased. We present a theory to explain this phenomenon. Our proposition is that, as the network becomes more strongly clustered, topological loop structure can emerge, generating a set of eigenvalues that are close to zero. As a result, the dynamics of synchronization tends to be dominated by the loop structure. As the clustering coefficient is increased, the size of the dominant loop increases, leading to continuous degradation of the network synchronizability. We provide analysis and numerical evidence to support the proposition and we speculate that the loop structure can provide a platform for controlling dynamical processes on scale-free networks with high clustering coefficients.

Original language | English |
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Article number | 056106 |

Number of pages | 6 |

Journal | Physical Review. E, Statistical, Nonlinear and Soft Matter Physics |

Volume | 79 |

Issue number | 5 |

DOIs | |

Publication status | Published - May 2009 |

### Keywords

- complex networks
- eigenvalues and eigenfunctions
- network topology
- nonlinear dynamical systems
- synchronisation
- synchronization
- statistics

## Cite this

*Physical Review. E, Statistical, Nonlinear and Soft Matter Physics*,

*79*(5), [056106]. https://doi.org/10.1103/PhysRevE.79.056106