### 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 |
---|---|

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

**Emergence of loop structure in scale-free networks and dynamical consequences.** / Ma, Xiaojuan; Huang, Liang; Lai, Ying-Cheng; Zheng, Zhigang.

Research output: Contribution to journal › Article

*Physical Review. E, Statistical, Nonlinear and Soft Matter Physics*, vol. 79, no. 5, 056106. https://doi.org/10.1103/PhysRevE.79.056106

}

TY - JOUR

T1 - Emergence of loop structure in scale-free networks and dynamical consequences

AU - Ma, Xiaojuan

AU - Huang, Liang

AU - Lai, Ying-Cheng

AU - Zheng, Zhigang

PY - 2009/5

Y1 - 2009/5

N2 - 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.

AB - 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.

KW - complex networks

KW - eigenvalues and eigenfunctions

KW - network topology

KW - nonlinear dynamical systems

KW - synchronisation

KW - synchronization

KW - statistics

U2 - 10.1103/PhysRevE.79.056106

DO - 10.1103/PhysRevE.79.056106

M3 - Article

VL - 79

JO - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics

JF - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics

SN - 1539-3755

IS - 5

M1 - 056106

ER -