### Abstract

We consider the question of determining how the topological structure influences a consensus dynamical processes taking place on a network. By considering a large dataset of real-world networks we first determine that the removal of edges according to their communicability angle—an angle between

position vectors of the nodes in an Euclidean communicability space—increases the average time of consensus by a factor of 5:68 in real-world networks. The edge betweenness centrality also identifies—in a smaller proportion—those critical edges for the consensus dynamics, i.e., its removal increases the

time of consensus by a factor of 3:70. We justify theoretically these findings on the basis of the role played by the algebraic connectivity and the isoperimetric number of networks on the dynamical process studied, and their connections with the properties mentioned before. Finally, we study the role played by global topological parameters of networks on the consensus dynamics. We determine that the network density and the average distance-sum—an analogous of the node degree for shortest-path distances, account for more than 80% of the variance of the average time of consensus in the real-world networks studied.

position vectors of the nodes in an Euclidean communicability space—increases the average time of consensus by a factor of 5:68 in real-world networks. The edge betweenness centrality also identifies—in a smaller proportion—those critical edges for the consensus dynamics, i.e., its removal increases the

time of consensus by a factor of 3:70. We justify theoretically these findings on the basis of the role played by the algebraic connectivity and the isoperimetric number of networks on the dynamical process studied, and their connections with the properties mentioned before. Finally, we study the role played by global topological parameters of networks on the consensus dynamics. We determine that the network density and the average distance-sum—an analogous of the node degree for shortest-path distances, account for more than 80% of the variance of the average time of consensus in the real-world networks studied.

Original language | English |
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Article number | 052809 |

Number of pages | 10 |

Journal | Physical Review E |

Volume | 92 |

Issue number | 5 |

DOIs | |

Publication status | Published - 19 Nov 2015 |

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### Keywords

- capillary flow
- topology
- vector spaces
- algebraic connectivity
- dynamical process
- edge betweenness centrality
- isoperimetric number
- network consensus
- real-world networks
- topological parameters
- topological structure
- dynamics

### Cite this

Estrada, E., Vargas Estrada, E., & Ando, H. (2015). Communicability angles reveal critical edges for network consensus dynamics.

*Physical Review E*,*92*(5), [052809]. https://doi.org/10.1103/PhysRevE.92.052809