### Abstract

Language | English |
---|---|

Pages | 89-119 |

Number of pages | 31 |

Journal | Physics Reports |

Volume | 514 |

Issue number | 3 |

Early online date | 17 Jan 2012 |

DOIs | |

Publication status | Published - May 2012 |

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### Keywords

- quantitative measurement
- communicability
- network correlation

### Cite this

*Physics Reports*,

*514*(3), 89-119. https://doi.org/10.1016/j.physrep.2012.01.006

}

*Physics Reports*, vol. 514, no. 3, pp. 89-119. https://doi.org/10.1016/j.physrep.2012.01.006

**The physics of communicability in complex networks.** / Estrada, Ernesto; Hatano, Naomichi; Benzi, Michele.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The physics of communicability in complex networks

AU - Estrada, Ernesto

AU - Hatano, Naomichi

AU - Benzi, Michele

PY - 2012/5

Y1 - 2012/5

N2 - A fundamental problem in the study of complex networks is to provide quantitative measures of correlation and information flow between different parts of a system. To this end, several notions of communicability have been introduced and applied to a wide variety of real-world networks in recent years. Several such communicability functions are reviewed in this paper. It is emphasized that communication and correlation in networks can take place through many more routes than the shortest paths, a fact that may not have been sufficiently appreciated in previously proposed correlation measures. In contrast to these, the communicability measures reviewed in this paper are defined by taking into account all possible routes between two nodes, assigning smaller weights to longer ones. This point of view naturally leads to the definition of communicability in terms of matrix functions, such as the exponential, resolvent, and hyperbolic functions, in which the matrix argument is either the adjacency matrix or the graph Laplacian associated with the network. Considerable insight on communicability can be gained by modeling a network as a system of oscillators and deriving physical interpretations, both classical and quantummechanical, of various communicability functions. Applications of communicability measures to the analysis of complex systems are illustrated on a variety of biological, physical and social networks. The last part of the paper is devoted to a review of the notion of locality in complex networks and to computational aspects that by exploiting sparsity can greatly reduce the computational efforts for the calculation of communicability functions for large networks.

AB - A fundamental problem in the study of complex networks is to provide quantitative measures of correlation and information flow between different parts of a system. To this end, several notions of communicability have been introduced and applied to a wide variety of real-world networks in recent years. Several such communicability functions are reviewed in this paper. It is emphasized that communication and correlation in networks can take place through many more routes than the shortest paths, a fact that may not have been sufficiently appreciated in previously proposed correlation measures. In contrast to these, the communicability measures reviewed in this paper are defined by taking into account all possible routes between two nodes, assigning smaller weights to longer ones. This point of view naturally leads to the definition of communicability in terms of matrix functions, such as the exponential, resolvent, and hyperbolic functions, in which the matrix argument is either the adjacency matrix or the graph Laplacian associated with the network. Considerable insight on communicability can be gained by modeling a network as a system of oscillators and deriving physical interpretations, both classical and quantummechanical, of various communicability functions. Applications of communicability measures to the analysis of complex systems are illustrated on a variety of biological, physical and social networks. The last part of the paper is devoted to a review of the notion of locality in complex networks and to computational aspects that by exploiting sparsity can greatly reduce the computational efforts for the calculation of communicability functions for large networks.

KW - quantitative measurement

KW - communicability

KW - network correlation

U2 - 10.1016/j.physrep.2012.01.006

DO - 10.1016/j.physrep.2012.01.006

M3 - Article

VL - 514

SP - 89

EP - 119

JO - Physics Reports

T2 - Physics Reports

JF - Physics Reports

SN - 0370-1573

IS - 3

ER -