## Abstract

The average Watts-Strogatz clustering coecient and the network transitivity

are widely used descriptors for characterizing the transitivity of relations in

real-world graphs (networks). These indices are bounded between zero and one, with low values indicating poor transtivity and large ones indicating a high proportion of closed triads in the graphs. Here, we prove that these two indices diverge for windmill graphs when the number of nodes tends to infinity. We also give evidence that this divergence occurs in many real-world networks, especially in citation and collaboration graphs. We obtain analytic expressions for the eigenvalues and eigenvectors of the adjacency and the Laplacian matrices of the windmill graphs. Using this information we show the main characteristics of two dynamical processes when taking place on windmill graphs: synchronization and epidemic spreading. Finally, we show that many of

the structural and dynamical properties of a real-world citation network are well reproduced by the appropriate windmill graph, showing the potential of these graphs as models for certain classes of real-world networks.

are widely used descriptors for characterizing the transitivity of relations in

real-world graphs (networks). These indices are bounded between zero and one, with low values indicating poor transtivity and large ones indicating a high proportion of closed triads in the graphs. Here, we prove that these two indices diverge for windmill graphs when the number of nodes tends to infinity. We also give evidence that this divergence occurs in many real-world networks, especially in citation and collaboration graphs. We obtain analytic expressions for the eigenvalues and eigenvectors of the adjacency and the Laplacian matrices of the windmill graphs. Using this information we show the main characteristics of two dynamical processes when taking place on windmill graphs: synchronization and epidemic spreading. Finally, we show that many of

the structural and dynamical properties of a real-world citation network are well reproduced by the appropriate windmill graph, showing the potential of these graphs as models for certain classes of real-world networks.

Original language | English |
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Pages (from-to) | 249-263 |

Number of pages | 15 |

Journal | Linear Algebra and its Applications |

Volume | 488 |

Early online date | 17 Nov 2015 |

DOIs | |

Publication status | Published - 1 Jan 2016 |

## Keywords

- clustering coecients
- citation networks
- collaboration graphs
- real-world networks
- graph spectra
- windmill graphs