## Abstract

It is well known that many random graphs with infinite variance degrees are ultra-small. More precisely, for configuration models and preferential attachment models where the proportion of vertices of degree at least k is approximately k -(τ - 1) with τ ∈ (2,3), typical distances between pairs of vertices in a graph of size n are asymptotic to 2 log log n | log ( τ - 2 ) | and 4 log log n | log ( τ - 2 ) | , respectively. In this paper, we investigate the behavior of the diameter in such models. We show that the diameter is of order log log n precisely when the minimal forward degree d fwd of vertices is at least 2. We identify the exact constant, which equals that of the typical distances plus 2 / log d fwd . Interestingly, the proof for both models follows identical steps, even though the models are quite different in nature.

Original language | English |
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Pages (from-to) | 444-498 |

Number of pages | 55 |

Journal | Random Structures and Algorithms |

Volume | 54 |

Issue number | 3 |

DOIs | |

Publication status | Published - May 2019 |

## Keywords

- configuration model
- diameter
- preferential attachment model
- random graphs
- scale free
- ultra-small