## Abstract

We prove that, via an appropriate scaling, the degree of a fixed vertex in the Barabási–Albert model appeared at a large enough time converges in distribution to a Yule process. Using this relation we explain why the limit degree distribution of a vertex chosen uniformly at random (as the number of vertices goes to infinity), coincides with the limit distribution of the number of species in a genus selected uniformly at random in a Yule model (as time goes to infinity). To prove this result we do not assume that the number of vertices increases exponentially over time (linear rates). On the contrary, we retain their natural growth with a constant rate superimposing to the overall graph structure a suitable set of processes that we call the planted model and introducing an ad-hoc sampling procedure.

Original language | English |
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Article number | 125177 |

Journal | Applied Mathematics and Computation |

Volume | 378 |

Early online date | 3 Apr 2020 |

DOIs | |

Publication status | Published - 1 Aug 2020 |

## Keywords

- Barabási–Albert model
- Discrete - and continuous-time models
- Planted model
- Preferential attachment random graphs
- Yule model