We prove that, via an appropriate scaling, the degree of a fixed vertex in the Barabàsi-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.

On the continuous-time limit of the Barabàsi-Albert random graph

Federico Polito;Laura Sacerdote
2020

Abstract

We prove that, via an appropriate scaling, the degree of a fixed vertex in the Barabàsi-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.
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https://arxiv.org/pdf/1607.04183
Barabàsi-Albert model, Preferential attachment random graphs, Planted model, Discrete- and continuous-time models, Yule model
Angelica Pachon, Federico Polito, Laura Sacerdote
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2318/1730660
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