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June 2018 Typical distances in the directed configuration model
Pim van der Hoorn, Mariana Olvera-Cravioto
Ann. Appl. Probab. 28(3): 1739-1792 (June 2018). DOI: 10.1214/17-AAP1342

Abstract

We analyze the distribution of the distance between two nodes, sampled uniformly at random, in digraphs generated via the directed configuration model, in the supercritical regime. Under the assumption that the covariance between the in-degree and out-degree is finite, we show that the distance grows logarithmically in the size of the graph. In contrast with the undirected case, this can happen even when the variance of the degrees is infinite. The main tool in the analysis is a new coupling between a breadth-first graph exploration process and a suitable branching process based on the Kantorovich–Rubinstein metric. This coupling holds uniformly for a much larger number of steps in the exploration process than existing ones, and is therefore of independent interest.

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Pim van der Hoorn. Mariana Olvera-Cravioto. "Typical distances in the directed configuration model." Ann. Appl. Probab. 28 (3) 1739 - 1792, June 2018. https://doi.org/10.1214/17-AAP1342

Information

Received: 1 November 2015; Revised: 1 August 2017; Published: June 2018
First available in Project Euclid: 1 June 2018

zbMATH: 1393.05101
MathSciNet: MR3809476
Digital Object Identifier: 10.1214/17-AAP1342

Subjects:
Primary: 05C80
Secondary: 60B10

Keywords: branching processes , Couplings , directed configuration model , Kantorovich–Rubinstein distance , random digraphs , typical distances

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.28 • No. 3 • June 2018
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