The Annals of Applied Probability

Typical distances in the directed configuration model

Pim van der Hoorn and Mariana Olvera-Cravioto

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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.

Article information

Ann. Appl. Probab., Volume 28, Number 3 (2018), 1739-1792.

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

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Digital Object Identifier

Primary: 05C80: Random graphs [See also 60B20]
Secondary: 60B10: Convergence of probability measures

Random digraphs directed configuration model typical distances branching processes couplings Kantorovich–Rubinstein distance


van der Hoorn, Pim; Olvera-Cravioto, Mariana. Typical distances in the directed configuration model. Ann. Appl. Probab. 28 (2018), no. 3, 1739--1792. doi:10.1214/17-AAP1342.

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