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Upper large deviations for power-weighted edge lengths in spatial random networks

Part of: Geometric probability and stochastic geometry Graph theory Limit theorems Stochastic processes

Published online by Cambridge University Press:  05 June 2023

Christian Hirsch Open the ORCID record for Christian Hirsch [Opens in a new window]  and
Daniel Willhalm
Christian Hirsch*
Affiliation:
Aarhus University
Daniel Willhalm*
Affiliation:
University of Groningen and CogniGron
*
*Postal address: Department of Mathematics, Ny Munkegade 118, 8000 Aarhus C, DK. Email address: hirsch@math.au.dk
**Postal address: Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, Nijenborgh 9, 9747 AG Groningen, NL. CogniGron (Groningen Cognitive Systems and Materials Center), Nijenborgh 4, 9747 AG Groningen, NL. Email address: d.willhalm@rug.nl
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Abstract

We study the large-volume asymptotics of the sum of power-weighted edge lengths $\sum_{e \in E}|e|^\alpha$ in Poisson-based spatial random networks. In the regime $\alpha > d$, we provide a set of sufficient conditions under which the upper-large-deviation asymptotics are characterized by a condensation phenomenon, meaning that the excess is caused by a negligible portion of Poisson points. Moreover, the rate function can be expressed through a concrete optimization problem. This framework encompasses in particular directed, bidirected, and undirected variants of the k-nearest-neighbor graph, as well as suitable $\beta$-skeletons.

Keywords

Large deviationscondensationk-nearest-neighbor graphβ-skeleton

MSC classification

Primary: 60F10: Large deviations
Secondary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes 05C99: None of the above, but in this section
Type
Original Article
Information
Advances in Applied Probability , Volume 56 , Issue 1 , March 2024 , pp. 34 - 70
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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