Electronic Communications in Probability

Escape of resources in a distributed clustering process

Jacob van den Berg, Marcelo Hilário, and Alexander Holroyd

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In a distributed clustering algorithm introduced by Coffman, Courtois, Gilbert and Piret [1], each vertex of $\mathbb{Z}^d$ receives an initial amount of a resource, and, at each iteration, transfers all of its resource to the neighboring vertex which currently holds the maximum amount of resource. In [4] it was shown that, if the distribution of the initial quantities of resource is invariant under lattice translations, then the flow of resource at each vertex eventually stops almost surely, thus solving a problem posed in [2]. In this article we prove the existence of translation-invariant initial distributions for which resources nevertheless escape to infinity, in the sense that the the final amount of resource at a given vertex is strictly smaller in expectation than the initial amount. This answers a question posed in [4].

Article information

Electron. Commun. Probab., Volume 15 (2010), paper no. 40, 442-448.

Accepted: 30 September 2010
First available in Project Euclid: 6 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 68M14: Distributed systems

Clustering process random spanning tree

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van den Berg, Jacob; Hilário, Marcelo; Holroyd, Alexander. Escape of resources in a distributed clustering process. Electron. Commun. Probab. 15 (2010), paper no. 40, 442--448. doi:10.1214/ECP.v15-1567. https://projecteuclid.org/euclid.ecp/1465243983

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