Abstract
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].
Highlights
1.1 Definitions and statement of the main resultConsider, for d ≥ 1, the d-dimensional integer lattice
In a distributed clustering algorithm introduced by Coffman, Courtois, Gilbert and Piret [1], each vertex of 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
An(x) is chosen uniformly at random among the vertices around x that maximize Cn. Apart from those possible tie breaks, all the randomness is contained in the initial configuration
Summary
For d ≥ 1, the d-dimensional integer lattice. This is the graph with vertex set d , and edge set comprising all pairs of vertices (x, y) (= ( y, x)) with |x − y| = 1. Note that an(x) is the vertex to which the resources located at x at time n (if any) will be transferred during the (n + 1)-th step of the evolution. We say that there is a tie in x at time n if Cn(x) > 0 and the cardinality of Mn(x) is strictly greater than one In case this occurs, an(x) is chosen uniformly at random among the vertices around x that maximize Cn. Note that, apart from those possible tie breaks, all the randomness is contained in the initial configuration. Note that, when two or more vertices transfer their resources to the same vertex, these resources are added up This algorithm models a clustering process in the lattice starting from a disordered initial configuration. There exists a translation-invariant distribution for the initial configuration C0(x); x ∈ d such that, for each x ∈ d , C∞(x) < C0(x)
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