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On the Search Efficiency of Parallel Lévy Walks on ${\mathbb Z}^2$

Andrea Clementi 1 Francesco d'Amore 2 George Giakkoupis 3 Emanuele Natale 2
2 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
3 WIDE - the World Is Distributed Exploring the tension between scale and coordination
Inria Rennes – Bretagne Atlantique , IRISA-D1 - SYSTÈMES LARGE ÉCHELLE
Abstract : Motivated by the \emph{Lévy flight foraging hypothesis} -- the premise that the movement of various animal species searching for food resembles a \emph{Lévy walk} -- we study the search efficiency of parallel Lévy walks on the infinite 2-dimensional grid. We assume that $k$ independent identical discrete-time Lévy walks, with exponent parameter $\alpha \in(1,+\infty)$, start simultaneously at the origin, and we are interested in the time $h_{\alpha,k,\ell}$ until some walk visits a given target node at distance $\ell$ from the origin. First, we observe that the total work, i.e., the product $k\cdot h_{\alpha,k,\ell}$, is at least $\Omega(\ell^2)$, for any combination of the parameters $\alpha,k,\ell$. Then we provide a comprehensive analysis of the time and work, for the complete range of these parameters. Our main finding is that for any $\alpha$, there is a specific choice of $k$ that achieves optimal work, $\tilde{\mathcal{O}}\left(\ell^2\right)$, whereas all other choices of $k$ result in sub-optimal work. In particular, in the interesting super-diffusive regime of $2 < \alpha < 3$, the optimal value for $k$ is $ \tilde \Theta\left(\ell^{1-(\alpha-2)}\right)$. Our results should be contrasted with several previous works showing that the exponent $\alpha = 2$ is optimal for a wide range of related search problems on the plane. On the contrary, in our setting of multiple walks which measures efficiency in terms of the natural notion of work, no single exponent is optimal: for each $\alpha$ (and $\ell$) there is a specific choice of $k$ that yields optimal efficiency.
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  • HAL Id : hal-02530253, version 3
  • ARXIV : 2004.01562

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Andrea Clementi, Francesco d'Amore, George Giakkoupis, Emanuele Natale. On the Search Efficiency of Parallel Lévy Walks on ${\mathbb Z}^2$. [Research Report] Inria & Université Cote d'Azur, CNRS, I3S, Sophia Antipolis, France; Università degli Studi di Roma "Tor Vergata"; Univ Rennes, Inria, CNRS, IRISA, France. 2020. ⟨hal-02530253v3⟩

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