Intermittent random walks for an optimal search strategy: one-dimensional case

Oshanin, Gleb; Wio, Horacio S.; Lindenberg, Katja; Burlatsky, S. F.

doi:10.1088/0953-8984/19/6/065142

Cited by 71 publications

(109 citation statements)

References 34 publications

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“…As an alternative to Lévy search, intermittent strategies have been introduced to improve the efficiency of diffusive search (17)(18)(19)(20)(21). Intermittent search requires that the searcher occasionally shifts focus from the search and concentrates on fast relocation.…”

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confidence: 99%

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Lévy strategies in intermittent search processes are advantageous

Lomholt

Koren

Metzler

et al. 2008

Proc. Natl. Acad. Sci. U.S.A.

245

230

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Intermittent search processes switch between local Brownian search events and ballistic relocation phases. We demonstrate analytically and numerically in one dimension that when relocation times are Lévy distributed, resulting in a Lévy walk dynamics, the search process significantly outperforms the previously investigated case of exponentially distributed relocation times: The resulting Lévy walks reduce oversampling and thus further optimize the intermittent search strategy in the critical situation of rare targets. We also show that a searching agent that uses the Lévy strategy is much less sensitive to the target density, which would require considerably less adaptation by the searcher.random processes | optimization | Lévy walk | movement ecology R andom search processes occur in many areas, from chemical reactions of diffusing reactants (1) to the foraging behavior of bacteria and animals (2, 3). Of general importance is the search efficiency. Brownian search in one and two dimensions involves frequent returns to an area, leading to oversampling. Higher efficiency, can be achieved, for instance, by facilitated diffusion in gene regulation (4) or by controlled motion in foraging (2, 3). From theoretical and data analysis Lévy strategies, in which the searching agent performs excursions whose length is drawn from distributions with a heavy tailfor 0 < α < 2 were shown to be advantageous (5-16); occasional long excursions assist in exploring previously unvisited areas and significantly reduce oversampling.As an alternative to Lévy search, intermittent strategies have been introduced to improve the efficiency of diffusive search (17-21). Intermittent search requires that the searcher occasionally shifts focus from the search and concentrates on fast relocation. The relocation phase implies that the searcher is wasting time in the short run because the target cannot be spotted during it. However, the overall search efficiency is improved by introducing the searcher to previously unexplored areas (17-21).In refs. 18 and 20 relocation events were assumed to occur in a random direction for exponentially distributed time spans, giving rise to a Markovian process. We show here analytically and numerically in one dimension that this is only a partial solution to oversampling, as eventually the central limit theorem (CLT) reduces the process to a Brownian random walk with jumps on the scale of vτ 2 , where τ 2 is the typical time spent in a relocation event. In practice, revisits can be reduced by adjusting the average time spent in search and relocation phases to the density of targets. Lévy strategies, on the other hand, fundamentally circumvent the CLT, and we here demonstrate a twofold advantage of them over the exponential distribution: Lévy walk intermittent processes find the target faster than exponential strategies in the critical case of rare targets, and their performance is much less dependent on adapting to the target density. Intermittent Search with Lévy RelocationsGeneralizing the model from ref. 20, w...

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confidence: 99%

“…The relocation phase implies that the searcher is wasting time in the short run because the target cannot be spotted during it. However, the overall search efficiency is improved by introducing the searcher to previously unexplored areas (17)(18)(19)(20)(21).…”

mentioning

confidence: 99%

Lévy strategies in intermittent search processes are advantageous

Lomholt

Koren

Metzler

et al. 2008

Proc. Natl. Acad. Sci. U.S.A.

245

230

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“…How does the message find its way, let alone so efficiently? Searching and navigation problems such as this [3,4,5,6,7,8,9,10,11,12,13] are relevant to several disciplines, from sociology, to efficient algorithms in computer science, to the understanding of the foraging behavior of insects and animals.…”

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confidence: 99%

Asymptotic Behavior of the Kleinberg Model

Carmi

Carter²,

Sun³

et al. 2009

Phys. Rev. Lett.

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We study Kleinberg navigation (the search of a target in a d-dimensional lattice, where each site is connected to one other random site at distance r, with probability ∼ r −α ) by means of an exact master equation for the process. We show that the asymptotic scaling behavior for the delivery time T to a target at distance L scales as T ∼ ln 2 L when α = d, and otherwise as T ∼ L x , with, and x = 1 for α > d + 1. These values of x exceed the rigorous lower-bounds established by Kleinberg. We also address the situation where there is a finite probability for the message to get lost along its way and find short delivery times (conditioned upon arrival) for a wide range of α's.

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“…Similar dependence of the optimal parameters on the observation time has been recently reported in Refs. [56,57], which optimised the number of distinct sites visited by intermittent random walks. Note that ξ opt → 2 as ǫ → 0, but for any finite ǫ it is greater than 2.…”

Section: The Variance For α = 2 and Arbitrary ǫmentioning

confidence: 99%

Distribution of the least-squares estimators of a single Brownian trajectory diffusion coefficient

Boyer¹,

Dean²,

Mejía‐Monasterio³

et al. 2013

J. Stat. Mech.

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Abstract. In this paper we study the distribution function P (u α ) of the estimatorst dt, which optimise the least-squares fitting of the diffusion coefficient D f of a single d-dimensional Brownian trajectory B t . We pursue here the optimisation further by considering a family of weight functions of the form ω(t) = (t 0 + t) −α , where t 0 is a time lag and α is an arbitrary real number, and seeking such values of α for which the estimators most efficiently filter out the fluctuations. We calculate P (u α ) exactly for arbitrary α and arbitrary spatial dimension d, and show that only for α = 2 the distribution P (u α ) converges, as ǫ = t 0 /T → 0, to the Dirac delta-function centered at the ensemble average value of the estimator. This allows us to conclude that only the estimators with α = 2 possess an ergodic property, so that the ensemble averaged diffusion coefficient can be obtained with any necessary precision from a single trajectory data, but at the expense of a progressively higher experimental resolution. For any α = 2 the distribution attains, as ǫ → 0, a certain limiting form with a finite variance, which signifies that such estimators are not ergodic.

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Intermittent random walks for an optimal search strategy: one-dimensional case

Cited by 71 publications

References 34 publications

Lévy strategies in intermittent search processes are advantageous

Lévy strategies in intermittent search processes are advantageous

Asymptotic Behavior of the Kleinberg Model

Distribution of the least-squares estimators of a single Brownian trajectory diffusion coefficient

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