Uncovering the statistical patterns that characterize the trajectories humans follow during their daily activity is not only a major intellectual challenge, but also of importance for public health 1-5 , city planning 6-8 , traffic engineering 9, 10 and economic forecasting 11 . For example, quantifiable models of human mobility are indispensable for predicting the spread of biological pathogens 1-5 or mobile phone viruses 12 .In the past few years the availability of mobile phone records, GPS data, and other datasets capturing aspects of human mobility have given a new empirically driven momentum to the subject. While the available datasets significantly differ in their reach and resolution, the results appear to agree on a number of quantitative characteristics of human mobility. For example, both dollar bill tracking 13 and mobile phone data 14 indicate that the

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...

Exact results for the first passage time and leapover statistics of symmetric and one-sided Lévy flights (LFs) are derived. LFs with a stable index alpha are shown to have leapover lengths that are asymptotically power law distributed with an index alpha for one-sided LFs and, surprisingly, with an index alpha/2 for symmetric LFs. The first passage time distribution scales like a power law with an index 1/2 as required by the Sparre-Andersen theorem for symmetric LFs, whereas one-sided LFs have a narrow distribution of first passage times. The exact analytic results are confirmed by extensive simulations.

We introduce and explore a model of an ensemble of enzymes searching, in parallel, a circular DNA strand for a target site. The agents performing the search combine local scanning-conducted by a 1D motion along the strandand random relocations on the strand-conducted via a confined motion in the medium containing the strand. Both the local scan mechanism and the relocation mechanism are considered general. The search durations are analysed, and their limiting probability distributions-for long DNA strands-are obtained in closed form. The results obtained (i) encompass the cases of single, parallel and massively parallel searches, taking place in the presence of either finite-mean or heavytailed relocation times, (ii) are applicable to a wide spectrum of local scan mechanisms including linear, Brownian, selfsimilar, and sub-diffusive motions, (iii) provide a quantitative theoretical justification for the necessity of the relocation mechanism, and (iv) facilitate the derivation of optimal relocation strategies.