Tight Bounds for the Cover Times of Random Walks with Heterogeneous Step Lengths

Search patterns of randomly oriented steps of different lengths have been observed on all scales of the biological world, ranging from the microscopic to the ecological, including in protein motors, bacteria, T-cells, honeybees, marine predators, and more. Through different models, it has been demonstrated that adopting a variety in the magnitude of the step lengths can greatly improve the search efficiency. However, the precise connection between the search efficiency and the number of step lengths in the repertoire of the searcher has not been identified. Motivated by biological examples in one-dimensional terrains, a recent paper studied the best cover time on an n-node cycle that can be achieved by a random walk process that uses k step lengths. By tuning the lengths and corresponding probabilities the authors therein showed that the best cover time is roughly n 1+Θ(1/k). While this bound is useful for large values of k, it is hardly informative for small k values, which are of interest in biology. In this paper, we provide a tight bound for the cover time of such a walk, for every integer k > 1. Specifically, up to lower order polylogarithmic factors, the upper bound on the cover time is a polynomial in n of exponent 1+ 1/(2k−1). For k = 2, 3, 4 and 5 the exponent is thus 4/3 , 6/5 , 8/7 , and 10/9 , respectively. Informally, our result implies that, as long as the number of step lengths k is not too large, incorporating an additional step length to the repertoire of the process enables to improve the cover time by a polynomial factor, but the extent of the improvement gradually decreases with k

Random Walks with Multiple Step Lengths

International audienceIn nature, search processes that use randomly oriented steps of different lengths have been observed at both the microscopic and the macroscopic scales. Physicists have analyzed in depth two such processes on grid topologies: Intermittent Search, which uses two step lengths, and Lévy Walk, which uses many. Taking a computational perspective, this paper considers the number of distinct step lengths k as a complexity measure of the considered process. Our goal is to understand what is the optimal achievable time needed to cover the whole terrain, for any given value of k. Attention is restricted to dimension one, since on higher dimensions, the simple random walk already displays a quasi linear cover time. We say X is a k-intermittent search on the one dimensional n-node cycle if there exists a probability distribution $p = (pi) k i=1$ , and integers L1, L2,. .. , L k , such that on each step X makes a jump $±Li$ with probability pi, where the direction of the jump (+ or −) is chosen independently with probability 1/2. When performing a jump of length Li, the process consumes time Li, and is only considered to visit the last point reached by the jump (and not any other intermediate nodes). This assumption is consistent with biological evidence, in which entities do not search while moving ballistically. We provide upper and lower bounds for the cover time achievable by k-intermittent searches for any integer k. In particular, we prove that in order to reduce the cover time $Θ(n 2)$ of a simple random walk to linear in n up to logarithmic factors, roughly log n log log n step lengths are both necessary and sufficient, and we provide an example where the lengths form an exponential sequence. In addition, inspired by the notion of intermittent search, we introduce the Walk or Probe problem, which can be defined with respect to arbitrary graphs. Here, it is assumed that querying (probing) a node takes significantly more time than moving to a random neighbor. Hence, to efficiently probe all nodes, the goal is to balance the time spent walking randomly and the time spent probing. We provide preliminary results for connected graphs and regular graphs

Marches de Lévy intermittentes, et leurs applications aux recherches biologiques

Throughout the last two decades, a type of trajectories has been found to be almost ubiquitous in biological searches: the Lévy Patterns. Such patterns are fractal, with searches happening in self-similar clusters. Their hallmark is that their step-lengths are distributed in a power-law with some exponent μ ∈ (1, 3). This discovery lead to two intriguing questions: first, do these patterns emerge from an internal mechanism of the searcher, or from the interaction with the environment? Second, and independently of the previous question: what do these searchers have in common? When can we expect to see a Lévy Pattern of exponent μ? And how much does the knowledge of μ inform on the biological situation? This dissertation is an attempt at shedding some light on the topic, especially when the searcher can only detect targets intermittently, by studying the Lévy. Walk model, a random walk model in which the lengths of the steps are drawn according to a power-law of exponent μ. In the first chapter, I will provide more background in the foraging literature, especially in the Lévy Foraging literature. I will also provide the definitions of the probability models – Markov Chains, random walks on Euclidean spaces and, to a minor extent, on graphs – we will need in the theoretical analyses. In Chapter 2, I will present general facts about random walks on Euclidean spaces: how to analyse their search performances based on pointwise probability bounds, what is the distance achieved by a random walk with a general step-length distribution, and a useful monotonicity property. I will also study, as both a preliminary to the more involved proofs of later chapter, and for its own sake, a model of intermittent search on general graphs. Chapter 3 returns to the Lévy Walks, and contains an analysis of their efficiency when detection is intermittent, and targets appear in various sizes. In particular, I show that the much-debated inverse-square Lévy Walk is uniquely efficient in this setting. This is based on a joint work with Amos Korman (Guinard and Korman, 2020a), to be published. The question of how animals can perform Lévy Patterns has been much debated. Among possible solutions, it has been suggested that animals could approximate a Lévy distribution by having k different modes of movement, where k = 2, 3. Chapter 4, which condenses (Boczkowski et al., 2018a) and its refinement, (Guinard and Korman, 2020b), proves tight bounds for the performances of such an algorithm, and shows, in accordance with the literature, that having k = 3 modes may be sufficiently efficient in biological scenariosCes deux dernières décennies, la recherche en éthologie, et plus spécifiquement celle de l’investigation des comportements des animaux lorsqu’ils traquent de la nourriture, ont exhibé qu’une famille de motifs de trajectoires, connue sous le nom de Motifs de Lévy, est prévalente à travers le règne animal et même au-delà. Dans ces motifs auto-similaires, la longueur d’un pas en ligne droite suit une distribution de puissance, d’exposant μ ∈ (1, 3). Ces découvertes ont posé notamment deux questions: la première est de savoir si ces motifs émergent spontanément, par un mécanisme interne à l’organisme biologique, ou si ils sont la résultante d’interactions avec l’environnement (de la même manière que le mouvement Brownien s’explique par la collision de particules). La seconde est de savoir quelles informations pertinentes sur le plan biologique ces motifs, et notamment l’exposant μ, nous révèlent. À travers cette thèse, j’essaie d’apporter des éléments de réponse à ces questions complexes en étudiant le modèle des marches de Lévy, des marches aléatoires dont la longueur des pas est donnée par une loi de puissance, et qui génère, naturellement, des motifs de Lévy. Plus particulièrement, je l’étudie dans le contexte où la détection d’une cible ne peut être faite que de manière intermittente. Dans le premier chapitre, je parle plus en détail desdites recherches en éthologie, et je donne les bases mathématiques des modèles probabilistes de cette thèse (chaı̂ne de Markov, marches aléatoires dans les espaces euclidiens et, dans une mesure moins importante, dans des graphes). Au second chapitre, je discute des propriétés générales des marches aléatoires en espace euclidiens: comment obtenir des bornes sur les temps de recherche d’une marche aléatoire lorsque l’on en connaı̂t la distribution du marcheur; des bornes sur la distance parcourue par un marcheur après un certain temps; ainsi qu’une propriété utile de monotonie. En introduction aux preuves plus complexes des chapitres suivants, j’étudie un modèle de recherche intermittente sur un graphe. Au troisième chapitre, je montre comment les performances des marches de Lévy,dans le modèle intermittent de détection, dépend de manière cruciale de la taille des cibles, et je montre que ces considérations sont opérantes à un niveau biologiquement pertinent. Ce chapitre est basé sur un travail commun avec Amos Korman, à paraı̂tre (Guinard and Korman, 2020a). L’ultime chapitre est consacré à la question suivante: quelles sont les performances d’un agent incapable d’exécuter une marche de Lévy, mais qui peut en réaliser une approximation en utilisant k différentes longueurs fixées ? De tels modèles ont été suggérés en biologie avec k = 2, 3, et je montre notamment qu’utiliser seule-ment trois modes est efficace pour un espace d’une taille biologiquement pertinente.Ce chapitre est basé sur (Boczkowski et al., 2018a) et (Guinard and Korman, 2020b)

Marches de Lévy intermittentes, et leurs applications aux recherches biologiques

Throughout the last two decades, a type of trajectories has been found to be almost ubiquitous in biological searches: the Lévy Patterns. Such patterns are fractal, with searches happening in self-similar clusters. Their hallmark is that their step-lengths are distributed in a power-law with some exponent μ ∈ (1, 3). This discovery lead to two intriguing questions: first, do these patterns emerge from an internal mechanism of the searcher, or from the interaction with the environment? Second, and independently of the previous question: what do these searchers have in common? When can we expect to see a Lévy Pattern of exponent μ? And how much does the knowledge of μ inform on the biological situation? This dissertation is an attempt at shedding some light on the topic, especially when the searcher can only detect targets intermittently, by studying the Lévy. Walk model, a random walk model in which the lengths of the steps are drawn according to a power-law of exponent μ. In the first chapter, I will provide more background in the foraging literature, especially in the Lévy Foraging literature. I will also provide the definitions of the probability models – Markov Chains, random walks on Euclidean spaces and, to a minor extent, on graphs – we will need in the theoretical analyses. In Chapter 2, I will present general facts about random walks on Euclidean spaces: how to analyse their search performances based on pointwise probability bounds, what is the distance achieved by a random walk with a general step-length distribution, and a useful monotonicity property. I will also study, as both a preliminary to the more involved proofs of later chapter, and for its own sake, a model of intermittent search on general graphs. Chapter 3 returns to the Lévy Walks, and contains an analysis of their efficiency when detection is intermittent, and targets appear in various sizes. In particular, I show that the much-debated inverse-square Lévy Walk is uniquely efficient in this setting. This is based on a joint work with Amos Korman (Guinard and Korman, 2020a), to be published. The question of how animals can perform Lévy Patterns has been much debated. Among possible solutions, it has been suggested that animals could approximate a Lévy distribution by having k different modes of movement, where k = 2, 3. Chapter 4, which condenses (Boczkowski et al., 2018a) and its refinement, (Guinard and Korman, 2020b), proves tight bounds for the performances of such an algorithm, and shows, in accordance with the literature, that having k = 3 modes may be sufficiently efficient in biological scenariosCes deux dernières décennies, la recherche en éthologie, et plus spécifiquement celle de l’investigation des comportements des animaux lorsqu’ils traquent de la nourriture, ont exhibé qu’une famille de motifs de trajectoires, connue sous le nom de Motifs de Lévy, est prévalente à travers le règne animal et même au-delà. Dans ces motifs auto-similaires, la longueur d’un pas en ligne droite suit une distribution de puissance, d’exposant μ ∈ (1, 3). Ces découvertes ont posé notamment deux questions: la première est de savoir si ces motifs émergent spontanément, par un mécanisme interne à l’organisme biologique, ou si ils sont la résultante d’interactions avec l’environnement (de la même manière que le mouvement Brownien s’explique par la collision de particules). La seconde est de savoir quelles informations pertinentes sur le plan biologique ces motifs, et notamment l’exposant μ, nous révèlent. À travers cette thèse, j’essaie d’apporter des éléments de réponse à ces questions complexes en étudiant le modèle des marches de Lévy, des marches aléatoires dont la longueur des pas est donnée par une loi de puissance, et qui génère, naturellement, des motifs de Lévy. Plus particulièrement, je l’étudie dans le contexte où la détection d’une cible ne peut être faite que de manière intermittente. Dans le premier chapitre, je parle plus en détail desdites recherches en éthologie, et je donne les bases mathématiques des modèles probabilistes de cette thèse (chaı̂ne de Markov, marches aléatoires dans les espaces euclidiens et, dans une mesure moins importante, dans des graphes). Au second chapitre, je discute des propriétés générales des marches aléatoires en espace euclidiens: comment obtenir des bornes sur les temps de recherche d’une marche aléatoire lorsque l’on en connaı̂t la distribution du marcheur; des bornes sur la distance parcourue par un marcheur après un certain temps; ainsi qu’une propriété utile de monotonie. En introduction aux preuves plus complexes des chapitres suivants, j’étudie un modèle de recherche intermittente sur un graphe. Au troisième chapitre, je montre comment les performances des marches de Lévy,dans le modèle intermittent de détection, dépend de manière cruciale de la taille des cibles, et je montre que ces considérations sont opérantes à un niveau biologiquement pertinent. Ce chapitre est basé sur un travail commun avec Amos Korman, à paraı̂tre (Guinard and Korman, 2020a). L’ultime chapitre est consacré à la question suivante: quelles sont les performances d’un agent incapable d’exécuter une marche de Lévy, mais qui peut en réaliser une approximation en utilisant k différentes longueurs fixées ? De tels modèles ont été suggérés en biologie avec k = 2, 3, et je montre notamment qu’utiliser seule-ment trois modes est efficace pour un espace d’une taille biologiquement pertinente.Ce chapitre est basé sur (Boczkowski et al., 2018a) et (Guinard and Korman, 2020b)

Intermittent inverse-square Lévy walks are optimal for finding targets of all sizes

International audienceLévy walks are random walk processes whose step lengths follow a long-tailed power-law distribution. Because of their abundance as movement patterns of biological organisms, substantial theoretical efforts have been devoted to identifying the foraging circumstances that would make such patterns advantageous. However, despite extensive research, there is currently no mathematical proof indicating that Lévy walks are, in any manner, preferable strategies in higher dimensions than one. Here, we prove that in finite two-dimensional terrains, the inverse-square Lévy walk strategy is extremely efficient at finding sparse targets of arbitrary size and shape. Moreover, this holds even under the weak model of intermittent detection. Conversely, any other intermittent Lévy walk fails to efficiently find either large targets or small ones. Our results shed new light on the Lévy foraging hypothesis and are thus expected to affect future experiments on animals performing Lévy walks

Random Walks with Multiple Step Lengths

International audienceIn nature, search processes that use randomly oriented steps of different lengths have been observed at both the microscopic and the macroscopic scales. Physicists have analyzed in depth two such processes on grid topologies: Intermittent Search, which uses two step lengths, and Lévy Walk, which uses many. Taking a computational perspective, this paper considers the number of distinct step lengths k as a complexity measure of the considered process. Our goal is to understand what is the optimal achievable time needed to cover the whole terrain, for any given value of k. Attention is restricted to dimension one, since on higher dimensions, the simple random walk already displays a quasi linear cover time. We say X is a k-intermittent search on the one dimensional n-node cycle if there exists a probability distribution $p = (pi) k i=1$ , and integers L1, L2,. .. , L k , such that on each step X makes a jump $±Li$ with probability pi, where the direction of the jump (+ or −) is chosen independently with probability 1/2. When performing a jump of length Li, the process consumes time Li, and is only considered to visit the last point reached by the jump (and not any other intermediate nodes). This assumption is consistent with biological evidence, in which entities do not search while moving ballistically. We provide upper and lower bounds for the cover time achievable by k-intermittent searches for any integer k. In particular, we prove that in order to reduce the cover time $Θ(n 2)$ of a simple random walk to linear in n up to logarithmic factors, roughly log n log log n step lengths are both necessary and sufficient, and we provide an example where the lengths form an exponential sequence. In addition, inspired by the notion of intermittent search, we introduce the Walk or Probe problem, which can be defined with respect to arbitrary graphs. Here, it is assumed that querying (probing) a node takes significantly more time than moving to a random neighbor. Hence, to efficiently probe all nodes, the goal is to balance the time spent walking randomly and the time spent probing. We provide preliminary results for connected graphs and regular graphs