Lévy walk process in self-organization of pedestrian crowds

Robust spatial self-organization in crowds of asynchronous pedestrians

Human crowds display various self-organized collective behaviours, such as the spontaneous formation of unidirectional lanes in bidirectional pedestrian flows. In addition, parts of pedestrians' footsteps are known to be spontaneously synchronized in one-dimensional, single-file crowds. However, footstep synchronization in crowds with more freedom of movement remains unclear. We conducted experiments on bidirectional pedestrian flows (24 pedestrians in each group) and examined the relationship between collective footsteps and self-organized lane formation. Unlike in previous studies, pedestrians did not spontaneously synchronize their footsteps unless following external auditory cues. Moreover, footstep synchronization generated by external cues disturbed the flexibility of pedestrians' lateral movements and increased the structural instability of spatial organization. These results imply that, without external cues, pedestrians marching out of step with each other can efficiently self-organize into robust structures. Understanding how asynchronous individuals contribute to ordered collective behaviour might bring innovative perspectives to research fields concerned with self-organizing systems.

One-dimensional Lévy quasicrystal

Space-fractional quantum mechanics (SFQM) is a generalization of the standard quantum mechanics when the Brownian trajectories in Feynman path integrals are replaced by Lévy flights. We introduce Lévy quasicrystal by discretizing the space-fractional Schrödinger equation using the Grünwald-Letnikov derivatives and adding on-site quasiperiodic potential. The discretized version of the usual Schrödinger equation maps to the Aubry-André (AA) Hamiltonian, which supports localization-delocalization transition even in one dimension. We find the similarities between Lévy quasicrystal and the AA model with power-law hopping, and show that the Lévy quasicrystal supports a delocalization-localization transition as one tunes the quasiperiodic potential strength and shows the coexistence of localized and delocalized states separated by mobility edge. Hence, a possible realization of SFQM in optical experiments should be a new experimental platform to test the predictions of AA models in the presence of power-law hopping.

How Lévy Flights Triggered by the Presence of Defectors Affect Evolution of Cooperation in Spatial Games

Cooperation among individuals has been key to sustaining societies. However, natural selection favors defection over cooperation. Cooperation can be favored when the mobility of individuals allows cooperators to form a cluster (or group). Mobility patterns of animals sometimes follow a Lévy flight. A Lévy flight is a kind of random walk but it is composed of many small movements with a few big movements. The role of Lévy flights for cooperation has been studied by Antonioni and Tomassini, who showed that Lévy flights promoted cooperation combined with conditional movements triggered by neighboring defectors. However, the optimal condition for neighboring defectors and how the condition changes with the intensity of Lévy flights are still unclear. Here, we developed an agent-based model in a square lattice where agents perform Lévy flights depending on the fraction of neighboring defectors. We systematically studied the relationships among three factors for cooperation: sensitivity to defectors, the intensity of Lévy flights, and population density. Results of evolutionary simulations showed that moderate sensitivity most promoted cooperation. Then, we found that the shortest movements were best for cooperation when the sensitivity to defectors was high. In contrast, when the sensitivity was low, longer movements were best for cooperation. Thus, Lévy flights, the balance between short and long jumps, promoted cooperation in any sensitivity, which was confirmed by evolutionary simulations. Finally, as the population density became larger, higher sensitivity was more beneficial for cooperation to evolve. Our study highlights that Lévy flights are an optimal searching strategy not only for foraging but also for constructing cooperative relationships with others.

Functional duality in group criticality via ambiguous interactions

Critical phenomena are wildly observed in living systems. If the system is at criticality, it can quickly transfer information and achieve optimal response to external stimuli. Especially, animal collective behavior has numerous critical properties, which are related to other research regions, such as the brain system. Although the critical phenomena influencing collective behavior have been extensively studied, two important aspects require clarification. First, these critical phenomena never occur on a single scale but are instead nested from the micro- to macro-levels (e.g., from a Lévy walk to scale-free correlation). Second, the functional role of group criticality is unclear. To elucidate these aspects, the ambiguous interaction model is constructed in this study; this model has a common framework and is a natural extension of previous representative models (such as the Boids and Vicsek models). We demonstrate that our model can explain the nested criticality of collective behavior across several scales (considering scale-free correlation, super diffusion, Lévy walks, and 1/f fluctuation for relative velocities). Our model can also explain the relationship between scale-free correlation and group turns. To examine this relation, we propose a new method, applying partial information decomposition (PID) to two scale-free induced subgroups. Using PID, we construct information flows between two scale-free induced subgroups and find that coupling of the group morphology (i.e., the velocity distributions) and its fluctuation power (i.e., the fluctuation distributions) likely enable rapid group turning. Thus, the flock morphology may help its internal fluctuation convert to dynamic behavior. Our result sheds new light on the role of group morphology, which is relatively unheeded, retaining the importance of fluctuation dynamics in group criticality.

Parallelization of Array Method with Hybrid Programming: OpenMP and MPI

For parallelization of applications with high processing times and large amounts of storage in High Performance Computing (HPC) systems, shared memory programming and distributed memory programming have been used; a parallel application is represented by Parallel Task Graphs (PTGs) using Directed Acyclic Graphs (DAGs). For the execution of PTGs in HPC systems, a scheduler is executed in two phases: scheduling and allocation; the execution of the scheduler is considered an NP-complete combinatorial problem and requires large amounts of storage and long processing times. Array Method (AM) is a scheduler to execute the task schedule in a set of clusters; this method was programmed sequentially, analyzed and tested using real and synthetic application workloads in previous work. Analyzing the proposed designs of this method in this research work, the parallelization of the method is extended using hybrid OpenMP and MPI programming in a server farm and using a set of geographically distributed clusters; at the same time, a novel method for searching free resources in clusters using Lévy random walks is proposed. Synthetic and real workloads have been experimented with to evaluate the performance of the new parallel schedule and compare it to the sequential schedule. The metrics of makespan, waiting time, quality of assignments and search for free resources were evaluated; the results obtained and described in the experiments section show a better performance with the new version of the parallel algorithm compared to the sequential version. By using the parallel approach with hybrid programming applied to the extraction of characteristics of the PTGs, applied to the search for geographically distributed resources with Lévy random walks and applied to the metaheuristic used, the results of the metrics are improved. The makespan is decreased even when the loads increase, the times of the tasks in the waiting queue are decreased, the quality of assignments in the clusters is improved by causing the tasks with their subtasks to be assigned in the same clusters or in cluster neighbors and, finally, the searches for free resources are executed in different geographically distributed clusters, not sequentially.

Levy geometric graphs

We present a family of graphs with remarkable properties. They are obtained by connecting the points of a random walk when their distance is smaller than a given scale. Their degree (number of neighbors) does not depend on the graph's size but only on the considered scale. It follows a gamma distribution and thus presents an exponential decay. Levy flights are particular random walks with some power-law increments of infinite variance. When building the geometric graphs from them, we show from dimensional arguments that the number of connected components (clusters) follows an inverse power of the scale. The distribution of the size of their components, properly normalized, is scale invariant, which reflects the self-similar nature of the underlying process. This allows to test if a graph (including nonspatial ones) could possibly result from an underlying Levy process. When the scale increases, these graphs never tend towards a single cluster, the giant component. In other words, while the autocorrelation of the process scales as a power of the distance, they never undergo a phase transition of percolation type. The Levy graphs may find applications in community detection and in the analysis of collective behaviors as in face-to-face interaction networks.

Infinite density and relaxation for Lévy walks in an external potential: Hermite polynomial approach

Lévy walks are continuous-time random-walk processes with a spatiotemporal coupling of jump lengths and waiting times. We here apply the Hermite polynomial method to study the behavior of LWs with power-law walking time density for four different cases. First we show that the known result for the infinite density of an unconfined, unbiased LW is consistently recovered. We then derive the asymptotic behavior of the probability density function (PDF) for LWs in a constant force field, and we obtain the corresponding qth-order moments. In a harmonic external potential we derive the relaxation dynamic of the LW. For the case of a Poissonian walking time an exponential relaxation behavior is shown to emerge. Conversely, a power-law decay is obtained when the mean walking time diverges. Finally, we consider the case of an unconfined, unbiased LW with decaying speed v(τ)=v_{0}/sqrt[τ]. When the mean walking time is finite, a universal Gaussian law for the position-PDF of the walker is obtained explicitly.

Modeling Social Distancing and Quantifying Epidemic Disease Exposure in a Built Environment

As we transition away from pandemic-induced isolation and social distancing, there is a need to estimate the risk of exposure in built environments. We propose a novel metric to quantify social distancing and the potential risk of exposure to airborne diseases in an indoor setting, which scales with distance and the number of people present. The risk of exposure metric is designed to incorporate the dynamics of particle movement in an enclosed set of rooms for people at different immunity levels, susceptibility due to age, background infection rates, intrinsic individual risk factors (e.g., comorbidities), mask-wearing levels, the half-life of the virus and ventilation rate in the environment. The model parameters have been selected for COVID-19, although the modeling framework applies to other airborne diseases. The performance of the metric is tested using simulations of a real physical environment, combining models for walking, path length dynamics, and air-conditioning replacement action. We have also created a visualization tool to help identify high-risk areas in the built environment. The resulting software framework is being used to help with planning movement and scheduling in a clinical environment ahead of reopening of the facility, for deciding the maximum time within an environment that is safe for a given number of people, for air replacement settings on air-conditioning and heating systems, and for mask-wearing policies. The framework can also be used for identifying locations where foot traffic might create high-risk zones and for planning timetabled transitions of groups of people between activities in different spaces. Moreover, when coupled with individual-level location tracking (via radio-frequency tagging, for example), the exposure risk metric can be used in real-time to estimate the risk of exposure to the coronavirus or other airborne illnesses, and intervene through air-conditioning action modification, changes in timetabling of group activities, mask-wearing policies, or restricting the number of individuals entering a given room/space. All software are provided online under an open-source license.

A single-agent extension of the SIR model describes the impact of mobility restrictions on the COVID-19 epidemic

Mobility restrictions are successfully used to contain the diffusion of epidemics. In this work we explore their effect on the epidemic growth by investigating an extension of the Susceptible-Infected-Removed (SIR) model in which individual mobility is taken into account. In the model individual agents move on a chessboard with a Lévy walk and, within each square, epidemic spreading follows the standard SIR model. These simple rules allow to reproduce the sub-exponential growth of the epidemic evolution observed during the Covid-19 epidemic waves in several countries and which cannot be captured by the standard SIR model. We show that we can tune the slowing-down of the epidemic spreading by changing the dynamics of the agents from Lévy to Brownian and we investigate how the interplay among different containment strategies mitigate the epidemic spreading. Finally we demonstrate that we can reproduce the epidemic evolution of the first and second COVID-19 waves in Italy using only 3 parameters, i.e , the infection rate, the removing rate, and the mobility in the country. We provide an estimate of the peak reduction due to imposed mobility restrictions, i. e., the so-called flattening the curve effect. Although based on few ingredients, the model captures the kinetic of the epidemic waves, returning mobility values that are consistent with a lock-down intervention during the first wave and milder limitations, associated to a weaker peak reduction, during the second wave.

Mutual anticipation can contribute to self-organization in human crowds

Human crowds provide paradigmatic examples of collective behavior emerging through self-organization. Understanding their dynamics is crucial to help manage mass events and daily pedestrian transportation. Although recent findings emphasized that pedestrians' interactions are fundamentally anticipatory in nature, whether and how individual anticipation functionally benefits the group is not well understood. Here, we show the link between individual anticipation and emergent pattern formation through our experiments of lane formation, where unidirectional lanes are spontaneously formed in bidirectional pedestrian flows. Manipulating the anticipatory abilities of some of the pedestrians by distracting them visually delayed the collective pattern formation. Moreover, both the distracted pedestrians and the nondistracted ones had difficulties avoiding collisions while navigating. These results imply that avoidance maneuvers are normally a cooperative process and that mutual anticipation between pedestrians facilitates efficient pattern formation. Our findings may influence various fields, including traffic management, decision-making research, and swarm dynamics.

Lévy noise-driven escape from arctangent potential wells

The escape from a potential well is an archetypal problem in the study of stochastic dynamical systems, representing real-world situations from chemical reactions to leaving an established home range in movement ecology. Concurrently, Lévy noise is a well-established approach to model systems characterized by statistical outliers and diverging higher order moments, ranging from gene expression control to the movement patterns of animals and humans. Here, we study the problem of Lévy noise-driven escape from an almost rectangular, arctangent potential well restricted by two absorbing boundaries, mostly under the action of the Cauchy noise. We unveil analogies of the observed transient dynamics to the general properties of stationary states of Lévy processes in single-well potentials. The first-escape dynamics is shown to exhibit exponential tails. We examine the dependence of the escape on the shape parameters, steepness, and height of the arctangent potential. Finally, we explore in detail the behavior of the probability densities of the first-escape time and the last-hitting point.

Development of an automatic turntable-type multiple T-maze device and observation of pill bug behavior

In recent years, various animal observation instruments have been developed to support long-term measurement and analysis of animal behaviors. This study proposes an automatic observation instrument that specializes for turning behaviors of pill bugs and aims to obtain new knowledge in the field of ethology. Pill bugs strongly tend to turn in the opposite direction of a preceding turn. This alternation of turning is called turn alternation reaction. However, a repetition of turns in the same direction is called turn repetition reaction and has been considered a malfunction of turn alternation. In this research, the authors developed an automatic turntable-type multiple T-maze device and observed the turning behavior of 34 pill bugs for 6 h to investigate whether turn repetition is a malfunction. As a result, most of the pill bug movements were categorized into three groups: sub-diffusion, Brownian motion, and Lévy walk. This result suggests that pill bugs do not continue turn alternation mechanically but elicit turn repetition moderately, which results in various movement patterns. In organisms with relatively simple nervous systems such as pill bugs, stereotypical behaviors such as turn alternation have been considered mechanical reactions and variant behaviors such as turn repetition have been considered malfunctions. However, our results suggest that a moderate generation of turn repetition is involved in the generation of various movement patterns. This study is expected to provide a new perspective on the conventional view of the behaviors of simple organisms.

Lévy walk dynamics in an external harmonic potential

Lévy walks (LWs) are spatiotemporally coupled random-walk processes describing superdiffusive heat conduction in solids, propagation of light in disordered optical materials, motion of molecular motors in living cells, or motion of animals, humans, robots, and viruses. We here investigate a key feature of LWs-their response to an external harmonic potential. In this generic setting for confined motion we demonstrate that LWs equilibrate exponentially and may assume a bimodal stationary distribution. We also show that the stationary distribution has a horizontal slope next to a reflecting boundary placed at the origin, in contrast to correlated superdiffusive processes. Our results generalize LWs to confining forces and settle some longstanding puzzles around LWs.