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In this paper, we study the flocking behavior of a locally interacting multi‐agent system with a unit speed constraint. Using the spectral gap of a connected stochastic matrix, together with an elaborate estimation on perturbations of a linearized system, we obtain a sufficient condition imposed only on initial data and model parameters to guarantee flocking and further show that the system achieves a consensus at an exponential rate. [ABSTRACT FROM AUTHOR]

Multiple automatic guided vehicles are widely involved in industrial intelligence. Path planning is crucial for their successful application. However, achieving robust and efficient path planning of multiple automatic guided vehicles for real-time implementation is challenging. In this paper, we propose a two-layer strategy for multi-vehicle path planning. The approach aims to provide fast computation and operation efficiency for implementation. The start–destination matrix groups all the vehicles, generating a dynamic virtual leader for each group. In the first layer, the hybrid A* algorithm is employed for the path planning of the virtual leaders. The second layer is named leader–follower; the proposed Weight-Leader-Vicsek model is applied to navigate the vehicles following their virtual leaders. The proposed method can reduce computational load and achieve real-time navigation by quickly updating the grouped vehicles' status. Collision and deadlock avoidance is also conducted in this model. Vehicles in different groups are treated as dynamic obstacles. We validated the method by conducted simulations through MATLAB to verify its path-planning functionality and experimented with a localization sensor. [ABSTRACT FROM AUTHOR]

Collective behaviors in nature and human societies have been intensively studied in recent decades. The Vicsek model is one of the typical models that explain self-ordered particle systems well. In the original Vicsek model, the neighbor strategy takes all its neighbors' mean directions into account when updating particles' directions, which leads to a longer convergence time and higher computation cost due to the excess number of neighbors. In this paper, we introduce a new neighbor strategy to the Vicsek model. It defines that each particle will only select a certain number of particles with the farthest distance that fall into its vision sector as its neighbors. In addition, we classify the Vicsek model as the static model and the dynamic model according to whether the features of particles in the model are constant or not. Moreover, we design a new rule to apply the new neighbor strategy to dynamic Vicsek models. The simulation results indicate that our new neighbor strategy can significantly decrease the average number of particles' neighbors but still be able to further enhance the Vicsek model's convergence performance. The comparative results found that the static and dynamic model applied with the new neighbor strategy outperforms the models that only apply view restriction or remote neighbor strategy in noiseless and noisy conditions. [ABSTRACT FROM AUTHOR]

For biological groups, the behaviors of individuals will have an impact on the alignment efficiency of the collective movement. Motivated by Vicsek's pioneering research on self-organized particles and other related works about flocking behaviors, we propose two mathematical models based on the local information of individuals to include more realistic details in the interaction mechanism between individuals and the rest of the group during the flocking process. The local information of the individual refers to the local consistency, representing the degree of alignment with its neighbors. These two models are the self-introspection model, where the process of orientation adjustment of one individual is ruled by the degree of local consistency with the neighborhood, and the credit evaluation model, where the average orientation of the neighborhoods is weighed using the local consistency of the interacting individuals. Different metrics are calculated to analyze the effects of the model parameters and flocking parameters on groups. Simulation calculations indicate that the two improved models have certain advantages in terms of alignment efficiency for the group. Finally, the optimal model parameters are determined, and the effects of random noise on groups with a single behavior and mixed behaviors are analyzed. The results confirm that individuals with mixed behaviors still possess robustness against noise. This research would contribute to the further interdisciplinary cooperation that involves biology, ethology, and multi-agent complex systems. [ABSTRACT FROM AUTHOR]

We consider a spatially homogeneous Kolmogorov–Vicsek model in two dimensions, which describes the alignment dynamics of self-driven stochastic particles that move on a plane at a constant speed, under space-homogeneity. In [A. Figalli, M.-J. Kang and J. Morales, Global well-posedness of spatially homogeneous Kolmogorov–Vicsek model as a gradient flow, Arch. Rational Mech. Anal. 227 (2018) 869–896] Alessio Figalli and the authors have shown the existence of global weak solutions for this two-dimensional model. However, no time-asymptotic behavior is obtained for the two-dimensional case, due to the failure of the celebrated Bakery and Emery condition for the logarithmic Sobolev inequality. We prove exponential convergence (with quantitative rate) of the weak solutions towards a Fisher-von Mises distribution, using a new condition for the logarithmic Sobolev inequality. [ABSTRACT FROM AUTHOR]

We consider interacting particle dynamics with Vicsek-type interactions, and their macroscopic Partial Differential Equation (PDE) limit, in the non-mean-field regime; that is, we consider the case in which each particle/agent in the system interacts only with a prescribed subset of the particles in the system (for example, those within a certain distance). In this non-mean-field regime the influence between agents (i.e. the interaction term) can be normalized either by the total number of agents in the system (global scaling) or by the number of agents with which the particle is effectively interacting (local scaling). We compare the behavior of the globally scaled and the locally scaled systems in many respects, considering for each scaling both the PDE and the corresponding particle model. In particular, we observe that both the locally and globally scaled particle system exhibit pattern formation (i.e. formation of traveling-wave-like solutions) within certain parameter regimes, and generally display similar dynamics. The same is not true of the corresponding PDE models. Indeed, while both PDE models have multiple stationary states, for the globally scaled PDE such (space-homogeneous) equilibria are unstable for certain parameter regimes, with the instability leading to traveling wave solutions, while they are always stable for the locally scaled one, which never produces traveling waves. This observation is based on a careful numerical study of the model, supported by further analysis. [ABSTRACT FROM AUTHOR]

Multiple automatic guided vehicles are widely involved in industrial intelligence. Path planning is crucial for their successful application. However, achieving robust and efficient path planning of multiple automatic guided vehicles for real-time implementation is challenging. In this paper, we propose a two-layer strategy for multi-vehicle path planning. The approach aims to provide fast computation and operation efficiency for implementation. The start–destination matrix groups all the vehicles, generating a dynamic virtual leader for each group. In the first layer, the hybrid A* algorithm is employed for the path planning of the virtual leaders. The second layer is named leader–follower; the proposed Weight-Leader-Vicsek model is applied to navigate the vehicles following their virtual leaders. The proposed method can reduce computational load and achieve real-time navigation by quickly updating the grouped vehicles’ status. Collision and deadlock avoidance is also conducted in this model. Vehicles in different groups are treated as dynamic obstacles. We validated the method by conducted simulations through MATLAB to verify its path-planning functionality and experimented with a localization sensor.

Collective behaviors in nature and human societies have been intensively studied in recent decades. The Vicsek model is one of the typical models that explain self-ordered particle systems well. In the original Vicsek model, the neighbor strategy takes all its neighbors’ mean directions into account when updating particles’ directions, which leads to a longer convergence time and higher computation cost due to the excess number of neighbors. In this paper, we introduce a new neighbor strategy to the Vicsek model. It defines that each particle will only select a certain number of particles with the farthest distance that fall into its vision sector as its neighbors. In addition, we classify the Vicsek model as the static model and the dynamic model according to whether the features of particles in the model are constant or not. Moreover, we design a new rule to apply the new neighbor strategy to dynamic Vicsek models. The simulation results indicate that our new neighbor strategy can significantly decrease the average number of particles’ neighbors but still be able to further enhance the Vicsek model’s convergence performance. The comparative results found that the static and dynamic model applied with the new neighbor strategy outperforms the models that only apply view restriction or remote neighbor strategy in noiseless and noisy conditions.

For biological groups, the behaviors of individuals will have an impact on the alignment efficiency of the collective movement. Motivated by Vicsek’s pioneering research on self-organized particles and other related works about flocking behaviors, we propose two mathematical models based on the local information of individuals to include more realistic details in the interaction mechanism between individuals and the rest of the group during the flocking process. The local information of the individual refers to the local consistency, representing the degree of alignment with its neighbors. These two models are the self-introspection model, where the process of orientation adjustment of one individual is ruled by the degree of local consistency with the neighborhood, and the credit evaluation model, where the average orientation of the neighborhoods is weighed using the local consistency of the interacting individuals. Different metrics are calculated to analyze the effects of the model parameters and flocking parameters on groups. Simulation calculations indicate that the two improved models have certain advantages in terms of alignment efficiency for the group. Finally, the optimal model parameters are determined, and the effects of random noise on groups with a single behavior and mixed behaviors are analyzed. The results confirm that individuals with mixed behaviors still possess robustness against noise. This research would contribute to the further interdisciplinary cooperation that involves biology, ethology, and multi-agent complex systems.

There have been numerous studies on collective behavior, among which communication between agents can have a great impact on both the payoff and the cost of making decisions. Research usually focuses on how to improve the collective synchronization rate or accelerate the process of cooperation under given communication cost constraints. In this context, evolutionary game theory (EGT) and reinforcement learning (RL) arise as essential frameworks for tackling this intricate problem. In this study, an adapted Vicsek model is introduced, wherein agents exhibit varying movement patterns contingent on their chosen strategies. Each agent gains a payoff determined by the advantages of collective motion juxtaposed with the cost of communicating with neighboring agents. Individuals choose the objective agents based on the Q-learning strategy and then adapt their strategies following the Fermi rule. The research reveals that the utmost level of cooperation and synchronization can be attained at an optimal communication radius after applying Q-learning. Similar conclusions have been drawn from research on the influence of random noise and relative cost. Different cost functions were considered in the study to demonstrate the robustness of the proposed model and conclusions under a wide range of conditions. (https://github.com/WangchengjieT/VM-EGT-Q) • An improved VM model is proposed to study the cooperative behavior by incoperating EGT and Q-learning. • The experiments show that incorporating Q-learning into EGT helps the cooperation behavior of agents in Vicsek. • A new cost is proposed and the similar simulation results verified the robustness of the proposed model. [ABSTRACT FROM AUTHOR]

In this work, we explore how the emergence of collective motion in a system of particles is influenced by the structure of their domain. Using the Vicsek model to generate flocking, we simulate two-dimensional systems that are confined based on varying obstacle arrangements. The presence of obstacles alters the topological structure of the domain where collective motion occurs, which, in turn, alters the scaling behavior. We evaluate these trends by considering the scaling exponent and critical noise threshold for the Vicsek model, as well as the associated diffusion properties of the system. We show that obstacles tend to inhibit collective motion by forcing particles to traverse the system based on curved trajectories that reflect the domain topology. Our results highlight key challenges related to the development of a more comprehensive understanding of geometric structure's influence on collective behavior.

In this paper we provide a local Cauchy theory both on the torus and in the whole space for general Vicsek dynamics at the kinetic level. We consider rather general interaction kernels, nonlinear viscosity, and nonlinear friction. Particularly, we include normalized kernels which display a singularity when the flux of particles vanishes. Thus, in terms of the Cauchy theory for the kinetic equation, we extend to more general interactions and complete the program initiated in [I. M. Gamba and M.-J. Kang, Arch. Ration. Mech. Anal., 222 (2016), pp. 317-342] (where the authors assume that the singularity does not take place) and in [A. Figalli, M.-J. Kang, and J. Morales, Arch. Ration. Mech. Anal., 227 (2018), pp. 869-896] (where the authors prove that the singularity does not happen in the spatially homogeneous case). Moreover, we derive an explicit lower time of existence as well as a global existence criterion that is applicable, among other cases, to obtain a long time theory for nonrenormalized kernels and for the original Vicsek problem without any a priori assumptions. On the second part of the paper, we also establish the mean-field limit in the large particle limit for an approximated (regularized) system that coincides with the original one whenever the flux does not vanish. Based on the results proved for the limit kinetic equation, we prove that for short times, the probability that the dynamics of this approximated particle system coincides with the original singular dynamics tends to one in the many particle limit. [ABSTRACT FROM AUTHOR]

Recent years have witnessed a growing interest in using machine learning to predict and identify phase transitions (PTs) in various systems. Here we adopt convolutional neural networks (CNNs) to study the PTs of Vicsek model, solving the problem that traditional order parameters are insufficiently able to do. Within the large-scale simulations, there are four phases, and we confirm that all the PTs between two neighboring phases are first-order. We have successfully classified the phase by using CNNs with a high accuracy and identified the PT points, while traditional approaches using various order parameters fail to obtain. These results indicate the great potential of machine learning approach in understanding the complexities in collective behaviors, and in related complex systems in general.

The present study investigates how combined information from audition and vision impacts group-level behavior. We consider a modification to the original Vicsek model that allows individuals to use auditory and visual sensing modalities to gather information from neighbors in order to update their heading directions. Moreover, in this model, the information from visual and auditory cues can be weighed differently. In a simulation study, we examine the sensitivity of the emergent group-level behavior to the weights that are assigned to each sense modality in this weighted composite model. Our findings suggest combining sensory cues may play an important role in the collective behavior and results from the composite model indicate that the group-level features from pure audition predominate.

Synchronized movement of (both unicellular and multicellular) systems can be observed almost everywhere. Understanding of how organisms are regulated to synchronized behavior is one of the challenging issues in the field of collective motion. It is hypothesized that one or a few agents in a group regulate(s) the dynamics of the whole collective, known as leader(s). The identification of the leader (influential) agent(s) is very crucial. This article reviews different mathematical models that represent different types of leadership. We focus on the improvement of the leader-follower classification problem. It was found using a simulation model that the use of interaction domain information significantly improves the leader-follower classification ability using both linear schemes and information-theoretic schemes for quantifying influence. This article also reviews different schemes that can be used to identify the interaction domain using the motion data of agents.

We analyse the asymptotic behavior for kinetic models describing the collective behavior of animal populations. We focus on models for self-propelled individuals, whose velocity relaxes toward the mean orientation of the neighbors. The self-propelling and friction forces together with the alignment and the noise are interpreted as a collision/interaction mechanism acting with equal strength. We show that the set of generalized collision invariants, introduced in [1], is equivalent in our setting to the more classical notion of collision invariants, i.e., the kernel of a suitably linearized collision operator. After identifying these collision invariants, we derive the fluid model, by appealing to the balances for the particle concentration and orientation. We investigate the main properties of the macroscopic model for a general potential with radial symmetry.

Synchronized movement of (both unicellular and multicellular) systems can be observed almost everywhere. Understanding of how organisms are regulated to synchronized behavior is one of the challenging issues in the field of collective motion. It is hypothesized that one or a few agents in a group regulate(s) the dynamics of the whole collective, known as leader(s). The identification of the leader (influential) agent(s) is very crucial. This article reviews different mathematical models that represent different types of leadership. We focus on the improvement of the leader-follower classification problem. It was found using a simulation model that the use of interaction domain information significantly improves the leader-follower classification ability using both linear schemes and information-theoretic schemes for quantifying influence. This article also reviews different schemes that can be used to identify the interaction domain using the motion data of agents. [ABSTRACT FROM AUTHOR]

In this article we investigate the phase transition phenomena that occur in a model of self-organisation through body-attitude coordination. Here, the body attitude of an agent is modelled by a rotation matrix in R 3 as in Degond et al. (Math Models Methods Appl Sci 27(6):1005–1049, 2017). The starting point of this study is a BGK equation modelling the evolution of the distribution function of the system at a kinetic level. The main novelty of this work is to show that in the spatially homogeneous case, self-organisation may appear or not depending on the local density of agents involved. We first exhibit a connection between body-orientation models and models of nematic alignment of polymers in higher-dimensional space from which we deduce the complete description of the possible equilibria. Then, thanks to a gradient-flow structure specific to this BGK model, we are able to prove the stability and the convergence towards the equilibria in the different regimes. We then derive the macroscopic models associated with the stable equilibria in the spirit of Degond et al. (Arch Ration Mech Anal 216(1):63–115, 2015, Math Models Methods Appl Sci 27(6):1005–1049, 2017). [ABSTRACT FROM AUTHOR]

Abstract Since the pioneering work by Vicsek and his collaborators on the motion of self-propelled particles, most of the subsequent studies have focused on the onset of ordered states through a phase transition driven by particle density and noise. Usually, the particles in these systems are placed within periodic boundary conditions and interact via short-range velocity alignment forces. However, when the periodic boundaries are eliminated, letting the particles move in open space, the system is not able to organize into a coherently moving group since even small amounts of noise cause the flock to break apart. While the phase transition has been thoroughly studied, the conditions to keep the flock cohesive in open space are still poorly understood. Here we extend the Vicsek model of collective motion by introducing long-range alignment interactions between the particles. We show that just a small number of these interactions is enough for the system to build up long lasting ordered states of collective motion in open space and in the presence of noise. This finding was verified for other models in addition to the Vicsek one, suggesting its generality and revealing the importance that long-range interactions can have for the cohesion of the flock.

Starting from a particle model describing self-propelled particles interacting through nematic alignment, we derive a macroscopic model for the particle density and mean direction of motion. We first propose a mean-field kinetic model of the particle dynamics. After diffusive rescaling of the kinetic equation, we formally show that the distribution function converges to an equilibrium distribution in particle direction, whose local density and mean direction satisfies a cross-diffusion system. We show that the system is consistent with symmetries typical of a nematic material. The derivation is carried over by means of a Hilbert expansion. It requires the inversion of the linearized collision operator for which we show that the generalized collision invariants, a concept introduced to overcome the lack of momentum conservation of the system, plays a central role. This cross-diffusion system poses many new challenging questions. [ABSTRACT FROM AUTHOR]

Our ability to understand and control the emergence of order in swarming systems is a fundamental challenge in contemporary science. The standard Vicsek model (SVM) — a minimal model for swarming systems of self-propelled particles — describes a large population of agents reaching global alignment without the need of central control. Yet, the emergence of order in this model takes time and is not robust to noise. In many real-world scenarios, we need a decentralized protocol to guide a swarming system (e.g., unmanned vehicles or nanorobots) to reach an ordered state in a prompt and noise-robust manner. Here, we find that introducing a simple adaptive rule based on the heading differences of neighboring particles in the Vicsek model can effectively speed up their global alignment, mitigate the disturbance of noise to alignment, and maintain a robust alignment under predation. This simple adaptive model of swarming systems could offer new insights in understanding the prompt and flexible formation of animals and help us design better protocols to achieve fast and robust alignment for multi-agent systems. [ABSTRACT FROM AUTHOR]

Collective motion organized by autonomous motile elements is ubiquitous across scales, from motor proteins, flagellated sperms, to animals and fishes. Such motile materials are called active matter, and collective dynamics of active matter attracts broad interest from physics and biology due to their potential in exploiting universal feature of ordered structure and dynamics in biological systems. Here, we describe collective ordered phase of bacterial vortices under confinement with designed geometries. The transition of ordered vortices can be controlled by geometric rule derived from Vicsekstyle model. This finding may bring design principle of collective ordered phase of confined self-propelled particles. [ABSTRACT FROM AUTHOR]

The collective motion and compact spatial organization of active agents have attracted attention due to their widespread occurrence in active matter systems as well as their valuable applications in unmanned aerial vehicles (UAVs). In our work, a Vicsek-like model with a spatial perception cone is designed to implement spatial compaction and orientational synchronization. It is observed that the co-existent condensations in both coordinate and momentum spaces can occur within a limited range of the perception cone angle. The degrees of condensations are significantly stronger than those of previous models. Furthermore, two mechanisms are declared to explain the condensations. The results may offer some clues to help design robotic systems or UAVs with directed collective behavior. • A Vicsek-like model with a spatial perception cone is introduced. • Coexistent condensations in both coordinate and momentum spaces can occur within a limited range of perception cone angle. • Two mechanisms are declared to explain the condensations. [ABSTRACT FROM AUTHOR]

Flocking navigation and obstacle avoidance in complex environments remain challenging for multiple unmanned aerial vehicle (multi-UAV) systems, especially when only one UAV (termed as information UAV) knows the predetermined path and the communication range is limited. To this end, we propose a hierarchical weighting Vicsek model (HWVEM). In this model, a hierarchical weighting mechanism and an obstacle avoidance mechanism are designed. Based on the hierarchical weighting mechanism, all the UAVs are divided into different layers, and assigned with different weights according to the layer to which they belong. The purpose is to align the rest of UAVs with the information UAV more efficiently. Subsequently, the obstacle avoidance mechanism that utilizes only the local information is developed to ensure the system safety in an environment filled with obstacles differing in size and shape. A series of simulations have been conducted to demonstrate the high performance of HWVEM in terms of convergence time, success rate, and safety.

Flocking navigation, involving alignment-guaranteed path following and collision avoidance against obstacles, remains to be a challenging task for drones. In this paper, we investigate how to implement flocking navigation when only one drone in the swarm masters the predetermined path, instead of all drones mastering their routes. Specifically, this paper proposes a hierarchical weighting Vicsek model (WVEM), which consists of a hierarchical weighting mechanism and a layer regulation mechanism. Based on the hierarchical mechanism, all drones are divided into three layers and the drones at different layers are assigned with different weights to guarantee the convergence speed of alignment. The layer regulation mechanism is developed to realize a more flexible obstacle avoidance. We analyze the influence of the WVEM parameters such as weighting value and interaction radius, and demonstrate the flocking navigation performance through a series of simulation experiments.

Clustering of cells is an essential component of many biological processes from tissue formation to cancer metastasis. We develop a minimal, Vicsek-based model of cellular interactions that robustly and accurately captures the variable propensity of different cells to form groups when confined. We calibrate and validate the model with experimental data on clustering affinities of four lines of tumor cells. We then show that cell clustering or separation tendencies are retained in environments with higher cell number densities and in cell mixtures. Finally, we calibrate our model with experimental measurements on the separation of cells treated with anti-clustering agents and find that treated cells maintain their distances in denser suspensions. We show that the model reconstructs several cell interaction mechanisms, which makes it suitable for exploring the dynamics of cell cluster formation as well as cell separation. Insight: We developed a model of cellular interactions that captures the clustering and separation of cells in an enclosure. Our model is particularly relevant for microfluidic systems with confined cells and we centered our work around one such emerging assay for the detection and research on clustering breast cancer cells. We calibrated our model using the existing experimental data and used it to explore the functionality of the assay under a broader set of conditions than originally considered. Future usages of our model can include purely theoretical and computational considerations, exploring experimental devices, and supporting research on small to medium-sized cell clusters., (© The Author(s) 2024. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com.)

This study proposes a collective motion and self-organization control of a swarm of 10 UAVs, which are divided into two clusters of five agents each. A cluster is a group of UAVs in a dedicated area and multiple clusters make a swarm. This paper designs the 3D model of the whole environment by applying graph theory. To address the aforesaid issues, this paper designs a hybrid meta-heuristic algorithm by merging the particle swarm optimization (PSO) with the multi-agent system (MAS). First, PSO only provides the best agents of a cluster. Afterward, MAS helps to assign the best agent as the leader of the nth cluster. Moreover, the leader can find the optimal path for each cluster. Initially, each cluster contains agents at random positions. Later, the clusters form a formation by implementing PSO with the MAS model. This helps in coordinating the agents inside the nth cluster. However, when two clusters combine and make a swarm in a dynamic environment, MAS alone is not able to fill the communication gap of n clusters. This study does it by applying the Vicsek-based MAS connectivity and synchronization model along with dynamic leader selection ability. Moreover, this research uses a B-spline curve based on simple waypoint defined graph theory to create the flying formations of each cluster and the swarm. Lastly, this article compares the designed algorithm with the NSGA-II model to show that the proposed model has better convergence and durability, both in the individual clusters and inside the greater swarm.

In recent years, the well-developed Vicsek model has attracted more and more attention. The topological distance interaction among the neighboring self-propelled particles allows particle interact with fixed number of neighbors (FNN). This paper proposed the modified FNN rule (MFNN) to select the more effective neighboring particles, in which the fixed-number neighbors are uniform distributed in all directions around The MFNN rule can optimize the distribution of neighboring information for the collective motion. The results of the simulations demonstrate that the MFNN-based Vicsek model will converge much faster and will also enhance the capability of the anti-noise interference. • A modified fixed number of neighbors (MFNN) rule is proposed for Vicsek model to select more effective neighboring particles. • The proposed MFNN rule can optimize the distribution of neighboring information for the swarm motion. • The MFNN-based Vicsek model can converge much faster and also enhance the capability of the anti-noise interference. [ABSTRACT FROM AUTHOR]

• A proportional-integral (PI) control strategy is introduced to the Vicsek model. • The control parameters can be adjusted to optimize the system synchronization. • The control strategy improves the robustness of the system in the noise environment. • The effect of heterogeneous control parameters on system synchronization is investigated. This paper introduces an adaptive control strategy to the Vicsek model and investigates the collective behavior of self-propelled agents from the perspective of control theory. In the model, each agent adopts the classic proportional-integral (PI) control strategy to adjust its direction based on the deviation from the average direction of local neighbors. It is found that the integral term of the controller can eliminate the steady-state error, but strong integral effect will cause fluctuation of the system and prolong the convergence time. Therefore, the moderate integral control can adjust the headings of all agents to be synchronized after a short transient time. However, the proportional term incurs over-regulation of the directions, thus impeding the global convergence to be achieved. The robustness of the synchronizability with respect to different noise intensities is also investigated. Even in the strong noise environments, the control model can still maintain a high degree of directional consistency by adopting the conservative adjustment strategy. Furthermore, the diversity of control parameter is analyzed by considering the integral parameter following the normal distribution. It reveals that inconsistent adjustment magnitude is unfavorable to the synchronization of the system. [ABSTRACT FROM AUTHOR]

We investigate the influence of obstacles on the collective motion of self-propelled objects in the framework of the 2D Vicsek model. The obstacles are arranged in a square lattice and have a circular shape. We show that by increasing the radius of the obstacles the collective motion of the self-propelled objects can be altered from super-diffusive to diffusive. For obstacles with small radii, the system is composed of large clusters moving in one preferred direction, for larger radii, the system is composed of small clusters moving randomly in different directions. [Display omitted] • Collective motion of self propelled objects in an array of obstacles. • Structural transition from super diffusive to diffusive regime. • Controlling the collective motion by changing the size of obstacles. [ABSTRACT FROM AUTHOR]

A crisis in financial markets can be considered as a collective behaviour phenomenon. The collective behaviour is a complex behaviour which exists among a group of animals. The Vicsek model has been adapted to represent this complexity. A unique phase space has been introduced to represent all possible results of the model. The return of the transaction volumes versus the return of the closed price of each share has been used within the defined phase space. The findings show that the direction of the resultant velocity vectors of all share in this phase space act in the same direction when the financial crisis happens. By monitoring the market’s collective behaviour, it will be possible to gain more knowledge about the condition of the market days in crisis. This research aims to investigate the collective behaviour of stocks using the Vicsek model to study the condition of the market during the days in crisis.

We study the temporal correlations in the evolution of the order parameter ϕ t for the Vicsek model with vectorial noise by estimating its Hurst exponent H with detrended fluctuation analysis (DFA). We present results on this parameter as a function of noise amplitude η introduced in simulations. We also compare with well known order–disorder phase transition for that same noise range. We find that – regardless of detrending degree – H spikes at the known coexistence noise for phase transition, and that this is due to nonstationarities introduced by the transit of the system between two well defined states with lower exponents. We statistically support this claim by successfully synthesizing equivalent cases derived from a transformed fractional Brownian motion (TfBm). [ABSTRACT FROM AUTHOR]