PHYSICAL REVIEW E 79, 052102 共2009兲

Optimal view angle in collective dynamics of self-propelled agents Bao-Mei Tian,1 Han-Xin Yang,1 Wei Li,2 Wen-Xu Wang,3 Bing-Hong Wang,1,4 and Tao Zhou1,5,* 1

Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China 2 Department of Electrical Engineering, University of Texas, Arlington, Texas 76011, USA 3 Department of Electronic Engineering, Arizona State University, Tempe, Arizona 85287-5706, USA 4 The Research Center for Complex System Science, University of Shanghai for Science and Technology, Shanghai 200093, China 5 Department of Physics, University of Fribourg, Chemin du Musée 3, CH-1700 Fribourg, Switzerland 共Received 22 June 2008; revised manuscript received 4 January 2009; published 18 May 2009兲 We study a system of self-propelled agents with the restricted vision. The field of vision of each agent is only a sector of disk bounded by two radii and the included arc. The inclination of these two radii is characterized by the view angle. The consideration of restricted vision is closer to the reality because natural swarms usually do not have a panoramic view. Interestingly, we find that there exists an optimal view angle, leading to the fastest direction consensus. The value of the optimal view angle depends on the density, the interaction radius, the absolute velocity of swarms, and the strength of noise. Our findings may invoke further efforts and attentions to explore the underlying mechanism of the collective motion. DOI: 10.1103/PhysRevE.79.052102

PACS number共s兲: 05.60.Cd, 87.10.⫺e, 89.75.Hc, 02.50.Le

The collective motion of a group of autonomous agents 共or particles兲 关1–8兴 has attracted much attention in the past decade. One of the most remarkable characteristics of systems, such as flocks of birds, schools of fish, and swarms of locusts, is the emergence of collective states in which the agents move in the same direction. A particularly simple and popular model to describe such behavior was proposed by Vicsek et al. 关9兴. Due to simplicity and efficiency, the Vicsek model 共VM兲 has been intensively investigated in recent years 关10–22兴. In the VM, N agents move synchronously in a squareshaped cell of linear size L with the periodic boundary conditions. The initial directions and positions of the agents are randomly distributed in the cell, and each agent has the same absolute velocity v0. Agents i and j are neighbors at time ជ 共k兲 − Xជ 共k兲储 ⱕ R, where Xជ 共k兲 denotes step k if and only if 储X i j i the position of agent i on a two-dimensional 共2D兲 plane at time step k and R is the sensor radius. The direction of agent i at time step k + 1 is

␪i共k + 1兲 = 具␪i共k兲典R + ⌬␪ ,

ជ 共k + 1兲 = Xជ 共k兲 + v ei␪i共k兲⌬t, X i i 0



␪i共k + 1兲 = angle



e

j苸⌫i共k+1兲

neighbor set Γi(k+1,ω) of agent i

共1兲

where 具␪i共k兲典R denotes the average direction of agent i’s neighbors 共include itself兲, ⌬␪ denotes noise 共in the following discussions, ⌬␪ = 0 without special mention兲. To be more specific, let ⌫i共k兲 be the set of neighbors of agent i at time step k, the VM is then described as 关16,17兴

i␪ j共k兲

k + 1, which is the average direction of agents in the neighbor set ⌫i共k + 1兲. v0ei␪i共k兲 represents the velocity of agent i at time step k with constant speed v0 and direction ␪i共k兲. In the VM and most other models of self-propelled particles, the field of vision for every agent is a complete disk 共2D case兲 or a sphere 关three-dimensional 共3D兲 case兴 characterized only by its sensor radius R. In reality, however, most animals are incapable of complete view. For example, the cyclopean retinal field of human is about 180° and the cyclopean retinal field of tawny owl is 201° 关23兴. It is thus more reasonable to assume limited view angles of agents 关3,24兴, instead of the omnidirectional views, in swarm models to better mimic the real collective behaviors. In this Brief Report, we investigate the VM in which agents have limited view angles ␻, with ␻ 苸 共0 , 2␲兴. As illustrated in Fig. 1, the field of vision of every agent is only a sector of disk bounded by two radii and the included arc, the left 共right兲 boundary of vision and the heading of agent i have inclination ␻ / 2, that is, for every agent, the field of



heading of agent i

field of vision left boundary

共2兲 ,

ω

共3兲

view angle ω

R

i␪i共k兲

is the unitary complex directional vector of where e agent i, ei␪i共k兲 = cos共␪i共k兲兲 + i sin共␪i共k兲兲, where ␪i共k兲 苸 关0 , 2␲兲. Here the function angle共·兲 denotes the angle of a complex number. ␪i共k + 1兲 is the moving direction of agent at time step

*Corresponding author: [email protected]. 1539-3755/2009/79共5兲/052102共4兲

right boundary

FIG. 1. 共Color online兲 Illustration of the nonomnidirectional view of agent i at time step k + 1 in a 2D plane. 052102-1

©2009 The American Physical Society

PHYSICAL REVIEW E 79, 052102 共2009兲

BRIEF REPORTS

FIG. 3. The optimal view angle ␻opt as functions of the swarm number N, sensor radius R, and absolute velocity v0, respectively. For the left panel: R = 0.6, v0 = 0.04; for the middle panel: R = 0.6, N = 400; and for right panel: v0 = 0.04, N = 400. The lattice size is fixed as L = 10. Each data point is obtained by averaging over 500 different realizations. Note that the resolution of view angle in our simulation is set to be ␲ / 12.

FIG. 2. 共Color online兲 共a兲 The order parameter ⌽共k , ␻兲 as a function of time step k for different values of view angle ␻. Here N = 400, R = 0.6, and v0 = 0.04. 共b兲 The transient time step ␶ as a function of the view angle ␻. The symbols correspond to 䊏: R = 0.6, v0 = 0.02, N = 400; 夝: R = 0.6, v0 = 0.04, N = 400; 䉱: R = 0.6, v0 = 0.04, N = 500; 䉲: R = 0.8, v0 = 0.04, N = 400. Each data point is obtained by averaging over 500 different realizations.

view is symmetric about its current moving direction. Thus rule 共3兲 in the VM can be modified as



␪i共k + 1兲 = angle



j苸⌫i共k+1,␻兲



ei␪ j共k兲 ,

共4兲

where ⌫i共k + 1 , ␻兲 denotes the neighbor set of agent i with view angle ␻. When ␻ = 2␲, rule 共4兲 degenerates to the original Vicsek model 共3兲. To give a quantitative discussion, we define an order parameter 1 ⌽共k, ␻兲 = N

冏兺 冏 N

ei␪i共k兲 ,

0 ⱕ ⌽共k, ␻兲 ⱕ 1,

tremely rare cases 共for example, the cases may occur when R or ␻ is too small兲. To quantify the speed of direction consensus, we study the transient time step ␶, which is defined as the time step when the order parameter first surpasses a certain value ⌽0. Here we take ⌽0 = 0.99 and we have checked that qualitative results are not changed when ⌽0 is large enough. We then investigate the effects of the view angle ␻ on the transient process. As shown in Fig. 2共a兲, the order parameter ⌽共k , ␻兲 reaches 1 faster when the view angle ␻ = 3␲ / 2, compared with ␻ = 2␲ and ␻ = 5␲ / 6. Figure 2共b兲 shows the transient time step ␶ as a function of ␻ for different values of parameters. One can find that ␶ is not a monotonic function of ␻ and there exists an optimal view angle, leading to the shortest transient time. Figure 3 shows the optimal view angle ␻opt as functions of the parameters: the swarm number N, the sensor radius R, and the absolute velocity v0, respectively. One can see that the optimal view angle ␻opt decreases with the increasing of N and v0, and converges to a fixed value when N or v0 is large enough. ␻opt increases as the sensor radius R increases. In particular, when R is close to the lattice size L, agents with panoramic view will be globally coupled and the directions of the swarm can reach consensus in only one time step. We next investigate whether more communications are needed for faster convergence. We define ni共k , ␻兲 as the number of i’s neighbors, and the average number of neighbors 具n共k , ␻兲典 over all agents at time step k is

共5兲

N

1 具n共k, ␻兲典 = 兺 ni共k, ␻兲. N i=1

i=1

for system 共4兲 at time step k with view angle ␻, obviously, 0 ⱕ ⌽共k , ␻兲 ⱕ 1. In noiseless case, the order parameter ⌽共k , ␻兲 can approach 1 when the evolution is long enough, except for ex-

共6兲

In Fig. 4, we report this average neighboring number for different ␻. Combining Figs. 2共a兲 and 4, it is interesting to find that agents with optimal view angle ␻ = 3␲ / 2 have the

052102-2

PHYSICAL REVIEW E 79, 052102 共2009兲

BRIEF REPORTS

FIG. 4. 共Color online兲 The average number of neighbors 具n共k , ␻兲典 as a function of time step k for different view angle ␻. Here the parameters N, L, R, and v0 are the same with the parameters in Fig. 2共a兲. Each data point is obtained by averaging over 500 different realizations.

least number of neighbors in the steady state, compared with ␻ = 2␲, ␻ = 5␲ / 6, and ␻ = ␲. This result indicates the existence of superfluous communications in the VM, which may counteract the direction consensus. In the following, we focus on the noise effects associated with the restriction of view angle. The noise is introduced to the view angle model as



␪i共k + 1兲 = angle ei␰

ei␪ 共k兲冊 , 兺 j苸⌫ 共k+1,␻兲 j

共7兲

i

where the moving direction of each agent is perturbed by a random number ␰ chosen with a uniform probability from the interval 关−␩ , ␩兴. In the presence of noise, the order parameter ⌽共k , ␻ , ␩兲 will fluctuate and never remain fixed at a

FIG. 5. 共Color online兲 The statistically stable order parameter ⌽stable共␻ , ␩兲 as a function of the view angle ␻ for different noise ␩. 1 3000 Here ⌽stable共␻ , ␩兲 = 500 兺k=2501 ⌽共k , ␻ , ␩兲. N = 400, L = 10, R = 0.6, v0 = 0.04. Each data point is obtained by averaging over 500 different realizations.

FIG. 6. 共Color online兲 The transient time step ␶ as a function of the view angle ␻ for different values of the noise ␩. N = 400, L = 10, R = 0.6, v0 = 0.04. Each data point is obtained by averaging over 500 different realizations.

certain value; therefore we adopt a statistically stable order parameter in terms of ⌽stable共␻ , ␩兲, which is an average of the consecutive series of ⌽共k , ␻ , ␩兲 over many time steps after a sufficiently long transient time. Figure 5 shows that ⌽stable共␻ , ␩兲 increases as ␻ increases if the noise is kept constant and decreases as the noise increases. In the noisy case, we define the transient time step ␶ as the time step when the order parameter first exceeds 0.99⌽stable共␻ , ␩兲 for each run. For ␩ = 0, ⌽stable共␻ , 0兲 approaches 1; thus this definition of ␶ is applicable in the absence of noise. From Fig. 6, one can find that there still exists an optimal view angle ␻opt leading to the shortest transient time step in the presence of noise and the value of the optimal view angle decreases as the noise increases. In conclusion, we have studied the effects of restricted vision of a group of self-propelled agents. The field of vision of every agent is only a sector of disk and the included arc represents the view angle. It is interesting to find that there exists an optimal angle resulting in the fastest direction consensus. The value of the optimal view angle increases as the sensor radius increases, while it decreases as the swarm number, the absolute velocity, or the noise strength increases. Another interesting phenomenon is that agents with optimal view angle have the least number of neighbors in the steady state. Our studies indicate the existence of superfluous communications in the Vicsek model, which indeed hinder the direction consensus. Moreover, our results may be useful in designing the manmade swarms such as autonomous mobile robots. We thank Hai-Tao Zhang and Ming Zhao for their valuable comments. This work was funded by the National Basic Research Program of China 共973 Program No. 2006CB705500兲, the National Natural Science Foundation of China under Grants No. 10635040 and No. 10805045, and the Specialized Research Fund for the Doctoral Program of Higher Education of China 共Grant No. 20060358065兲.

052102-3

PHYSICAL REVIEW E 79, 052102 共2009兲

BRIEF REPORTS 关1兴 J. K. Parrish, Science 284, 99 共1999兲. 关2兴 H. Levine, W. J. Rappel, and I. Cohen, Phys. Rev. E 63, 017101 共2000兲. 关3兴 I. D. Couzin, J. Krause, R. James, G. D. Ruxton, and N. R. Franks, J. Theor. Biol. 218, 1 共2002兲. 关4兴 I. D. Couzin, J. Krause, N. R. Franks, and S. A. Levin, Nature 共London兲 433, 513 共2005兲. 关5兴 J. Buhl, D. J. T. Sumpter, I. D. Couzin, J. J. Hale, E. Despland, E. R. Miller, and S. J. Simpson, Science 312, 1402 共2006兲. 关6兴 M. R. D’Orsogna, Y. L. Chuang, A. L. Bertozzi, and L. S. Chayes, Phys. Rev. Lett. 96, 104302 共2006兲. 关7兴 D. Grunbaum, Science 312, 1320 共2006兲. 关8兴 A. Kolpas, J. Moehlis, and I. G. Kevrekidis, Proc. Natl. Acad. Sci. U.S.A. 104, 5931 共2007兲. 关9兴 T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, Phys. Rev. Lett. 75, 1226 共1995兲. 关10兴 L. Moreau, IEEE Trans. Autom. Control 50, 169 共2005兲. 关11兴 F. Cucker and S. Smale, IEEE Trans. Autom. Control 52, 852 共2007兲. 关12兴 G. Grégoire and H. Chaté, Phys. Rev. Lett. 92, 025702 共2004兲.

关13兴 C. Huepe and M. Aldana, Phys. Rev. Lett. 92, 168701 共2004兲. 关14兴 M. Aldana, V. Dossetti, C. Huepe, V. M. Kenkre, and H. Larralde, Phys. Rev. Lett. 98, 095702 共2007兲. 关15兴 M. Nagy, I. Daruka, and T. Vicsek, Physica A 373, 445 共2007兲. 关16兴 W. Li and X. F. Wang, Phys. Rev. E 75, 021917 共2007兲. 关17兴 W. Li, H. T. Zhang, M. Z. Q. Chen, and T. Zhou, Phys. Rev. E 77, 021920 共2008兲. 关18兴 W. Li, IEEE Trans. Syst., Man, Cybern., Part B: Cybern. 38, 1084 共2008兲. 关19兴 H. Chaté, F. Ginelli, G. Grégoire, and F. Raynaud, Phys. Rev. E 77, 046113 共2008兲. 关20兴 H. T. Zhang, M. Z. Q. Chen, and T. Zhou, Phys. Rev. E 79, 016113 共2009兲. 关21兴 L. Q. Peng, Y. Zhao, B. M. Tian, J. Zhang, B. H. Wang, H. T. Zhang, and T. Zhou, Phys. Rev. E 79, 026113 共2009兲. 关22兴 J. Zhang, Y. Zhao, B. M. Tian, L. Q. Peng, H. T. Zhang, B. H. Wang, and T. Zhou, Physica A 388, 1237 共2009兲. 关23兴 G. R. Martin, J. Comp. Physiol., A 174, 787 共1994兲. 关24兴 A. Huth and C. Wissel, J. Theor. Biol. 156, 365 共1992兲.

052102-4

Optimal view angle in collective dynamics of self-propelled agents

May 18, 2009 - 1Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China .... Higher Education of China Grant No.

201KB Sizes 0 Downloads 269 Views

Recommend Documents

Optimal view angle in collective dynamics of self-propelled agents
May 18, 2009 - fastest direction consensus. The value of the optimal view angle depends on the density, the interaction radius, the absolute velocity of swarms, ...

Collective chemotactic dynamics in the presence of self ... - NYU (Math)
Oct 22, 2012 - them around the domain. The dynamics is one of .... [22] S. Park, P. M. Wolanin, E. A. Yuzbashyan, P. Silberzan, J. B.. Stock, and R. H. Austin, ...

Evolutionary dynamics of collective action in N-person ...
the population. As shown in appendix A, random sampling of individuals leads to groups whose compo- sition follows a binomial distribution (Hauert et al. 2006),.

Collective dynamics of 'small-world' networks
Jun 4, 1998 - and social networks lie somewhere between these two extremes. Here we ... most, a linear effect on C; henceC(p) remains practically unchanged for small p even though L(p) drops rapidly. The important implica- tion here is ...

Evolutionary Dynamics of Collective Action in N-person ...
IRIDIA/CoDE, Université Libre de Bruxelles, Av. F. Roosevelt 50, CP 194/6, Brussels, ... Here we introduce a model in which a threshold less than the total group.

Collective chemotactic dynamics in the presence of self-generated ...
Oct 22, 2012 - [7] to study swimmer transport and rotation in a given background ... *Corresponding author: [email protected] and-tumble dynamics.

Collective Dynamics in a Binary Mixture of ...
May 5, 2015 - FS ij are (purely repulsive) steric or excluded volume interactions .... opposite-spin and same-spin rotors within distance r λ+(r) = 1. N. XN i=1.

Collective dynamics of 'small-world' networks
Jun 4, 1998 - We call them ... *Present address: Paul F. Lazarsfeld Center for the Social Sciences, Columbia .... Although small-world architecture has not.

The micro dynamics of collective violence
2011). For most people it is difficult to overcome their fear of, and inhibi- tions towards .... Connectivity and heterogeneity imply that for large social networks to syn- chronize .... M. Baas, F. S. Ten Velden, E. Van Dijk, and S. W. Feith (2010).

Collective Reputation and the Dynamics of Statistical ...
Sep 6, 2016 - disadvantaged group fails to coordinate on the good equilibrium. ..... identical fundamentals with respect to investment cost and information technology. ..... a training subsidy program that can reduce the human capital acquisition ...

Collective Reputation and the Dynamics of Statistical ...
R({aτ }∞ t ) = ∫ ∞ t β(aτ )e. −(δ+λ)(τ−t) dτ. Thus, the rate of human capital acquisition among ..... skill investment rate under the introduced subsidy program.

Collective Reputation and the Dynamics of Statistical ...
Young Chul Kim∗†. Korea Development Institute. Glenn C. Loury‡. Brown University. October 25, 2010. Abstract. Previous literature on statistical discrimination explained stereotypes based on the existence of multiple equilibria, in which princi

The Collective Dynamics of Smoking in a Large Social ...
May 22, 2008 - From the Department of Health Care .... For our study, we used the offspring cohort as the .... tistically meaningful difference between the pro-.

Optimal Monetary Policy with Heterogeneous Agents
horse for policy analysis in macro models with heterogeneous agents.1 Among the different areas spawned by this literature, the analysis of the dynamic aggregate ef ...... Under discretion (dashed blue lines in Figure 1), time-zero inflation is 4.3 p

Optimal Monetary Policy with Heterogeneous Agents -
See Auclert (2016) for a recent analysis of the Fisherian redistributive channel ... to the World interest rate.9 We find that inflation rises slightly on impact, as the ... first-best and the constrained-effi cient allocation in heterogeneous-agents

Optimal Monetary Policy with Heterogeneous Agents -
to the World interest rate.9 We find that inflation rises slightly on impact, as the central bank tries to ... first-best and the constrained-effi cient allocation in heterogeneous-agents models. In ... as we describe in the online appendix. Beyond .

1999_Modeling of failure in cement based angle ply ...
Connect more apps... Try one of the apps below to open or edit this item. 1999_Modeling of failure in cement based angle ply laminates_acers99_fracture.pdf.

Evolution of Cooperation in a Population of Selfish Adaptive Agents
Conventional evolutionary game theory predicts that natural selection favors the ... In both studies the authors concluded that games on graphs open a window.

Seasons- Angle of Sunlight.pdf
There was a problem previewing this document. Retrying... Download. Connect more apps... Try one of the apps below to open or edit this item. Seasons- Angle ...

Evolution of Cooperation in a Population of Selfish Adaptive Agents
of cooperation on graphs and social networks. Nature 441, 502–505 (2006). 10. Watts, D.J.: Small worlds: The dynamics of networks between order and random- ness. Princeton University Press, Princeton (1999). 11. Amaral, L.A., Scala, A., Barthelemy,

Cheap 360 degree fisheye full angle view 1.3MP 960P WIFI P2P ...
Cheap 360 degree fisheye full angle view 1.3MP 960P ... Bulb Camera Wireless Home Security WIFI Camera.pdf. Cheap 360 degree fisheye full angle view ...

Collective Churn Prediction in Social Network
Jun 11, 2011 - social network service [1]–[4]. Threats arising from churn have substantial impact on the profitability of service providers as retaining an existing ...

Collective frequency variation in network ...
Apr 25, 2016 - systems and show that for generic directed networks the collective frequency of the ensemble is not the same as the mean of the individuals' ...