Individual behaviour in group formation Wenfeng Feng
Yang Li
Yan Jia
Henan Polytechnic University Email:
[email protected] Telephone:13939119371
Henan Polytechnic University Email:
[email protected] Telephone:13839169343
Henan Polytechnic University Telephone:13673693173
Abstract—An analysis for individual behaviour in group formation is shown in this paper.Here,we find that,during group formation, individuals adjust their state in the form of pulse.Some characteristics of this kind oscillation are studyed in this paper.After that,we try to make an explanation for this phenomenon,that is,when group is in dynamic equilibrium,it’s changes of state will be transmited to individuals in the form of fluctuation,we discuss the reason of this phenomenon and make a try to determine the natural frequency of a certain group. Index Terms—group formation,self-organization,pulse,Boid
I. I NTRODUCTION In nature,may animal groups display structural order1 ,such as fish schools ,bird flocks and so on.Most groups will experience a self-organizing process from disorder to order. In this process,individuals usually do simple Interactive with it’s nearest neighbors,and finally the whole group emerge some new behaviours. Due to the development of computer technology, the individual-based simulation has brought a great convenience to model the formation of group. Many model have been established2,3,4,6 and successfully explain some phenomena in group formation,such as foraging, migration ,avoiding disadvantages and so on. At present, most researches tend to study how different interactive mechanism lead to different group behavior,and few consider the individual’s behavior in the formation of group,that is,in the process of self-organizing,how individuals gradually adjust their states by it’s environment and go to a final steady state.This paper is to study this field and try to reveal the general rule in this process. II. T HE B OID M ODEL The Boid model are proposed by Craig Reynolds in 19872 ,it was used to simulate the formation of the flock of birds.There are two simple rules in this model5 .1)Individuals try to maintain a mini-mum distance between themselves and others every time. In other words,if the min distance between a individual and his neighbors is greater than the presupposed minimum,it will move to it’s neighbors.if not,it will move in the opposite direction. 2)if the first rule is not met, individuals will align themselves with their neighbours. In the Year 2002, Couzin et.al simulated this model with the help of computer5 .Consider a group with N Individuals(i=1...N),every individual has a position denoted by ci and a direction vector vi .For each step of time,the max angle a individual rotate is θ.Individuals can only see their neighbours within a in length.The max angle of view is α.There
are three areas, say the zone of repulsion(zor), the zone of orientation(zoo) and the zone of attraction(zoa).Obviously a = zor + zoo + zoa(see Fig.1).
Fig. 1.
The three areas with a indiviidual in the center
For each step,a Individual will scan it’s zor first,if there are any other companion ,it’s next direction is as below(eq.1). di (t + 1) = −
∑nr
rij (t) j̸=i |rij (t)|
(1)
Where rij = cj − ci is the direction vector the individual i towards j. Eq.1 shows that,if individual space is too crowded,one will move away. If there is nobody in zor,the individual will scan the zoo and zoa,and if there is anyone in it,the next direction is provided by eq.2 di (t + 1) = 12 (
∑no
vj (t) j=1 |vj (t)|
+
∑na
rij (t) j̸=i |rij (t)| )
(2)
Eq.2 shows that,when there are somebody in zoo,the individual tend to align with them.When zoa has anybody in it,the individual tend to move towards them.When both satisfy the condition,it will take the average. Once the di (t + 1) has been determined,the next step is to calculate the vi (t + 1).It will follow the below rule:when di (t + 1) is at the right hand of vi (t),and if the angle between them is larger than θthe next step will turn right at θ degree;if not,the di (t + 1) will be chosen as the next direction. And vice versa.
[5]shows that,when changing parameters(the zor,zoo,zoa,or the α),the group will emerge four different behaviors,say Swarm, Torus, Dynamic parallel group,and Highly parallel group.In next section we will study the different behavior in individual in the group. III. T HE INDIVIDUAL BEHAVIOR IN GROUP FORMATION To analyse the individual behavior in group formation, and for simplicity, we chose dynamic parallel group to study. Without loss of generality, two individuals are randomly selected,we chose y = di (t) ∗ vi (t − 1) as statistics,which denote the quantity of changes in two successive steps. We adjust the parameters and let the group go to dynamic parallel.After 1000 steps,we get the below figures(see Fig.2).
(a)
(b)
(c) (a)
(b)
Fig. 3. (a)The cure of original data.(b)The distribution shown in bars.(c)The distribution shown in line.The coordinates are A(-0.9,103), B(14.0,341), C(7.13,3), E(-0.7,97), F(0.6,84), G(14.5,341), H(15.5,308).
the normal conditions and other points in the line denote the moil.This means the individual has two main states,the normal state and the puls state, beyond that,it spend little time in moil.Compared with Fig.3(b), A is equivalent to E and F,and B the G ahd H. IV. D ISCUSSION
(c) Fig. 2. (a)The view of group.(b)The condition curve of first individual.(c)the condition curve of second individual. Where N=100, zor=0.3, zoo=1.6, zoa=0.3, s=0.15, α = 50◦ and θ = 246◦ .
From Fig.2,we can see that when group is approaching to be stable,individuals will change their states in the form of plus,and this kind of chang is regular.Because the two individuals are selected randomly,so this demonstration has some generality. To study the amplitude,we gather statistics of distribution of amplitude(see Fig.3). From Fig.3(c),we can see two peaks,A and B,and a valley,C.The point A means that,the count of pulses with amplitude of -0.9 is 103.Where A denotes the pulses , B denotes
Image the water surface.where every point on it only interact with it’s direct neighbors.When other individuals’ state have changed,the influence will spread in the way of fluctuation and the speed is only related to the medium. When a group is formed,the mechanism of information transmission will be established.Beacause the interaction is local,so the information transfering have the characteristic of time-delay. External state changes will be passed to the individual by fluctuations.Invividuals will approach to completely orderly or dynamic equilibrium through constant oscillations. When a group reaches stable state, It’s information transmission rate should be constant.Take Fig.5 for example ,point A represents the pulse,is equivalent to E and F, the frequency will be calculated out as this form, F = 181/1120 ≈ 0.16. When time-delay is small, The spread of information has a faster rate,the individual’s state will change frequently and has no regular.From macro-scalewe will see disorder.When timedelay is a little large,messages will hit and merge together,and the group will go to stability. When more extreme, individuals don’t contact with each other,the time-delay will be infi-
nite,and the group behavior will not exist,in this condition,the group will revert to many single individuals. V. C ONCLUSION The oscillations in group is widespread in nature7,8 , whether this phenomenon are universal in the group formation is remain under study. A variety of phenomena show that the information exchange between individual and group seems by the form of fluctuate.If we take the whole group as the medium of the spread of messages,then it is necessary to determine the medium’s inherent property, which can help us to better understand the relation between individual and entirety. R EFERENCES [1] Iain D. Couzin & Jens Krause, Self-Organization and Collective Behavior in Vertebrates. Elsevier Science, 2003. 2 [2] Reynolds,C.W., Flocks,herds,and schools:A distributed behavioral model in computer graphics. Proc.OfSIGGRAPH’87,21(4):25-34, 1987. 3 [3] Vicsek,T., Czirok,A.,Ben-Jaco,E. et al,Novel type of phase transition in a system of self-driven particles. Phys.Rev.Lett,75:1226-1229. 1995. 4 [4] Ali Jadbabaie,Jie Lin,and A. Stephen Morse, Coordination of groups of Mobile Autonomous Agents Using Nearest Neighbor Rules. IEEE TRANSACTIONS ON AUTOMATIC CONTROL,VOL.48,NO.6, 2003. 5 [5] I.D.Couzin et al, Collective memory and spatial sorting in animal groups. Journal of Theoretical Biology .218,1-11. 2002. 6 [6] Couzin ID, Krause J, Franks NR, Levin SA, Effective leadership and decision-making in animal groups on the move. Nature;433:513-516, 2005. 7 [7] Arthur Prindle and Jeff Hasty, . Science 21 May 2010:Vol. 328 no. 5981 pp. 987-988 8 [8] A. F. Taylor, M. R. Tinsley, F. Wang, Z. Huang, K. Showalter, . Science 323, 614 (2009).
