MODEL OF AGGREGATION
A Topological Approach
Masud Rana and Dongsheng Cai
Department of Computer Science, University of Tsukuba, Ibaraki, Japan
Keywords: Collective Behaviour, Aggregation, Topology, SPP Model, Flock.
Abstract: The aggregate motion of flocks of birds, a herd of land animals, Mexican wave forming in stadia are
beautiful and nice examples of collective behaviour. The aggregation is constructed by the action of each
individual, each action solely on basis of its local perception of the world. Scientists from different
backgrounds have tried to model collective behaviour. Most of the models are strictly metric (based on
Euclidian distance among individuals) but flocks of birds do not act on metric perception. In this paper we
proposed a model based on topological perspective to construct a flock of birds with large number of
individuals and checked flock’s density independent behaviour.
1 INTRODUCTION
Collective behaviour could be stated as “the way in
which an individual unit’s activity is dominated by
its neighbours so that all units simultaneously alter
their behaviour to a common pattern” (Vicsek,
2001). By acting collectively, individuals (both
organisms and non-living objects are considerable)
synchronize their signals or motion. The main
features of collective behaviour are that an
individual unit’s action is dominated by the
influence of its neighbours – the unit behaves
differently from the way it would behave on its own;
and that such systems show interesting ordering
phenomena as the units simultaneously change their
behaviour to a common pattern.
The aggregate motion of flock of birds, a herd of
land animals, a school of fish are beautiful and nice
examples of collective behaviour. People clapping in
phase during rhythmic applause, Mexican wave
forming in stadia (Farkas, Helbing and Vicsek,
2002) also demonstrates collective behaviour. Even
non-living objects like ferromagnets show collective
behaviour. These materials can undergo spontaneous
magnetization, in effect because they are made up of
a host ‘tiny magnets’ (Vicsek, 2001).
Collective behaviour of animals exhibits many
contrasts. In case of flock of birds, flocks are made
of discrete birds yet the overall motion seems fluid;
it is simple in concept yet is so visually complex, it
seems randomly arrayed and yet is magnificently
synchronized. The aggregation is constructed by the
action of each individual, each action solely on the
basis of its local perception of the world (Reynolds,
1987).
Scientists from different backgrounds have tried
to understand and model different aggregations:
school of fish (Inada and Kawachi, 2002), flock of
birds (Reynold, 1987, Bhattacharya and Vicsek,
2010), pedestrian behavior (Moussaid, Helbing and
Theraulaz, 2011). Reynolds (1987) first introduced a
flock of birds model in computer graphics (Reynolds,
1987). He named the individual units ‘boids’ related
to ‘bird-like’ or ‘bird-oid’. To simulate a flock, he
used three simple rules: (1) collision avoidance, (2)
velocity matching and, (3) flock centering. Their
simulation was confined to some tens to some
hundreds of individuals. These three rules seem
reasonable, but they are unable to reproduce a flock
once the boids separate a little far away. Again,
global consideration is not realistic.
Another simple model (SPP model) (Vicsek and
Czirok, 1995); (Gonci et al., 2008); (Vicsek, 2008)
showed that an individual need not to consider the
whole flock to produce collective behaviour. Only
interactions with local neighbours and directional
averaging with neighbours, while some
environmental noise exists, is enough to produce
collective motion. In their model, the individuals
which exist around a certain radius circle to a
reference individual, are considered the neighbours
355
Rana M. and Cai D..
MODEL OF AGGREGATION - A Topological Approach.
DOI: 10.5220/0003818603550360
In Proceedings of the International Conference on Computer Graphics Theory and Applications (GRAPP-2012), pages 355-360
ISBN: 978-989-8565-02-0
Copyright
c
2012 SCITEPRESS (Science and Technology Publications, Lda.)
of that reference individual. Therefore, collective
behaviours created in this model greatly depend on
density of the aggregation. However, recent field
study from European scientists (Ballerini et al.,
2008) confirmed that the starling flock’s behaviour
is density independent. They argued that birds’
behaviour depends on topological distance rather
than metric one.
In this paper, we tried to construct a bird flock of
large numbers. We take the basic SPP model for its
simplicity (Vicsek and Czirok, 1995), but include
cohesion and collision avoidance. Though the SPP
model is strictly metric, we would exclude the metric
perspective, instead, include the topological
perspective for the topological idea is supported
from empirical study (Ballerini et al., 2008). Finally,
we would check flock’s density independent
behaviour.
2 SPP MODEL
The particles that make action or motion without the
influence or action of any external force are called
self-propelled particles (Simha and Ramaswamy,
2002). In this sense, animals that produce collective
behavior in different sorts of aggregations, can be
pointed as self-propelled particles. Instead of the
three rules model of Reynolds (Reynolds, 1987), the
SPP model (Vicsek and Czirok, 1995) is based on
only one rule: at each time step, a given particle
driven with a constant absolute velocity assumes the
average direction of motion of the particles in its
neighborhood of radius r with some random
perturbation added. The analogy can be formulated
as follows: The rule corresponding to the
ferromagnetic interaction tending to align the spins
in the same direction is replaced by the rule of
aligning the direction of motion of particles.
Random perturbations are applied in analogy with
the temperature. Biological subjects have the
tendency to move as other subjects do in their
neighborhood (Brien, 1989). Therefore, the SPP
model can be useful to model the flock of birds and
other living organisms.
The simulations were carried out in a square
shaped cell of linear size L with periodic boundary
conditions. Interaction radius r was used as the unit
to measure distances (r = 1), while the time unit, ∆
was the time interval between two updating of
direction and positions. The initial condition: (1) at
time, =0, particles were randomly distributed in
the cell, (2) had the same absolute velocity,
and
(3) randomly distributed directions. The velocities of
particles {
} were determined at each time step, and
the position of ith particle is updated according to-
(
+
)
=
<
(
)
>
|<
(
)
>
|
+
(1)
(
+
)
=
(
)
+
( +)
(2)
Here <..>
denotes averaging of the velocities
within a circle of radius r surrounding particle i.
<v
(
t
)
>
/| < v
(
t
)
>
| provides a unit vector
pointing in the average direction of motion.
Perturbation is taken account by adding a random
angle corresponding to the average direction of
motion in the neighbourhood particle of i.
Perturbations are random values taken from a
uniform distribution in the interval of [−,].
The only parameters of the model is the density --
the number of particles in unit square (for 2
dimensions) or unit volume (for 3 dimensions) – the
velocity,
and the level of perturbation,<1. In
two dimensional simulation, Vicsek showed that, for
a wide velocity range (0.003 <
<0.3), and
higher density ( = 12.0) and smaller level of noise
or perturbation ( = 0.1), after some time steps, all
particles move in the almost same direction i.e.
synchronize themselves by locally interacting with
each others.
In the SPP Model, Vicsek introduced an order
parameter which denotes the level or ordered motion
of the aggregation. The ordered parameter, , is
determined as follows:
=
1
|
|
(3)
Where N is the number of particles,
is the velocity
of the i th particles. goes near to 1 when the
aggregation is ordered and equal to 1 for fully
ordered. In contrast, when is near to zero; it means
that the particles are randomly walking and showing
no collective behaviour.
3 METRIC OR TOPOLOGY
Topological distance: The word ‘topology’ is
derived from Greek word ‘topos’ which means place
or space, and ‘logos’ which means study or idea or
theory (http://en.wikipedia.org/wiki/Topology,
http://www.nn.iij4u.or.jp/~hsat/techterm/topos.html)
. Therefore topology can be understood as the study
of place or space. “Topology" the English form, was
first used in 1883 in Listing's obituary in the journal
GRAPP 2012 - International Conference on Computer Graphics Theory and Applications
356
Nature to distinguish "qualitative geometry from the
ordinary geometry in which quantitative relations
chiefly treated". In this paper, when we would talk
about ‘metric distance’, we would mean the
quantitative distance i.e. real distance. And when we
use ‘topological distance’, we would rank the
surrounding particles to a reference. The rank would
be 1, for the most nearest neighbour, 2 for the
second nearest neighbour and so on. These ranks
would be the topological distances. Therefore
topological distances would be discrete: 1, 2, 3,..
The important distinction is that topological distance
does not change with the density of aggregation i.e.
the first nearest neighbour’s rank would be 1
(topological distance = 1) no matter how far or how
near it is. In economics, for example, the relevant
quantity is not how many kilometers separate two
countries (metric distance), but rather than the
number of intermediate countries between them
(topological distance) (Henrikson, 2002).
4 BALLERINI’S FIELD STUDY
Ballerini et al., (2008), by reconstructing three-
dimensional positions of individual birds of few
thousand members showed that the interactions
among the birds do not depend on metric distance
rather than depend on topological one. Moreover,
each bird interacts with a fixed number of birds (6-7
birds). They tried to show that the topological
interaction can achieve more cohesion than the
metric one while robust cohesion is needed for
complex density and shape changes of flock not
breaking cohesion among birds.
The main goal of the interaction among
individuals is to maintain cohesion of the
aggregation. This is very strong biological
requirement, shaped by the evolutionary pressure for
survivor: stragglers and small groups are
significantly more prone to predation than animals
belonging to large and highly cohesive aggregation
(Vine, 1971). In topological model, cohesion among
individuals does no vary with density changes,
therefore more suitable to keep cohesion.
Ballerini et al., (2008) discussed about the
characterization of structure of birds within flock by
showing the spatial distribution of nearest
neighbours. Given a reference bird, they measured
the angular orientation of its nearest neighbours with
respect to the flock’s direction of motion. The
measurement shows an anisotropic characteristic and
the anisotropic characteristic tends to fade out as the
rank of the nearest neighbours increases. This means
that the anisotropic characteristic of flock is the
result of individual interaction.
5 RESULTS AND DISCUSSIONS
Ballerini et al., (2008) made a simple two
dimensional predator-prey model based on SPP
model to emphasize that the topological interaction
should show strong cohesion. However, we
reproduced the same results in two dimensional case
and extended it to three dimensional predator-prey
model. We have been successful to show that the
three dimensional model exhibits the same type of
cohesion as the two dimensional model does (Figure
1b and 1e).
5.1 Predator-prey Model
In the predator-prey model (two dimensional), we
used equation (1) and (2) to update prey’s velocity
and position. However, the perturbation or noise part
is replaced by the impulsive force from the predator
to prey. Predator’s velocity and direction remain
unchanged and does not have effect from preys. The
impulsive force from predator to prey is determined
as equation (4).
=
|
|
(4)
is the impulsive force to i th bird,
is the
magnitude of the impulsive force posed by the
predator and
is the distance vector from
predator to prey. For metric case, we used
interaction radius as 0.15 and in case of topological
situation, we assume that a bird interact with three
nearest neighbour -- for two dimensional case
individuals show optimum interaction when they
interact with three nearest neighbours (Inada and
Kawachi, 2002). For both metric and topological
case, we calculated the isolated individuals separated
by predator attack. Figure 1a shows that in metric
case, maximum probability is for three isolated
individuals while in topological case, Figure 1b
shows that the maximum probability is for zero
isolated individual. Again, the probability bars of
separated individual decay very quickly in contrast
with the metric interaction. Therefore, it shows that
metric interaction is prone to predator attack and
topological interaction produces more cohesion
among individuals in aggregation.
We can assume that the birds may have
preference while aligning with the neighbours. We
MODEL OF AGGREGATION - A Topological Approach
357
ran another simulation taking weighted average of
neighbour’s velocity. We modified equation (1) to
equation (5) to update velocity, and found that
cohesion increased (Figure 1d). We cannot say for
sure, but point out to birds may have preferences
among nearest neighbours. Same sort of
characteristic has been achieved for three
dimensional predator-prey simulation (Figure 1e).
(
+
)
=
<
(
)
>
|<
(
)
>
|
+
(5a)
where,<
v
(t)
>
v
(
t
)
+
∑
v
(t)/(1+j)
1+
∑
1/(1+j)
(5b)
Figure 1: The horizontal axis (in a, b, d and e) shows the
number isolated bird after attack and vertical axis shows
the probability of that number isolated bird(s). In the
model, we valued
= 0.25,
= 0.05. At =0, all birds
are initialized with the same direction and the predator is
at the opposite direction. (a) shows the probability of
isolated bird in metric case (maximum probability is
16.5% for 3 isolated birds), (b) shows the probability in
topological case, and the maximum probability is 52.4%
for zero isolated bird. (c) shows the image of the
simulation; (d) Comparison between non-preferred and
preferred velocity alignment. Preferred alignment shows
better cohesion. (e) shows the simulation result for three
dimensional topology case. Time step is 1000, number of
simulation is 1000. 1000 individuals, initially, are
distributed in 1 unit radius sphere. The parameter values
are,
= 0.50,
=0.05, and isolation determination
distance is 1.15.
In the simulations, the number of individuals is
200. Data is measured after 2000 time steps for each
simulation, and probability is taken after 2000
simulations done for both metric and topological
case. The prey, initially are distributed a radius 1
circle and predator’s vertical position is 0.9 from the
flock’s centre. Interaction range for metric case, i.e.
metric range is 0.15 and topological range is 3. We
considered a bird is isolated if no other bird is
present in 0.45 radius with respect to the reference
bird. In 3D simulation, this radius would be 1.15.
5.2 Density Independence
In topological interaction, interactions among
individuals should be density independent, i.e. they
should show the same sort of interaction results for
different densities in aggregation. We have run
simulations (the above two dimensional predator-
prey model) for different densities and demonstrate
that the characteristic of interaction vary negligibly
(Figure 2).
Figure 2: Predator-prey model has been tested for different
densities (different numbers of individuals are distributed
within the same area). Other parameters coincide with the
two dimensional topological model in section 5.1. We
used the same parameter values as section 5.1.
5.3 Compatibility of SPP Model
Figure 3: (a) Linear correlation between sparseness and
metric range (Ballerini), Pearson correlation = 0.78. (b)
Linear correlation between sparseness and metric range
(simulation), Pearson correlation = 0.98. We used the
same parameter values as section 5.1.
Is the SPP model is compatible to model bird flock?
To test this, we have considered one of Ballerini’s
field study’s result (Ballerini et al., 2008). They
defined a parameter called sparseness (
) – the
average first nearest neighbor distance of a flock –
which is inverse proportion to the density of the
flock; and metric range for topological interaction
GRAPP 2012 - International Conference on Computer Graphics Theory and Applications
358
(
) – the average n
(= 7) th nearest neighbour
distance of a flock – and found a strong linear
correlation (Figure 3a) between them. We will take
this as a test-stone to test the compatibility of SPP
model. For ten different initial sparseness of our
predator-prey model, we found that our simulations
showed that there remains strong linear correlation
between sparseness and metric range (Figure 3b).
5.4 Our Model
In SPP model, we could produce some trend of
flock’s behaviour (staying together under
perturbation and linear correlation between
sparseness and metric range). But as only directional
alignment has been considered, as time passes
cohesion will break down (Chate, Gregoire, Peruani
and Raynaud, 2008). In our predator-prey
simulation, we found that even though boids have
strong relation in alignment, the flock tends to get
sparser as time passes even when there is negligible
perturbation (Figure 4). Therefore, to model a flock
consisting large number of individuals we have to
consider some other interactive forces that are
presented among individuals. Gruler et al., (1999),
and Kemkemer et al., (2000) described that human
melanocytes - pigment cells of the skin – are also act
collectively without external force. That is why,
melanocytes can be said as SPPs. But melanocytes
do not show directional properties rather show
apolar characteristics. Melanocytes show nematic
arrangements (Figure 5) and their net motion is zero.
They interact with each other nematically. This can
be a vital interaction in different SPPs (Simha and
Ramaswamy, 2002). Vicsek model (1995) assumes
objects as point like while melanocyes are rod like.
Therefore, to model bird flock, we can consider
birds as a rod like objects that consider nematic
forces for cohesion and also tend to make directional
alignment. With this hypothesis, we will introduce a
topological model where both nematic forces and
tenderness for directional alignment would exist. By
modifying SPP model with topological essence, we
described the velocity update for each bird as
equation (6). The main difference of this equation
with Chate et. al. is that it only deals with
topological range where Chate et. al. considered
metric distance.
(
+∆
)
=
{
(
)
+(1−)
+
}
(6)
Here,
is the speed, N
is the number of neighbours
for interaction,
represents the nematic or cohesive
force to each other,
is the unit vector to from i
th
bird to j th neighbor. is the system’s noise level, z
represents the random unit vector.
is the velocity
of j neighbour. s represents a strategy parameter,
where, 01. It determines to what extent, a
bird is going to evaluate directional alignment and
cohesion. Vicsek’s (1995) SPP model does not
consider the prevention of collision among the
individuals. We introduced collision prevention by
imposing an infinite value to
and, setting
=
−
when the nearest neighbor(s) are too close.
makes a vector to a unit vector, i.e. =∥∥
⁄
.
In large flocks, some characteristics can be
found: density fluctuation, wave flow and complex
patterns. SPP model for large number of particles
shows density variance in the system both in two
and three dimensions (Chate et al. 2008). By
simulating a large number of individuals with our
proposed topological cohesive-directional alignment
model, we were able to produce real like flock
(Figure. 6). The simulated flock mainly showed two
properties of real flock: visual complexity and
density variations through flock. Though the flock
shows visual similarities, we must test the internal
structures of simulated flock. At this point, we could
argue that the proposed model is able to create visual
complexity and density variations in flocks.
Figure 4: Sparseness increases with time steps.
Figure 5: Human melanocytes on a glass surface. We can
see that these cells have nematic arrangements (Simha and
Ramaswamy, 2002).
We think that velocity alignment is responsible
MODEL OF AGGREGATION - A Topological Approach
359
for density variation and nematic cohesive force is
responsible for complex pattern. However, yet, we
have not been able to include wave flow in flock of
birds. We are working on this.
Figure 6: A snapshot of flock of birds in our simulation.
Number of individuals is 4096. Initially we distributed the
individual randomly in a box of length 7 and initial
directions were randomly taken. Individuals were updated
according to equation (6) and equation (2). Time step was
1500. Other parameters are:
= 0.5, = 0.001, =
0.94 ,
= 0.05,∆ = 1.0 , and collision prevention
distance = 0.25.
6 CONCLUSIONS
Though interactions among birds in a flock depend
on topological range and birds interact only local
perception of the world, previous models for bird
flock lacks these properties of birds’ behaviour. We
presented a model of bird flocks from topological
perspective. We took two important behaviours of
self-propelled particles to model the bird flock:
alignment and cohesion with neighbours. The
simulation result presents two important properties
of bird flocks: complexity in shapes and density
variations through flocks. We were also able test the
density independence characteristics of flock of
birds and bird’s preferential behaviour that might be
true. Still we need to check flocks’ internal structure
of flocks to compare simulated flocks with real
flocks. Again, we are unable to create wave passing
through flock. We are working on this topic.
REFERENCES
Vicsek, T., 2001. “A question of scale”, Nature. vol411.
Farkas, I., Helbing, D. and Vicsek, T., 2002. “Mexican
waves in an excitable medium”. Nature. 419, 131-132.
Reynolds, Craig W., 1987. “Flocks, Herds, and Schools: A
Distributed Behavioral Model”. ACM Computer
Graphics. volume 21, No.4.
Inada, Y. and Kawachi, K., 2002. “Order and Flexibility in
the Motion of Fish Schools”. Journal of Theretical
Biology. vol.214, issue 13.
Bhattacharya, K. and Vicsek, T., 2010. “Collective
decision making in cohesive flocks”. New Journal of
Physics. 12 093019.
Moussaid, M., Helbing, D. and Theraulaz, G., 2011, “How
simple rules determine pedestrian behavior and crowd
disaster”. PNAS. vol. 108, no. 17.
Vicsek, T. et al., 1995. “Novel Type of Phase Transition in
a System of Self-Driven Particles”. Phys. Rev. Lett. 75,
1226–1229.
Gönci, B. M. Nagy and Vicsek, T., 2008. “Phase transition
in the scalar noise model of collective motion in three
dimensions”. The European Physical Journal. Volume
157, Number 1, 53-59.
Vicsek, T., 2008. “Universal Patterns of Collective Motion
from Minimal Models of Flocking”. 2008 Second
IEEE International Conference on Self-Adaptive and
Self-Organizing Systems.
Ballerini, M. et al., 2008. “Interaction ruling animal
collective behavior depends on topological rather than
metric distance: Evidence from a field study”, PNAS.
vol. 105 no. 4 1232-1237.
Simha, R. Aditi and Sriram Ramaswamy, 2002.
“Statistical hydrodynamics of ordered suspensions of
self-propelled particles: waves, giant number fluctions
and instabilities”, Physica A. 306 (2002) 262-269.
O’Brien, D. P., 1989. “Analysis of the internal
arrangement of individuals within crustacean
aggregations (Euphausiacea, Mysidacea)”, J. Exp. Mar.
Biol. Ecol.,Vol. 128, pp. l-30.
http://en.wikipedia.org/wiki/Topology. (26 September,
2011).
http://www.nn.iij4u.or.jp/~hsat/techterm/topos.html (26
September, 2011).
Henrikson, A. K, 2002. “Distance and foreign policy: a
political geography approach”, Intl. Political Sci. Rev
23, 437.
Vine, I., 1971. “Risk of visual detection and pursuit by a
predator and the selective advantage of flocking
behaviour”. Journal of Theoretical Biology. Volume
30, Issue 2, Pages 405-422.
Chaté, H., Ginelli, F., Grégoire, G., Peruani, F. and
Raynaud, F., 2008. “Modeling collective motion:
variations on the Vicsek model”, The European
Physical Journal B – Condensed Matter and Complex
Systems. Volume 64, Numbers 3-4, 451-456.
Gruler, H., Dewald, U. and Eberhardt, M., “Nematic liquid
crystals formed by living amoeboid cells”. The
European Physical Journal B – Condensed Matter and
Complex Systems. Volume 11, Number 1, 187-192.
Kemkemer, R., Kling, D., Kaufmann, D. and Gruler, H.,
2000. “Elastic properties of nematoid arrangements
formed by amoeboid cells”. The European Physical
Journal E: Soft Matter and Biological Physics.
Volume 1, Numbers 2-3, 215-225.
Parrish, J. K. and Hamner, W., 1997. Animal groups in
three dimensions. Cambridge University Press.
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