IEEE Circuits and Systems Magazine - Q3 2019 - 12

2) Mathematical Models
a) The Vicsek Model
Although Boid model provides a good research framework for researching collective behavior of bird flocks,
it is still inconvenient for quantitative interpretation of
the collective behavior of huge flocks. In order to establish a quantitative interpretation model of self-organization motion in flocks, a statistical physics type approach to the study of flocks was introduced in 1995 by
Vicsek et al. [48].
With the presence of perturbations, the updating
rules of the Vicsek model are
x i (t + Dt ) = x i (t ) + v i (t ) Tt,
_
Z
]] / sin i j (t ) bb
j ! N i (t)
i i (t + Dt ) = arctan [
` + Ti.
] / cos i j (t ) b
\ j ! Ni (t)
a
where N i (t ) is defined by N i (t ) = { j : < x i (t ) - x j (t )< 1 r },
r is the neighbor radius, which is a neighbor set of bird
i (i ! I ). Ti represents a random noise with uniform
probability distribution over the interval [-h/2, h/2],
where h is a system parameter.
This extremely simple model allows one to simulate
thousands of flocking birds, and displays the emergence
of a second order type dynamical phase transition from
disordered state to ordered state as the level of noise
decreases or the average density increases [48]. In addition, the Vicsek model has the following features:
1) It is a simple nonlinear model which is capable of
simulating swarming behavior of bird flocks.
2) At each time step, suppose that the movement region of a bird is limited to a circular area with itself
as the center, i.e., its region of interaction (see Fig. 6).
3) Each bird within the flock has an absolutely constant velocity.
4) The headings of disordered birds will reach consensus under the condition of high density and
low noise.

Region
of Interaction

In contrust with the fixed metric distance in the
Vicsek model, Ballerini et  al. reconstructed the trajectories of individual birds in flocks and discovered that
each bird in flocks interacts with partners in a fixed topological distance, i.e. a fixed number of neighbors [60].
By exploiting empirical data, In 2011, Bode et al. provided evidences that flocking could occur in the presence
of a fixed number of neighbors [61].
Recently, other simple models with similar self-propelled agents which obey original interaction rules have
been introduced to qualitatively describe the collective
motion of bird flocks [62]-[70]. When comparing the standard order parameter individually, all these models seem
to go through a disorder to order phase transition with
the presence of noise decreasing. Typically, they all described groups of agents moving with an imposed velocity with a non-vanishing constant magnitude. Moreover,
these models studied the onset of collective motion of bird
flocks under the circumstance of two dimensional space,
analogous to the one investigated in the Vicsek model.
No matter what type the model is, discrete [63], [64], [66]
or continuous [68]-[70], authors showed that their model
can self-organize and exhibit coherence. What's more,
the collective motion of the flocks is an example of spontaneously broken symmetry which is presented without
preferred direction in the bird flocks [68].
Additionally, Huepe and Aldana proposed a more detailed quantitative description through analyzing birds
clustering in space [66]. They revealed that the distribution of cluster sizes approaches a power-law style as the
noise level is reduced. And this trend suggested a nontrivial critical behavior, that is, reversal occurs when
the phase transition threshold of noise is reached [66].
In 2010, Cavagna et  al. also studied the distribution of
flock clusters, and indicated that behavioral correlations between birds are scale-free [71].
Because the Vicsek model is nonlinear, it is difficult to
analyze collective behavior of bird flocks with it. To avoid
this problem, in 2003, Jadbabie et al. ignored the effects of
noise and proposed a linearized Vicsek model, they also
introduced the corresponding theoretical explanation for
the observed behaviors of flocks [72]. The linerized heading updating rule of agent was formulated as
i i (t +1) =

Figure 6. Consensus in Vicsek model, in which circle is the
region of interaction among agents.
12

IEEE CIRCUITS AND SYSTEMS MAGAZINE

1
; N i (t );

/

j ! N i (t)

i j (t ) =

N

/ c ij i j (t ),

j =1

where the meaning of i i (t ) and N i (t ) were stated previously, while c ij is the coupling coefficient of agent or
bird i (i ! I ), and it is defined as follows: if there is a
direct interaction between bird i and j, namely, bird j
is one of the neighbors of bird i, then, c ij ! 0, otherwise,
c ij = 0. Moreover, for each bird i ! I, satisfies / Nj = 1 c ij = 0.
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IEEE Circuits and Systems Magazine - Q3 2019

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