IEEE Circuits and Systems Magazine - Q3 2019 - 16

between birds in a flock only required few informed individuals, namely, only few birds are leaders, while the others are followers [96]. They also proved that the balanced
weighting term ~ will be of least importance for agents'
moving accuracy, regardless of whether the proportion of
informed agents is small or large, otherwise, the agents'
moving accuracy will be strongly and positively correlated
with the balanced weighting term when proportion of informed agents is a proper intermediate value [96].
In order to investigate which factors have a significant impact on the consensus decision making of flocks.
In 2008, Lü et  al. extended this model and revealed
some underlying theoretical mechanisms for consensus
decision making of bird flocks via numerical simulations [97]. Taking into account that the informed agents
may have different preferred directions, Lü et  al. created two subsets of informed individuals corresponding
to different preferred directions, respectively (Fig. 9).
They assumed that there are two sources of information
denoted as G 1 and G 2, and the angle between them is
denoted as i (see Fig. 9), which characterizes the degree of conflicts between informed agents within the
flocks. Suppose there are N k (k =1, 2 ) informed individuals with preference to information source G k (k =1, 2 ),
respectively. At time t, the distance between bird i and

j (i, j ! I ) is denoted by d ij (t ). And d 0 describes the
minimum separate distance while d 1 indicates the maximum response distance. In addition, the authors also
considered two action mechanisms between agents
within the flocks: reciprocal action and alignment (as
shown in Fig. 10).
Note that X i is the neighbor set of agent i, and its cardinal number is n i . The coupling coefficient of agent i is
d i, and the reciprocal coefficient between agent i and
j is c ij . Then defined that c ij =-(d 0 - d ij)/d 0 and d i = 0
when X i = { j ; d ij # d 0}, while c ij = (d ij - d 0)/(d 1 - d 0) and
d i =1 when X i = { j ; d 0 1 d ij # d 1}.
Because each agent tends to balance the conflicts
between reciprocal action and its alignment, thus
one can obtain the desired direction of agent i at
time t + Tt
h i (t + Tt ) =

1-b
ni

/

j ! Xi

c ij

v j (t )
b
+
;v j (t ); 1 + n i

G1

θ
S0

R2
G2

Figure 9. Illustration of agents in flock may differ in their preferred direction. Where the golden node is a flock, the red and
blue nodes are two preferred directions, respectively [97].

(a)

(b)

Figure 10. Illustration of the basic action mechanisms between
agents in flocks [97]. (a) Reciprocal action, (b) alignment.
16

IEEE CIRCUITS AND SYSTEMS MAGAZINE

j ! Xi

di

v j (t )
,
;v j (t );

j!i

where b is the alignment ratio for all agents.
Considering the information sources, the final direction is given by
hli (t + Tt ) = (1 - ~ i)

R1

/

h i (t + Tt )
+ ~ i Ti (t ),
; h i (t + Tt );

where Ti (t ) is the unit vector from x i (t ) to the information source, ~ i is the prefer weight of agent i  [97].
By using numerical simulation methods, they revealed
some underlying theoretical mechanisms for consensus
decision making of bird flocks. For example, it indicated
that for different sizes of bird flocks and different structures of information sources, the coherent flocks will
be restricted to a common narrow interval  [97]. The
simulations of this work also demonstrated that consensus decision making of flocks in the built model will
display a surprising degree of robustness or tolerance
against errors but extremely fragile in the case that the
information sources change suddenly  [97]. They further revealed that for small changes in the number of
two subsets, consensus decision making of flock will
still occur [97].
B. Information Transfer in Sharp
Turning of Bird Flocks
Considering a flock of birds threatened by their natural
enemies, birds sharply change their directions of motion
to avoid being attacked. However, the slightest uncertainty in changing directions may put the flock into a fragile
condition, which would result in decreasing cohesion
within the flock or even splitting the group. Therefore,
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IEEE Circuits and Systems Magazine - Q3 2019

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