IEEE Circuits and Systems Magazine - Q3 2019 - 14

continuous kinetic version [84], potential nonsymmetric
kinetic version [86], and the hydrodynamic limit of the kinetic Cucker-Smale model [88]. In 2011, Dalmao et al. even
studied the Cucker-Smale model under the conditions of
random interactions and hierarchical leadership [89].

velocity correlations are superior to position correlations because of the former is more accurate according
to obtainable information [20].
Analogously, define the directional correlation function for a bird and the rest of the flock as

III. Group Dynamics on Hierarchical
Network of Flocks
In this section, we introduce the group dynamics on
hierarchical network of bird flocks to investigate how
one bird affects its fellows and even the whole flock.
In 2010, Nagy et al. tracked the homing pigeons flock
(with up to 10 individuals) by utilizing high resolution
but lightweight GPS devices, and collected their trajectories data [20]. And then, they analyzed collected data
using directional correlation delay functions inspired by
methods in statistical physics [20].
Prior to analysis, define the criteria of occurring a
leading event, that is another bird copied a bird's movement direction with a time delay. To determine whether
the leader-follower relationship between arbitrary two
birds exists or not, the authors calculated the directional correlation delay for each pair of birds i and j
(i, j ! I ), which is depicted by

C i (x) = G v i (t ) · v j (t + x)Ht, j,

C ij (x) = G v i (t ) · v j (t + x)Ht , i, j ! I, i ! j,
where v i (t ) is the normalized velocity of bird i, G·H represents the average value over time t, and note that
C ij (x ) = C ji (-x ). Apparently, x )ij =-x )ji . Then one can calculate the directional correlation delay time x )ij which
maximizes the value of the correlation function C ij (x ).
If x )ij is positive, it indicates that bird j is flying in
the wake of bird i (i, j ! I ). Here, authors considered

E
C
F
A
B
D

G
Figure 8. Illustration of hierarchical network of bird flocks.
This network is divided into 4 layers. Node A denotes the
leader (the first layer), other nodes represent the followers
(other layers).
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IEEE CIRCUITS AND SYSTEMS MAGAZINE

and x )i is the value that maximizes this fuction.
Then Nagy et al. constructed a directed leader-follower network for each flight by considing individual birds
as nodes, while the edges represent inferred relations
between their movements, and the direction is determined by the pairwise value x )ij . The resultant network
includes only those edges whose directional correlation
values based on x )ij are greater than a given variable
threshold, namely, x min .
Furthermore, they found a well defined hierarchical
network under the application of experimental data and
concerning the roles of leading in pairwise interactions.
In this work, the whole flock was divided into some different roles (as illustrated in Fig. 8), which suggested
that the relational structure within a moving flock in
the real world maybe more sophisticated. What's more,
they unveiled that the spatial position of a pigeon is
strongly correlated with its position in the hierarchical
network [20]. They also revealed that pigeons response
more quickly to conspecific signals through the perception of left eyes. At the last, Nagy et al. proposed that the
hierarchical organizations of flight birds may be more
efficient than egalitarian organizations [20].
At the same year, Quera et al. developed an agentbased simulation model for flocking in a virtual world,
the result suggested that the interaction rules among
birds were actually the hierarchical leadership [90].
By using the directional correlation time delay method
analogous to Ref. [20], Flack et al. ranked all birds in a
flock, and thus each bird was labeled by its own hierarchical leadership rank. To examine the flexibility of individual's rank, they manipulated three different hierarchical networks with different homing experiences, and
then they found that the changes of leadership between
trained and untrained birds are relatively small, thus the
hierarchical leadership in flight pigeons is robust [91].
In 2013, Nagy et al. established a robust and scalable method which is applicable to many kinds of social
structures, and they confirmed that the context-specific
hierarchical leadership must be based on different individual competences [92]. In 2014, by applying high resolution spatiotemporal data collected from flight pigeons,
Zhang et al. showed that when birds move along smooth
trajectories, they will tend to follow the average direction
of their neighbors, while they will follow their leaders'
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IEEE Circuits and Systems Magazine - Q3 2019

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