IEEE Circuits and Systems Magazine - Q3 2019 - 19

However, the analysis methods for discrete-time
complex networks and continuous-time complex networks are quite different. Thereby, it is necessary for
researchers to develop some new methods to deal with
discrete-time complex networks. In the past few years,
they have established many methods to solve issues of
discrete-time complex networks, using techniques such
as stochastic matrix theory  [108]-[109], graph theory [110], [111], [117], convex analysis [113] and eigenvalues method [126]-[128].
For discrete-time MASs, in 2009, Chen et al. proposed
a consensus model under conditions of nonlinear local
rules and time-varying delays  [114]. In 2010, Chen and
other coauthors proposed an approach to analyze the
convergence rate in the case of time-varying delays, and
obtained an upper bound under the joint connectivity
condition [115]. They also introduced a general time-delayed MASs model in the presence of noise and studied
its robust consensus in 2011  [116]. In 2013, Chen et  al.
considered a class of nonlinear MASs with general timevarying delays, which is expressed as [99]
n

x i (t +1) = / c ij (t ) fij (x j (t - x ij (t ))), i ! I,

(17)

j =1

where fij (·) is a nonlinear function, x ij (t ) denotes the
time-varying delay and c ij (t ) has the same meaning as
which in Eq. (15).
They proved that if f is a continuous function on a
convex set A, and there exists a bounded convex subset
B 3 A whose elements are fixed points of f, for arbitrary
non-fixed point x, the distance between f (x) and B
is less than that between x and B, then the system
(Eq. (17)) would reach consensus (Eq. (16)) under the
condition of joint connectivity  [99]. A few years later,
in 2017, based on graphical method, Chen and Lü introduced time-delays to discrete-time distributed coordination algorithm which originally can not realize
consensus, and found that if the introduced delays are
co-prime with the period of the communication topology, then consensus will happen [117].
There is no doubt that the consensus algorithm depends on topology structure among agents, result in
many researchers focused on this field in order to receive new and meaningful outcome in the past decade.
For instance, compared with the work of Jadbabie et al.,
in which authors investigated flock consensus with undirected and jointly connected topology  [72], in 2004,
Li and Wang extended this model to an infinitely connected case [118]. In 2005, Blondel et al. analyzed convergence in the case of unbounded intercommunication
intervals as well [119]. In 2005, Lü and Chen introduced
the synchronization of flocks with time-varying topolTHIRD QUARTER 2019

ogy structure [120]. In the meantime, Ren and Beard investigated the consensus in the presence of limited and
changing interaction topology [121].
In 2006, Sarlette et al. explored synchronization algorithms under conditions of fixed, undirected interconnection and time-varying, and directed communication
topology, respectively [122]. At the same year, Xiao and
Wang studied the state consensus of flocks with switching topologies and time-varying delays  [123]. Delellis
et al. also studied the consensus with switching communication topology in 2010 [124]. In 2008, Li investigated
the swarm with a directed and weighted topology [125].
Consensus problem of dynamically changing topology with non-uniform time-delays were revealed by
Ref. [107], [126]. After Ren and Atkins's research about
second order consensus problem [126], in 2010, Yu and
his coauthors established a further work in which some
necessary and sufficient conditions for second order
consensus were unveiled [127]. An accurate formulation of the convergence to consensus of birds in three
dimensions was proposed by Cucker and Smale in 2007
[77], [78]. Recently, researchers also have developed approaches for the consensus of complex networks with
uncertain topological structures [102]-[104]. In 2018, Liu
et  al. addressed the event-based leader-following consensus of a class of multi-agent systems with switching
networks [129].
V. Conclusion
In this tutorial review article, we summarized the collective behaviors of bird flocks through social interactions among individuals from a multi-agent systems
viewpoint. Due to the different social interactions
among their agents, bird flocks present various collective behaviors. This article introduced some advances
achievements in this multidisciplinary realm. In particular, models for line formations and cluster formations
were investigated based on different social interactions,
a more sophisticated leader-follower behaviors model,
i.e., hierarchical group behaviors model and some of
its further researches were discussed. Moreover, collective decision making including consensus behavior
and information transfer in turning of bird flocks were
explored, and some consensus models in control theory
were exhibited.
Researches of collective behaviors have profound
significance and are destined to bring great advantages to deal with actual situations, such as the prediction of global displacement of a huge fish school.
These theories have numerous potential applications,
for instance, collective robotics. However, there still
exist several open unresolved issues, such as the
fact that it is still unclear what kind of leader-follower
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