IEEE Circuits and Systems Magazine - Q3 2020 - 62

Kuramoto-oscillator networks inevitably suffer from various forms of noise in
coupling/control processes. Nevertheless, the global stochastic synchronization is
successfully achieved without any restriction on initial phases and natural frequencies.
	

N

u i = C / b ij (i j (t) - i i (t)), i = 1, f, N, (34)
j=1

where C 2 0 is the control strength, bij is the element of
the adjacency matrix B ! R N # N that describes the structure of the control layer G c .
Two different types of noise in sinusoidal coupling
processes and control links are considered, respectively. The one is the noise entering the Kuramoto-oscillator
layer G k in the sinusoidal coupling process
v i (t) dw i (t) = u

N

/ p ij sin (i j (t) - i i (t)) dw ij (t), i = 1, f, N,

j=1
(35)

where u 2 0 is the intensity of noise, pij is the element of
the adjacency matrix P ! R N # N signifying the structure
of the network with noisy communication links. wij is an
one-dimensional standard Brownian motion, i.e., its derivative wo ij is the independent Gaussian white noise. We
note that Gwo ij (t)H = 0 and Gwo ij (t), wo kl (t l ) H = d ij d kl d (t - t l ),
where the angular brackets G · H denote the average over
different realizations of the noise, d ij (kl) is the Kronecker function, and d denotes the Dirac function. Another
one is that noise emerges in communication links on
the networked control layer G c , which can be set as
v i (t) dw i (t) = t

N

/ q ij (i j (t) - i i (t)) dw ij (t), i = 1, f, N, 	(36)

j=1

where t 2 0 represents the intensity of noise, qij is the
element of the adjacency matrix Q ! R N # N representing
the structure of the network with noisy communication
links. wij and wo ij have the same meanings as those in (35).
Specifically, there are two definitions including
stochastic phase agreement and stochastic frequency synchronization for identical and nonidentical
oscillators [70]:
Global stochastic phase agreement is achieved if for
any i i (0) ! R, there exists
	

P $ tlim
i i (t) - i j (t) = 0 . = 1, 6i, j = 1, 2, f, N, (37)
"+3

where P : F 7 [0, 1] is a probability measure, and F is
a v -algebra.
Global stochastic frequency synchronization is
achieved if for any i i (0) ! R, ~ i (0) ! R, there exists
	
62 	

P $ tlim
io i (t) - io j (t) = 0 . = 1, 6i, j = 1, 2, f, N. (38)
"+3

By using the stochastic Lyapunov stability approach,
the local and global connectivity criteria are proved for
achieving global stochastic asymptotic phase agreement
and frequency synchronization, respectively. Moreover, it
is numerically verified that the instantaneous frequency
of Kuramoto-oscillator networks is more sensitive than instantaneous phase to noise perturbation. More concretely, phase agreement is robust to large noise intensities.
While for frequency synchronization, the adverse effect
of noise existed in the networked control layer is more
prominent than that in the Kuramoto-oscillator layer.
6. Summary and Discussions
In this survey, we have retrospected the recent achievements on collective synchronization of Kuramoto-oscillator networks and its applications in engineering and
sociology. Apart from several synchronization phenomena observed in mono-layer and multi-layer Kuramotooscillator networks, we have elaborated the stability
analysis of Kuramoto-oscillator networks without and
with control (e.g., a pacemaker, networked control).
Although a variety of applications and abundant theoretical results have been reported for the Kuramoto-oscillator networks, there are still some limitations of the
existing development, and many potential directions
should be pointed out as follows:
■ ■ This review bridges the Kuramoto-oscillator
model and the circuit oscillator for the first time,
but so far we know that the theoretical results of
collective synchronization of Kuramoto-oscillator
networks have not been applied into the locking
of circuit oscillators. Moreover, opinion dynamics
modelled by the Kuramoto-oscillator networks
have not successfully explained some collective
phenomena. For example, the public reaching
an agreement on a particular problem instantaneously is analogous to the explosive synchronization, different opinions kept in distinct groups
parallels cluster synchronization, and so forth.
In the power grids constructed by the multi-rate
Kuramoto model, the problem that how to design
a novel control strategy for achieving the finitetime frequency restoration keeps open yet. To
sum up, there is still a long road for the applications of the Kuramoto-oscillator model in sociology, engineering, and other fields.
■■ Most of the past studies focused on a monolayer network, and there are few achievements

IEEE CIRCUITS AND SYSTEMS MAGAZINE 		

THIRD QUARTER 2020
IEEE Circuits and Systems Magazine - Q3 2020

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