IEEE Circuits and Systems Magazine - Q3 2020 - 55

Starting from a network of scientific coauthorship, as
a real social network, Pluchino et al. [149] have studied
the local coupling model:
	

xo i (t) = ~ i + K
di

/

a sin (x j - x i) e - a |x j - x i|, (19)

j ! Ni

where di and Ni are the degree and the neighborhood set
of individual i, respectively. Other variables and parameters in (19) have the same meaning as those in (18). The
results obtained in [149] indicate that the network topology is a fundamental factor in the evolution of scientific
coauthorship. On the other hand, the effort of [150] adopted the following model
	

xo i (t) = ~ i + c i · 1
di

/

a j sin (x j - x i), (20)

j ! Ni

where ci represents the ith individual's ability of accepting the influence a j from the neighbor j. Other variables
and parameters in (20) have the same meaning as those
in (19). Obviously, if d i = N and K = c i a j, then the
above equation (20) is exactly the classical Kuramoto
model (1). Simulation results show that larger c i, a j
(or K ) can urge the emergence of synchronization. Recently, by adopting the artificial bee colony scheme,
Xiao et al. [151] leverage the Kuramoto oscillators as
individuals in the social network for exploring online
public opinion synchronization. They find that in weak
coupling systems, via improving the connection cost
and adopting an optimized network construction protocol, the starting time and higher level of synchronization can be still reached earlier under certain cost
constraints [151]. In a strong coupling system, synchronization can be achieved at a relative low cost [151].
Moreover, implementing optimization technology can
arrive at a better balance between synchronization performance and the cost [151].
Finally, it should mention that the model of Kuramoto-oscillator networks has been applied into many other
fields, such as biological system [8], financial market
networks [10], and neuronal networks [152].

of multi-layer networks can refer to surveys [153],
[154]. Note that a multiplex network is a special type of
a multi-layer network, where different layers have the
same number of nodes, and the only possible type of
inter-layer connections are those in which a given node
is only connected to its counterpart nodes in the rest
layers [153], [154].
However, the majority achievements of synchronization of Kuramoto-oscillator networks mainly root in a
single-layer (or mono-layer) network, which ignores the
influence from other layers. The existing achievements
concerning the synchronization emerging in multi-layer
Kuramoto-oscillator networks are few [41]-[44], [155],
[156]. In accordance with the network architecture,
these collective phenomena are mainly classified into
intra-layer synchronization and inter-layer synchronization. For simplicity and generality, the existing efforts
often start with a duplex network (see Fig. 9), in which
layer 1 has the same number, N, of nodes as those of
layer 2, and all connections between these two layers
are one-to-one.
4.1. Intra-Layer Synchronization
Intra-layer synchronization refers that the states of all
oscillators in the same layer reach agreement eventually. The final states of different network layers may be
different or the same. Two independent networks (layers 1 and 2) with the same size N are constructed as
follows [41]:

	

Z
k i, 1
]o
] i i, 1 = ~ i, 1 + ma i, 1 / sin (i j, 1 - i i, 1),
j=1
[
(21)
k i, 2
] io i, 2 = ~ i, 2 + ma i, 2 / sin (i j, 2 - i i, 2),
]
j=1
\

2
1

3

i

N

4. Synchronization of Multi-Layer Networks
Multi-layer networks are ubiquitous in human society
and nature world. For example, the whole (multi-layer) public transportation network consists of airline,
railway, highway, and waterway networks (layers), in
which each city can be regarded as a node of the network. In ecosystems, a food web is divided into multiple levels (or layers) mainly containing prey-predator
groups, where individuals cooperate within each group
and compete between different groups. More examples
THIRD QUARTER 2020 		

Layer 1

Layer 2

2

3

1

N

i

Figure 9. A duplex network consisting of layers 1 and 2.

IEEE CIRCUITS AND SYSTEMS MAGAZINE	

55
IEEE Circuits and Systems Magazine - Q3 2020

Table of Contents for the Digital Edition of IEEE Circuits and Systems Magazine - Q3 2020
