IEEE Circuits and Systems Magazine - Q3 2020 - 56

model (21), there is no practical link between layers 1
and 2, but dependency links [157].
By taking the corresponding inter-layer links into
consideration, another work adopts the node dynamics
of layer 1 as follows [42]:

1

0.5



(R1, R2)

0.2

d

0.1
0

0

0

fc
0

io i, 1 = ~ i, 1 + m a

0.5
f

0.4

sin (i j, 1 - i i, 1) + m x sin (i i, 2 - i i, 1),
j=1
(22)

1
0.8

Figure 10. Intra-layer explosive synchronization for N = 1000
and f = 1 [41]. Squares and circles (triangles and stars) are
used for denoting the forward and backward transition of R1
(R2), and the inset shows the corresponding dependence of
1 d 2 on f for ten realizations. Layers 1 and 2 are fixed as
two independent random ER networks with the same average degree 12 and having the random homogeneous distribution of frequencies in the range [-1].

1
0.8

(r )

0.6

λx = 0.1
λx = 0.3
λx = 0.5
λx = 0.7
λx = 1

0.4
0.2
0
0.5

1

(λα)

1.5

2

Figure 11. Synchronization level r of layer 1 vs. intra-layer
coupling strength m a for different values of inter-layer coupling strength m x [42].

where m is the overall coupling strength, k i, 1 ^k i, 2 h is the
degree of node i for layer 1 (2). The parameters a i, 1 and
a i, 2 involve the coupling of the two layers, which are set
as a i, 1 = ri, 2 and a i, 2 = ri, 1, if the pair of nodes i is part of
the fraction f of coupled nodes (otherwise, the parameters are set as a i, 1 = a i, 2 = 1). f is the fraction of nodes
controlled adaptively by the local order parameters ri, 1
k
and ri, 2 , which are defined by ri, 1 e iz1 = (1/k i, 1) R j =i,11 e ii j,1
k
and ri, 2 e iz2 = (1/k i, 2) R j =i,21 e ii j,2 . That is to say, the oscillators in one layer are controlled by the corresponding
oscillators in another layer. The global order parameter
R1 (R2) of layer 1 (2) is defined by R 1 e iW1 = (1/N) R Nj = 1 e ii j,1
^ R 2 e iW 2 = (1/N) R Nj = 1 e ii j,2 h . Simulation results show that
there is a critical value fc for starting explosive synchronization in each intra-layer (see Fig. 10) [41]. Note that in
56 	

N

/ a ij,1

where the last term is the interaction between layers
1 and 2. m a and m x are the intra-layer and inter-layer
coupling strengths, respectively. From Fig. 11, one can
find that a larger inter-layer coupling strength can
advance the synchronization, whereas the explosive
synchronization happens by weakening the inter-layer
coupling strength.
Recent effort of [43] has investigated collective phenomena in heterogeneous multiplex networks, namely
the dynamics of each layer is different, which mimics
the interplay between neural activity and energy transport across brain regions. Layer 1 as neural activity is
modeled by the Kuramoto-oscillator network. Another
dynamical process, i.e., energy transport at layer 2, is
modeled by a continuous-time random walk. Numerical results show that the heterogeneous distribution of
walkers in layer 2 is responsible for the emergence of
intra-layer explosive synchronization in layer 1, where
it is not necessary to impose any external assumptions,
and the Kuramoto model is not coupled to other dynamical systems.
Other findings have emphasized the importance of
multiplexing or the impact of one layer on the dynamical evolution of other layers [158]. Multiplexing of two
second-order Kuramoto-oscillator networks that have
homogeneous degree distribution supports the firstorder transition in both the layers, thereby facilitating
explosive synchronization.
4.2 Inter-layer synchronization
Inter-layer synchronization refers that the states of each
oscillator with its counterparts in different layers reach
an agreement eventually no matter the final states of
oscillators in that layer. A duplex Kuramoto-oscillator
network [156] comes as
Z
] io i, 1 = ~ i, 1 +
]
[
o
 ]] i i, 2 = ~ i, 2 +
\

m1

k i, 1
m2

k i, 2

/

sin (i j, 1 - i i, 1) + m 12 sin (i i, 2 - i i, 1),

/

sin (i j, 2 - i i, 2) + m 21 sin (i i, 1 - i i, 2),
(23)

j ! K i, 1

j ! K i, 2

where the natural frequencies ~ i, 1 and ~ i, 2 are distributed by an even and symmetric probability density. In [156], m 1 = m 2 = m intra is the intra-layer coupling

IEEE CIRCUITS AND SYSTEMS MAGAZINE 		

THIRD QUARTER 2020
IEEE Circuits and Systems Magazine - Q3 2020

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