Producing Complex Networks Using Coupled
Oscillatory Circuits with Evolutionary Connections
Yoko Uwate† , Thomas Ott‡ and Yoshifumi Nishio††

Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: {uwate, nishio}@ee.tokushima-u.ac.jp‡
Zurich University of Applied Sciences
Einsiedlerstrasse 31a, 8820 Waedenswil, Switzerland
Email: thomas.ott@zhaw.ch

Abstract— In this study, we propose a method of generatingcomplex networks by exploiting synchronization between coupledoscillatory circuits. To each node of a 2D fully connected network
a van der Pol oscillator is assigned. We then study the topologicalevolution of the network in dependence on environmental conditions. These conditions are modeled by considering the distance
between the oscillators and some small frequency errors that areadded. By carrying out computer simulations, we confirm thatdifferent types of complex networks are obtained depending on
different environmental conditions.

I. I NTRODUCTION
The synchronization phenomena observed from coupledoscillators are suitable models to analyze high-dimensional
nonlinear phenomena [1], [2]. Many different coupled oscillatory networks have been proposed in the literature and aplethora of interesting synchronization phenomena have been
discovered [3], [4]. Synchronization in dynamical complex
networks has attracted a great deal of attention from various
fields such as biology and neuroscience [5]-[8], engineering [9]and physics [10]-[13], economy [14] and social sciences [15][17]. In particular, understanding the relation between the
topological structure and the functional behavior of a network
is considered a significant topic for practical applications inmany disciplines.
In a previous study, we have investigated synchronization
phenomena in dynamical polygonal circuit networks withswitching couplings [18]. We confirmed that new synchronization states can be produced when the network topology
is changed by switching connections in the proposed system.
Furthermore, we focused on brain networks as a particular
example of dynamical complex networks. We applied threedifferent types of coupling distributions: uniform, Gaussianand heavy tail. It was confirmed that the dynamical networkwith a heavy tail coupling distribution tended to avoid global
synchronization [19]. Although these results are interesting,the proposed dynamical network was restricted to on/off
couplings and the network topology did not grow up over time.
In this contribution, we want to study evolutionary processesof networks depending on environmental conditions.
We propose a method of generating complex networks byusing synchronization states of coupled oscillatory circuits. To

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this end, a van der Pol oscillator is assigned to each node ofa given network. Initially, all nodes are fully connected withvery weak coupling strengths. Additionally, small frequency
errors are added to all oscillators. We then apply two rules forgenerating complex networks, inspired by existing real social
networks. First, the connection between two nodes which arealike (that is, the frequencies of the two oscillators are very
close), becomes stronger. Hence, the synchronization state
between the two nodes is used to determine whether they arealike or not. Second, each node has an increasing influenceon other nodes in a certain neighborhood. We then let the
network evolve and investigate the final network structures,by changing the crucial parameters behind these two rules,
namely frequency errors and neighbourhood distance. This two
parameters define the environmental conditions.
By carrying out computer simulations, we confirm thatdifferent types of complex networks are obtained dependingon the environmental conditions. We show that the generated
networks vary substantially depending on these environmental
conditions, even when the initial state of the network is thesame. Hence, this model based on coupled oscillatory circuits
offers an interesting model to study evolutionary processes incomplex networks in general.
II. C OUPLED C IRCUITS N ETWORK
A. Circuits Model
The initial arrangement of 100 van der Poll oscillators isshown in Fig. 1. All circuits are connected to each other by
resistors. At the initial state, all nodes are fully connected withvery weak coupling strengths. The van der Pol oscillator andthe coupling method are shown in Fig. 2.
In the following, we introduce the circuit equations of the
network model. The vk − iRk characteristics of the nonlinear
resistor are approximated by the following third order polynomial equation,
ik = −g1 vk + g3 vk 3 (g1 , g3 > 0),(k = 1, 2, ...13).

(1)

The normalized circuit equations governing the circuit are
expressed as1[step-1] At the initial state, all nodes are fully connected withvery weak coupling strengths (γ0 =0.0003).
[step-2] After a transient phase, we apply two rules for asequence of generations. Each generation has length τg =10, 000. The conceptual diagram of the computer simulationis shown in Fig. 3.
• (Rule-1: Influence of synchronization) In order to check
whether two nodes are alike, we calculate the synchronization ratio for every pair of oscillators. If the synchronization ratio is larger than 90%, the corresponding
coupling strength becomes stronger with ∆γs .
In order to analyze the synchronization ratio, we define
a synchronization state as

0.9
0.8
0.7

y

0.6
0.5
0.4
0.3
0.2
0.1
0

0

Fig. 1.

0.1

0.2

0.3

0.4

0.5
x

NR

0.7

0.8

0.9

1

Circuit arrangement in 2D space (N =100).

ir

n

0.6

i

v

αkC

(a) van der Pol oscillator.

Fig. 2.

|xk − xn | < 0.1

L(Rule-2: Influence of neighborhood distance) In order toapply the second rule (each node increasingly influences
other nearby nodes), we calculate the distance between
all pairs of nodes. If the distance is smaller than 0.1, thecorresponding coupling strength ∆γd is increased.
[step-3] Step-2 is repeated until 100 generations are reached(G = 100).
[step-4] At the final state (G = 100), we check the synchronization ratio for every pair of oscillators. If the synchronization ratio is larger than 90%, the nodes are connected,
otherwise the connection is cancelled. Finally, we obtainthe network topology by affecting the frequency errors anddistance.

R•(b) Coupling method.

Circuit model.

[kth oscillator]
 

dxk
1 2
2


=
ω
ε
1
−
xk xk − yk
 dτ

3



n

X


yk
− (γ0 + ∆γs + ∆γd )
k=1



dy

k

= xk



 dτ
(k = 1, 2, ...N ).

(2)transient state G1 G2 G3 G4τtsτ

determinationsynchronization ratio > 90%
distance < 0.1

r
r
g1
g1
C
vk =
yk ,
xk , ik =
3g3
3g3 L
r
r
L
C
1
α = 2 , ε = g1
, γ=R
,
ω
C
L
r

In these equations, γ0 is the initial coupling strength. ∆γsand ∆γd denote the increasing coupling strengths influenced
in the evolution process by the synchronization ratio andthe neighborhood distance. ε denotes the nonlinearity of theoscillators and ω denotes the frequency error. The frequency
error is set to 10%. For the computer simulations, we used
a fourth-order Runge-Kutta method with step size h = 0.01.
The parameter of nonlinearity of this circuit model is fixedto ε = 0.1. The number of van der Pol oscillators is set to
N = 100.
B. Evolution Process of Network
We now explain the evolution process of the coupled circuits
network. The network topology of the coupled circuits evolves
according to the following steps.

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G100

τg τg τg τg ... τg

where√t = LCτ ,(k ∈ Sn ).

τts=1000,000
τg=10,000
Generation (G)=100

Fig. 3.

Evolution process.

Next, we consider the methods of increasing the couplingstrengths in each generation. Six types of methods are proposed as shown in Fig. 4. Figure 4 (a), (b) show the basic
methods. The coupling strength change is the same for the
influence of the synchronization and the distance condition.
The coupling strength change shown in Fig. 4 (a) is ∆γs =∆γd = 0.0001 and in Fig. 4 (b) is ∆γs = ∆γd = 0.0002.
In the case of Fig. 4 (c), (d), the coupling strength change
is different for the influence of the synchronization and the
distance condition. The coupling strength change of the distance condition is twice as large as for the synchronization
condition in Fig. 4 (c). While, in Fig. 4 (d), it is the other way
around. In the last two methods, the coupling strength change

for the distance and synchronization conditions varies withthe generation sequence, as shown in Fig. 4 (e), (f). These six
methods are named as Basic-1, 2, Constant-1, 2 and Varying-1,
2, respectively.
0.00030.0003synchronization (∆γs)
distance (∆γd)

synchronization (∆γs)
distance (∆γd)

0.000250.0002∆γ∆γ0.00020.000150.00010.0001

5e-05

0

20

40

60

80

0

100

0

40

20

(∆γs = ∆γd = 0.0001)(∆γs = ∆γd = 0.0002)

N ETWORK CHARACTERISTICS (N = 100)
Network

Avg. Degree

Basic-1
Basic-2
Constant-1
Constant-2
Varing-1
Varing-2

5.31
8.29
7.76
5.60
6.95
6.57

0.0003synchronization (∆γs)
distance (∆γd)

synchronization (∆γs)
distance (∆γd)

0.000250.0002∆γ

0.00020.000150.00010.000150.0001

5e-05

5e-05

0

20

40

60

80

0

100

0

20

generation

40

60

80

Avg. Path Length
3.75
3.90
4.19
2.87
3.58
4.11

100

(c) Constant-1.
(∆γs = 0.0001, ∆γd = 0.0002)(d) Constant-2.
(∆γs = 0.0002, ∆γd = 0.0001)

0.00030.0003synchronization (∆γs)
distance (∆γd)

0.00025

Avg. Clustering
Coefficient
0.76
0.79
0.77
0.80
0.79
0.76

generationsynchronization (∆γs)
distance (∆γd)

0.00025

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.0002∆γ

0.00020.25

probability

∆γ

100

TABLE I(b) Basic-2.

0.0003∆γ80(a) Basic-1.

0.00025060

generation

generation

probability

0

A comparison of Varying-1 and Varying-2 reveals that they
have a similar average degree and a similar average clustering
coefficient. However, Varying-1 has a smaller average path
length than Varying-2. Thus, Varying-1 also shows a smallworld topology. To produce small-world topology, the network
should be affected from synchronization at the end of thegenerations.
Figure 5 shows the average degree distributions of six methods for increasing the coupling strengths. From these figures,
we confirm that the graphs do not follow a power-low scaling.
The generated networks have no scale-free characteristics.

0.000150.05

0.000150.00010.0001

5e-05

5e-05

0
-5

0.05

0

5

10

20

15

25

30

35

0
-5

40

0

5

10

20

40

60

80

100

0

(a) Basic-1.
020

generation(e) Varing-1.

40

60

80

25

30

35

40

25

30

35

40

30

35

40

k

(b) Basic-2.

100

generation
0.25

0.25

0.2

0.2

0.15

0.15

(f) Varing-2.

Fig. 4. Six methods of increasing the coupling strength depending on theinfluence of synchronization and distance (horizontal axis: generation, verticalaxis: ∆γ).

probability

0

probability

0

20

15

k

0.1

0.05

0
-5

III. S IMULATION R ESULTS

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0.05

0

5

10

20

15

25

30

35

0
-5

40

0

5

10

k

(c) Constant-1.

(d) Constant-2.

0.25

0.25

0.2

0.15

probability

0.2

0.15

0.1

0.05

0
-5

20

15

k

probability

For the computer simulations, we use 100 van der Pol
oscillators as coupled circuits network and simulate for 20
different networks with frequency errors.
Table I summarizes the average network characteristics(average degree, average clustering coefficient and averagepath length). First, we compare the average degree of Basic-1
and Basic-2. When using a large coupling strength change,
the average degree of Basic-2 is larger the average degree ofBasic-1.
Next, we compare the average clustering coefficient andaverage path length of Constant-1 and Constant-2. Constant-2
has a large average clustering coefficient and a small averagepath length. Obviously, the network structure shows a smallworld topology as the network has a high clustering coefficientand a low path length. From this result, we can see that thesynchronization plays an important role for producing a smallworld network.

0.1

0.1

0.05

0

5

10

20

15

25

k

(e) Varing-1.

30

35

40

0
-5

0

5

10

20

15

25

k

(f) Varing-2.

Fig. 5. Average degree distribution (horizontal axis: degree k, vertical axis:
probability).

Figure 6 shows one example of the generated networks.
Generally, we obtain different network topologies dependingon the method for increasing the coupling strength. However,
it is difficult to pick up the network characteristics. So, we

1
1

1
1

0.9
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0.8
0.8

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yy

yy

use the layout algorithm called Fruchterman-Reingold Algorithm [20]. The idea of a force directed layout algorithm is
to consider a force between any two nodes. In this algorithm,
the nodes are represented by steel rings and the edges are
springs between them. The generated network structures using
Fruchterman-Reingold Algorithm are shown in Fig. 7. There
are several clusters in each network. We can see that the nodes
connect to nodes with similar degree. It is considered that thecongenic selectivity of these networks is high.

0.5
0.4

(c) Constant-1.

(d) Constant-2.

(e) Varing-1.

(f) Varing-2.

0.5

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0.2

0.2

0.1

0.1
0

0

0.1

0.2

0.3

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0.5
xx

0.7

0.6
0.6

0.8
0.8

0.9

11

0

0.1

0.2

(a) Basic-1.

0.3

0.4
0.4

0.5
xx

0.6
0.6

0.7

0.8
0.8

0.9

11

(b) Basic-2.

1
1

1
1

0.9
0.9

0.9

0.8
0.8

0.8

0.7

0.7

0.6

0.6
yy

yy

(b) Basic-2.

0.4

0

0.5
0.4

0.5
0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

0

0.1

0.2

0.3

0.4
0.4

0.5
xx

0.7

0.6
0.6

0.8
0.8

0.9

11

0

0.1

0.2

(c) Constant-1.

0.3

0.4
0.4

0.5
xx

0.6
0.6

0.7

0.8
0.8

0.9

11

(d) Constant-2.

1
1

1
1

0.9
0.9

0.9
0.9

0.8
0.8

0.8
0.8

0.7

0.7

0.6

0.6
yy

yy

(a) Basic-1.

0.5
0.4

0.5

Fig. 7. Produced network topologies using Fruchterman-Reingold algorithm.

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

0

0.1

0.2

0.3

0.4
0.4

0.5
xx

0.7

0.6
0.6

0.8
0.8

0.9

11

0

0.1

0.2

(e) Varing-1.
Fig. 6.

0.3

0.4
0.4

0.5
xx

0.6
0.6

0.7

0.8
0.8

0.9

11

(f) Varing-2.
Produced network topologies.

IV. C ONCLUSIONSIn this study, we have proposed a method of generatingcomplex networks by exploiting synchronization between coupled oscillatory circuits. To each node of a 2D fully connected
network a van der Pol oscillator was assigned. We studied
the topological evolution of the network in dependence on
environmental conditions. These conditions were modeled byconsidering the neighborhood distance of the oscillators and978-1-5386-4881-0/18/$31.00 ©2018 IEEE

some small frequency errors. By carrying out computer simulations, we confirmed that the network structure of Constant-2
showed a small-world topology depending on the two rules.
It was also considered that the congenic selectivity of theseproduced networks was high.
In our future work, we would like to investigate the evolutional processes for creating complex networks using the
proposed circuits system in detail. Furthermore, we would
like to model a mechanism of opinion formation in complex
networks by improving the proposed coupled nonlinear circuits
system.
ACKNOWLEDGMENT
This work was partly supported by JSPS Grant-in-Aid for
Scientific Research 16K06357.

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