Synchoronization Penomena of Coupled Oscillators
in Weighted Three-Dimensional Complex Networks
Daiki Akai, Kiichi Yamashita, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering
Tokushima University
2-1 Minami-Josanjima, Tokushima 7708506, Japan
Email:{akai, kiichi, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract— Complex networks have received a great deal ofattention and have been studied in a variety of fields. However,most of these studies consider two-dimensional networks andanalyze them by weighting edges. In this study, we extend thenetwork to three dimensions and analyze the network from theviewpoint of synchronization phenomena for complex networks
composed of van der Pol oscillators. Then, the coupling strength is
set using the Euclidean distance in three-dimensional space and wecompare synchronization rate with the two-dimensional network.
Keywords; Oscillator; Synchronization; Conplex networks

Figure 1. van der Pol oscillator.

I.
INTRODUCTION
Complex networks have been a focal point in various
disciplines, including sociology, biology, and engineering [1][3]. Within engineering, there's an evolving study of thesenetworks using circuit models, and the phenomenon ofsynchronization among these circuits has become a captivating
subject [4],[5]. While past research has demonstrated that a
network's structure affects its synchronization, much of thiswork has been centered on two-dimensional networks [6].
In this study, we built a three-dimensional scale-free networkmodel consisting of 100 van der Pol oscillators to delve into
synchronization phenomena. In this configuration, we
modulated the coupling strength of the network based on thenodes' degree and the edges' Euclidean distance. We anticipate
this will enhance our comprehension of synchronization
behaviors in a three-dimensional context.
II.

SYSTEM MODEL

Figure 1 shows the van der Pol oscillator used in this study.
This oscillator is a simple circuit consisting of only a capacitor,inductor, and nonlinear elements. Figure 2 shows the model ofthe three-dimensional scale-free network used Barabasi Albert
model (BA model) in this study. The circuit in Fig. 1
corresponds to the nodes in Fig. 2. When these van der Pol
oscillators are connected with resistors, a complex network isconstructed as shown in Fig. 2.

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Figure 2. Barabasi Albert model (BA model) in threedimensional space.
From the circuit depicted in Fig. 1, we derive thecharacteristic equation for the nonlinear element as well as thecircuit equation. Utilizing the normalization parameter andvariables, we can represent the circuit equation in its
normalized form as follows:

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Table 1. Average coupling strength and synchronization
rate in two-dimensional and three-dimensional networks.

The adjacency matrix of the network is denoted by 𝐸!" . It'sa matrix that confirms whether there's a connection between
node n and node k. If nodes n and k are linked, 𝐸!" is set to 1;otherwise, it's set to 0. The coupling intensity, 𝛾!" , isascertained by applying the parameter q in the following
manner:average coupling strength

average synchronization rate

2D

0.510

24.83 [%]

3D

0.507

28.61 [%]

Table 1 shows the average synchronization rates in twodimensional and three-dimensional. We confirm that the averagesynchronization rate increased by about 4% by changing thenetwork from two-dimensional to three-dimensional.

Here, 𝑑!" is the Euclidean distance of the edge between
node 𝑛 and node 𝑘, and 𝑤 is the degree of each node.
III. RESULTIn this study, the parameter of the van der Pol oscillator is set
to 𝜀 = 0.1 . Moreover, 𝛼 represents the small error of thecapacitor, in the range of [0.975:1.025] in increments of 0.0005.
The parameter 𝑞, which determines the coupling strength, is set
so that the coupling strength takes close to 𝛾 = 0.5 for eachnetwork.
Figure 3 shows the synchronization rates in two and threedimensions. The blue box shows the synchronization rate in twodimensions, and the green box shows the synchronization rate inthree-dimensions. In two-dimensional networks, synchronization
rates tend to be biased toward 0-10% and 90-100%. However,by changing from two dimensions to three dimensions, thefrequency of nodes with 0-10% synchronization rate decreased
and the frequency of nodes with 10% or more synchronization
rate increased.

IV. CONCLUSION
In this study, we constructed two-dimensional and threedimensional networks of the BA model with 100 van der Pol
oscillators to investigate the synchronization phenomena
between circuits. For the weighting of the network, we focusedon the Euclidean distance of the edges and the degree of the node.
As a result, there was a large tendency for synchronization rates
to be biased toward 0-10% and 90-100% in the two-dimensional
network, but this tendency was reduced when the network wasextended to three dimensions.
In the future, we would like to approach the threedimensional network closer to a brain model. Changing thenetwork over time, the coupling strength of the edges changes
with time. We can expect it to approach the synapses in the brain.
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Figure 3. Synchronization rate distribution in twodimensional and three-dimensional networks.

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