Rewiring Effect of High Synchronization Edges
in Complex Oscillator Networks
Haruka Sakohira, Kiichi Yamashita, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering
Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
E-mail: {sakohira, kiichi, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract— It is a random network that is constructed with 100 vander Pol oscillators. The synchronization phenomena between thecircuits of the networks are analyzed. Then, focusing on thesynchronization rate and changing the topology accordingly, thesimulation is performed. It was confirmed that thesynchronization rate changes when the topology is changed.

are small and the difference that the number of edges gatheredat each node is small.

Keywords; Switching; Complex networks; Synchronization;
oscillator

I.

INTRODUCTION

Complex networks are complex and huge networks that exist
in reality. It has three properties that are scale-free, small-world,and clustered, and it is able to be reproduced using nodes andedges. This has attracted attention from various fields, including
communications, engineering, economics, and others [1]-[3].
Especially in the engineering field, there have been interesting
research such as synchronization phenomena between circuits[4]. This synchronization phenomena are a phenomenon inwhich the circuits become aligned with each other under theinfluence of surrounding objects [5].
In the previous study, it was investigated the synchronization
phenomena which are caused by differences in the structure ofthe network. In this study, however, we focus on thesynchronization rate and investigate changes in synchronization
phenomena caused by differences in the synchronization rate. Arandom network is constructed using 100 van der Pol oscillators.
By using arbitrary synchronization rates as thresholds, weinvestigate how the synchronization phenomena change when
the network topology is changed by cutting edges. The network
topology is changed by setting the threshold value to 70~100%and cutting off edges with high synchronization rates above thethreshold value. In addition, the coupling strength is changed
from 𝛾 = 0.2 to 𝛾 = 0.5 for each threshold value, and how
the synchronization phenomena change is investigated.
II.

Figure 1. ER modelIII.

The van der Pol oscillator is used in this study. It is shown inFig.2. The circuit is a simple circuit that consists only of acapacitor, an inductor, and a nonlinear element. These van der
Pol oscillators are connected using resistors to reproduce thenetwork.

Figure 2. van der Pol oscillator

The normalized circuit equation is shown in Eq. (1).
100

SYSTEM MODEL

In this study, we use the Erdös Rényi model (ER model),which is one of the random networks. This network has 100
nodes, and the average degree is set to near 5.3. Also, the modelis shown in Fig.1. It has the property that the cluster coefficients

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CIRCUIT MODEL𝑑𝑥𝑛= 𝛼 {𝜀𝑥𝑛 (1 − 𝑥𝑛 2 ) − 𝑦𝑛 − ∑ 𝐸𝑛𝑘 𝛾(𝑥𝑛 − 𝑥𝑘 )}𝑑𝜏𝑛,𝑘=1(1)𝑑𝑦𝑛= 𝑥𝑛𝑑𝜏
{233(𝑛, 𝑘 = 1,2,∙ ∙ ∙ ,100).

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In this equation, 𝛼 is the small error of each capacitor and 𝐸
is the adjacency matrix of the network.
The network topology is changed by cutting edges and beinvestigated how the synchronization phenomena change.
IV.

Furthermore, the relationship between the averagesynchronization rate and total number of cutting edges is shownin Fig.5. It shows that the number of cutting edges increases
when the average synchronization rate is higher.

RESULTSFirst, we define synchronization in this study. We calculate
the synchronization rate when 𝜏 = 10000~20000, which isthe steady state of the voltage difference. We set the thresholdvalue as shown in Eq. (2) and judge that the nodes aresynchronized when they are within the range. In this equation, 𝑛and 𝑘 are defined as the number of circuits.
|𝑥 𝑛 − 𝑥 𝑘 | < 0.01.

(2)

Figure 5. The relationship between the average synchronization rate andtotal number of cutting edges

V.

CONCLUSION

In this study, we conducted research restructuring thenetwork through cutting edges with high synchronization rates.
Consequently, the average synchronization rate tends to behigher. It was also found an average synchronization rate
became higher when the number of cutting edges was larger.
Figure 3. Voltage difference waveform

The changes of the average synchronization rates when thenetwork topology is changed are shown in Fig.4. The simulation
was conducted with the threshold set to 100% and the couplingstrength changed from 𝛾 = 0.2 to 𝛾 = 0.5 in steps of 0.1. The
results show that the synchronization rate is higher when thecoupling strength is stronger. In addition, the synchronization
rates tend to increase when the topology is changed.

In the future, we would like to investigate the trends byconducting the simulation several times instead of just once. Inaddition, we would like to investigate dynamic simulations bymoving the simulation time and change the conditions ofconnecting the edges.
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Figure 4. Changes of the average synchronization rate

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