Synchronization Phenomena of Coupled Oscillatorswith Node and Edge Weights
in Two-Dimensional Complex Networks
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: {kiichi, uwate, nishio}@ee.tokushima-u.ac.jp

connecting these circuits with resistors, edges are created toform networks.

Abstract—Complex networks have attracted a great deal ofattention, and research is being conducted in various fields.
However, most of these studies have analyzed networks bycoupling strength is kept constant between nodes for allconnections. In this study, we focus on the Euclidean distance ofedges and the degree of nodes to give weight to the network, andwe set the coupling strength. We performed networks analysis interms of synchronization phenomena for complex networks
constructed with van der Pol oscillators by computer simulations.
Keywords; Oscillator; Synchronization; Complex networks

I.

INTRODUCTION

Complex networks have received much attention in various
fields such as sociology, biology and engineering [1]-[3].
Furthermore, in the field of engineering, complex networks
using circuits have been studied, and interesting phenomena
such as synchronization between circuits have been observed. Itwas confirmed that the topological structure of the network
influences the synchronization [4],[5]. However, most of thesestudies have examined synchronization phenomena in networks
where the coupling strength is kept constant for all theconnections [6].

Figure 1. van der Pol oscillator.

In this study, we construct a network model with scale-freeproperty in two-dimensional space using 100 van der Pol
oscillators. We analyze the network from the synchronization
phenomenon between circuits when the network is weighted byconsidering the degree of nodes and the Euclidean distance ofthe edges.

II.

Figure 2. Barabási Albert model (BA model).

The characteristic equation of the nonlinear element andcircuit equation are obtained from the circuit in Fig. 1. By using
the normalization parameter and the variables, the normalized
circuit equation described as follows:

SYSTEM MODEL⎧⎪

Figure 1 shows a van der Pol oscillator. This oscillator is a
simple circuit, consisting of only a capacitor, an inductor and anonlinear element. Figure 2 shows the Barabási Albert model(BA model) used in this study. For this model, five networks
with different topologies are constructed under the sameconditions. We set the average degree of node to be close to 4.0.
The circuit in Fig. 1 is considered as a single node. By

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⎨
⎪⎩

280

1,,1,2,∙ ∙ ∙ ,100 .

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First, Table 1 shows that each network has almost the same
average synchronization rate for each method of determining thecoupling strength. The average synchronization rate is about50% when only nodes and only edges are weighted. On the otherhand, it drops to less than about 30% when both are weighted.
Next, Figures 3(a) and 3(b) show that for the synchronization
rate distribution of BA ○1 among the five networks, thedistribution is biased toward larger and smaller values for the
coupling strength weighted for both node and edge than for the
coupling strength weighted for only nodes and only edges.

represents the adjacency matrix of the network. This isa matrix that indicates whether node n and node k are connectedor not.
1 if node n and node k are connected, and0 if they are not connected. The coupling strength
isdetermined by using the parameter # as follows:
1 &%'$#∙( (.

2Here,is the Euclidean distance of the edge between
node and node , and ( is the degree of each node. In thisstudy, we compare the following three cases:
when the only nodes are weighted (when the only edges are weighted (( (when both node and edge are weighted.

1)
2)
3)

1 in Eq. (2))
1 in Eq. (2))

IV.

In this study, we constructed multiple networks of BA modelusing 100 van der Pol oscillators and investigated thesynchronization phenomenon between circuits. Here, the degreeof nodes and the Euclidean distance of edges were used as thecoupling strength to weight the networks. The results show thatwhen both node and edge are weighted for all networks, thesynchronization rate decreases compared to when only each ofthem is weighted. We also find that when both are weighted, thesynchronization rate distribution is divided into two patterns:
one is close to perfect synchronization and the other is almost
out of synchronization.

The parameter of the van der Pol oscillator is set to0.1.
Moreover, represents the small error of the capacitor, in therange of [0.975:1.025] in increments of 0.0005. The parameter#, which determines the coupling strength, is set so that thecoupling strength takes close to0.5 for each network.
III.

RESULTS

The average synchronization rates for each network (BA○1~BA○5 ) for coupling strength determination method are shownin Table 1. The distribution of synchronization rates for Network
1 is also shown in Fig. 3.
TABLE I.

In the future, we would like to further investigate these twopatterns and create a clustering index to see if we can form
clusters with higher precision.

AVERAGE SYNCHRONIZATION RATE FOR EACH COUPLING
STRENGTH DETERMINATION METHOD.

REFERENCES[1]

coupling
strength

BA○
1

BA○
2

BA○
3

BA○
4

BA○
5avg

1)

49.95

49.82

50.73

53.73

48.61

50.57

2)

47.26

45.87

47.26

50.38

47.80

47.72

3)

26.15

31.31

21.49

30.17

25.96

27.02

[2]

[3]

[4]

[5]

(a) Only nodes and only edges.

(b) Both node and edge.

[6]Figure 3. Synchronization rate distribution by coupling strength
determination method in BA○
1 network.

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CONCLUSION

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