IEEE Workshop on Nonlinear Circuit Networks
December 3-4, 2021

Performance of Communication Network
Focusing on Complex Networks
Consisting of van der Pol Oscillators
Tsuyoshi Isozaki, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {isozaki, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—Complex networks have been the subject of great
attention because some characteristics of complex networks
are closely related to the real-world network. In this study,
we study the performance of communication networks fromperspective of complex networks. We first investigate the synchronization between nodes in complex networks, and then relate
the synchronization to communication network performance.
Synchronization in communication networks is very important
for the transmitting and receiving of information. If the network
is congested, it may take a long time to send information, or
the information you want to send may be lost in the process.
Therefore we aim to improve the efficiency of informationtransmission.

follows the binomial distribution and short distance between
nodes. Here, following the binomial distribution means thatthe number of nodes with degrees close to the average degree
is larger.

I. INTRODUCTION
Complex networks are modeled with nodes and edges basedon several properties of real-world networks. As example,
there are properties of small world and scale free. small world
means that the distance between nodes is short compared to thetotal number of nodes. The second is the property of scale free.
Scale free means that a small number of nodes, called hubs,
have large degree, while almost all nodes have relatively smalldegree. Further in our daily lives, transportation networks andthe Internet are examples of complex networks. Thus Complex
networks have attracted a lot of attention from various fields
such as sociology, biology and physics. In the field of engineering, the performance of communication networks has beenstudied with a focus on complex networks [1], [2]. Moreoverthe studies of synchronization between nodes in complex
networks have identified some interesting phenomena [3].
In this study, first, we build two network models using
van der Pol oscillators and calculate the synchronization
rate between nodes. Next, we set parameters related to thecommunication network based on the synchronization rate andevaluate the performance.

Fig. 1. ER random network.

Second, Fig. 2 shows the BA scale free network with
100 nodes. This network has properties such as the degree
distribution follows the power law, short distance between
nodes and growth and priority selection. Here, following thepower law distribution means that a few nodes are connected
to many other nodes and have a large degree. On the otherhand, most other nodes are connected to only a few nodes
and have a small degree. In addition, priority selection meansthat newly added nodes are more likely to be connected withnodes of higher degree during formation of the network.

II. NETWORK MODELSIn this study, we use two network models, Erdős Rényimodel (ER random network) [4] and Barabási Albert model(BA scale free network) [5]. Each of these network models
has some properties.
First, Fig. 1 shows the ER random network with 100 nodes.
This network has properties such as the degree distribution- 84 -Fig. 2. BA scale free network.

The conditions of the networks created in this case areshown in Table I. The number of nodes N , average nodedegree davg , maximum node degree dmax , minimum nodedegree dmin and average distance Davg of two networkmodels.

By using the parameters and the variables:√√√
g1
g1 Cx, i =y, t = LCτ
v=
g3
g3 L
√
√
L1 L
ε = g1, γ=.
CR C

TABLE IT HE CONDITIONS OF THE NETWORKS .
networkNdavg
dmaxdminDavg

ER random
100
4.08
91
3.38

The normalized circuit equations are described as follows:100∑ dxn2Enk γnk (xn − xk ) dτ = α εxn (1 − xn ) − yn −BA scale free100
3.92
32
22.98

n,k=1dyn= xn dτIn this study, we set the average node degree to be close to4.0 each network model. Further as properties of the network,
the maximum node degree of the BA scale free network islarger than that of the ER random network and the average
distance of the BA scale free network is shorter than that of
the ER random network.

(n, k = 1, 2, · · ·, 100),

where Enk represents the connected state of node n and nodek. If node n and node k are connected, Enk = 1 and if they
are not connected, Enk = 0. Further we set the parametersof van der Pol oscillator as ε = 0.1 and α = 1.0. Here, α
represents the small error on the capacitor, given randomly in
the [0.95:1.05] range for each circuit.

III. SYSTEM MODEL
A. Synchronization
Figure 3 shows van der Pol oscillator. This oscillator
consists of a capacitor, an inductor and a nonlinear element.
Further one circuit is considered to be a single node anda network is formed by connecting circuits to each otherusing resistors. In this study, 100 van der Pol oscillators areconnected to form each network.

In this study, we investigate the synchronization between
nodes in the network and calculate it as a percentage in thetwo network models.
In order to analyze synchronization, we define the synchronization as following equation.
|xn − xk | < 0.01,(4)where n and k are number of circuits. Figure 4 shows thedifferential voltage waveform observed in this study. The twolines in Fig. 4 represent the thresholds shown in Eq. (4). If thevoltage difference falls within these thresholds, it is consideredsynchronous. When calculating the synchronization rates, weuse the voltage differences in the range of 10000 to 20000 inτ to account for the time it takes to settle to a steady state.

nFig. 3. van der Pol oscillator.

When the circuits are connected by using resistors, thecircuit equations are given as follows:n,k=1din

= vnL
dt


xn-xk


100∑
1dvn


= −iL(n) − ig(n) −(vn − vk )C
dtR0.01-0.01(1)(n, k = 1, 2, · · ·, 100),

where the parameter ig is the equation for the nonlinear
element and described as follows:τFig. 4. The differential voltage waveform.

ig = −g1 v + g3 v 3 .

(2)- 85 -(3)IV. COMMUNICATION NETWORK OPERATIONIn this study, information is represented as packets. Moreover we relate oscillator synchronization to communication
network performance.

results of three different patterns of the number of transmitting
packets S in the BA scale free network. Further Fig. 6(a)shows the results under the SP routing and Fig. 6(b) shows
the results under the HS routing.

The number of generating packets R(a) SP routing
The average packet drop rate ρ̃Packets are generated by the host nodes and transmit
through the links one hop at a step until they reach thedestinations. Moreover each node in the network has a buffer,
the buffer size for node n being B(n). Then, the packets
transmission algorithm operates as follows:
1) Packets Generation
At each time step, hosts generate R packets.
2) Packets Transmission
The number of transmitting packets for node per step isS. At each time step, packets of each node are forwarded
to their destinations by one step according to the routing
algorithm.
3) Packets Dropped
If the total number of packets reaching one node is larger
than buffer size B(n), transmitted packets are dropped.
4) Packets Released
Packets already arrived at their destinations are released
from the buffer.
Further B(n) = µk(n), where k(n) is degree of the nthnode and µ is control parameter. The number of transmitting
packets S is set three different patterns. The first is set
randomly in the range of 1 to 10. The second is set to constant(here S = 5). The third is set based on the synchronization rate.

The average packet drop rate ρ̃

A. Packets transmission algorithm

The number of generating packets R(b) HS routing

Fig. 5. Three different patterns of the number of transmitting packets

S (ER random network).
The average packet drop rate ρ̃

B. Routing algorithms
In this study, we study the performance of the network under
two different algorithms, shortest path (SP) routing and highsynchronization (HS) routing. A shortest path refers to thepath with minimum hops from the source to the destination.
HS routing selects the path with the highest synchronization
rate between nodes.
C. Performance parameterWe define a performance parameter ρ to compare thenetwork performance. ρ is packet drop rate and it is describedthe equation as follows:(a) SP routing
The average packet drop rate ρ̃ρ=The number of generating packets R

T he number of dropped packets in time step. (5)T he number of generated packets in time step
V. RESULTS

We show the results of the relation between average packetdrop rates ρ̃ and the number of generating packets R.
First, we compare S under two routing algorithms in theER random network and the BA scale free network. Figure
5 shows the results of three different patterns of the numberof transmitting packets S in the ER random network. FurtherFig. 5(a) shows the results under the SP routing and Fig. 5(b)shows the results under the HS routing. Figure 6 shows theThe number of generating packets R(b) HS routing

Fig. 6. Three different patterns of the number of transmitting packets

S (BA scale free network).

- 86 -The average packet drop rate ρ̃As a result, we can see from Fig. 5(a) and Fig. 6(a) thatin the value of R is small such as 1 or 2, when S is set toconstant, ρ̃ is lower than the others S. However for valuesof R other than 1 or 2, ρ̃ is almost the same and has little
to do with the difference in the S settings. Further it canbeen seen from Fig. 5(b) and Fig. 6(b) that when S is set
based on the synchronization rate, ρ̃ is lower than the others
S. In this simulation, the average S of the three patterns isset to be almost the same. However, when S is set based on
the synchronization rate under the HS routing, the average
S becomes larger than the others. It is considered that thishas something to do with it. Therefore it can been seen thatwhen the average S is set to be the same, there is not muchdifference depending on how the S is set.

The number of generating packets R

The average packet drop rate ρ̃(a) SP routing

Next, we compare SP routing with HS routing for eachnetwork. In this case, S is set to constant. Figure 7 shows theresults for two different routing algorithms, and Fig. 7(a) forthe ER random network and Fig. 7(b) for the BA scale freenetwork.
The average packet drop rate ρ̃The number of generating packets R(b) HS routing

Fig. 8. Comparing the networks under the SP routing and the HSrouting.

From Fig. 8(a), we can see that ρ̃ is lower for the ER random
under the SP routing and from Fig. 8(b), it can be seen that
there is almost no difference under the HS routing. Therefore
it can been seen that the network structure affects the network
performance under certain routing algorithms.

The number of generating packets R(a) ER random network
The average packet drop rate ρ̃VI. CONCLUSION
In this study, we first investigated the synchronization
between nodes in complex networks, and then investigated
the network performance by relating the synchronization to
the efficiency of information transmission. It was confirmedthat when the average S is set to be the same, there is notmuch difference depending on how the S is set. Further it isconfirmed that the HS routing is inferior to the SP routing. Itis considered that this is due to the fact that the same node isselected multiple times as a transit point under the HS routing.
We would like to study on improving the HS routing.

The number of generating packets R(b) BA scale free network

Fig. 7. Comparing the routing algorithms in the ER random networkand the BA scale free network.

From Fig. 7, we can see that ρ̃ is lower under the SP routing
in both networks. Therefore it can been seen that the routing
algorithm affects the network performance.
Finally, we compare the ER random network with the BA
scale free network under two routing algorithms. In this case,
S is set to constant. Figure 8 shows the results of comparing
the two networks under the same routing algorithm, andFig. 8(a) shows the result under the SP routing and Fig. 8(b)shows the result under the HS routing.

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