Information Transmission Focusing onComplex Networks Consisting of Oscillators
Tsuyoshi Isozaki, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering
Tokushima University
2-1 Minami-Josanjima, Tokushima 7708506, Japan
E-mail: {isozaki, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract— Complex networks have been the subject of great
attention. In this study, we investigate synchronization phenomenain the complex networks using van der Pol oscillators and theperformanceofinformationtransmission
when thesynchronization rate was taken into account in the number oftransmitting packets.
Keywords; Oscillator; Synchronization; Complex networks

I.

(a) ER random.

INTRODUCTIO

Figure 2: Network models.

Complex networks have attracted a lot of attention fromvarious fields such as sociology, biology and physics. In thefield of engineering, the performance of communication
networks has been studied with a focus on complex networks[1],[2],[3]. Moreover the studies of synchronization between
nodes in complex networks have identified some interesting
phenomena [4],[5].

Table 1: Node number 𝑁, Average node degree 𝑑𝑎𝑣𝑔 andMaximum node degree 𝑑𝑚𝑎𝑥 of two network models.
network𝑁𝑑𝑎𝑣𝑔
𝑑𝑚𝑎𝑥

In this study, we first investigate the synchronization between
nodes in complex networks, and then investigate the network
performance by relating the synchronization to the efficiency ofinformation transmission.

II.

(b) BA scale-free.

III.

ER random
100
4.08
9.0

BA scale-free
100
3.92
32.0

INFORMATION TRANSMISSION ALGORITHM

In this study, information is represented as packets. Packets
are generated by the host nodes and transmit through the links
one hop at a step until they reach the destinations. Also, eachnode in the network has a buffer 𝐵 . Then, the packets
transmission algorithm operates as follows:

NETWORK MODEL

Figure 1 shows van der Pol oscillator. We use two networkmodels in this study. Figure 2(a) shows Erdos-Renyi model (ERrandom) and Fig. 2(b) shows Barabasi-Albert model (BA scalefree). Table 1 shows each network properties. In this study, we
set the average node degree to be close to 4.0 each networkmodel. Further the network models composed of oscillators areused to calculate the synchronization rate between each node.

1)

Packets GenerationAt each time step, hosts generate 𝑅 packets.
2)

Packets Transmission

The number of transmitting packets for node per step is 𝑆. At
each time step, packets of each node are forwarded to theirdestinations by one step according to the shortest path routing
algorithm.
3)

Packets Dropped

If the total number of packets reaching one node is larger thanbuffer size 𝐵, transmitted packets are dropped.
4)

Figure 1: Oscillator.

Packets Released

Packets already arrived at their destinations are released fromthe buffer.

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on the synchronization rate and 𝜎̃ is small when 𝑆 is set torandom in the both network models.

Further we consider two performance parameters to compare
the packets transmission performance. The first, packet droprate 𝜌 , the second is arrival rate 𝜎 , and it is defined theseequations for these parameters are as follows:𝜌=

𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑟𝑜𝑝𝑝𝑒𝑑 𝑝𝑎𝑐𝑘𝑒𝑡𝑠 𝑖𝑛 𝑡𝑖𝑚𝑒 𝑠𝑡𝑒𝑝
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒𝑑 𝑝𝑎𝑐𝑘𝑒𝑡𝑠 𝑖𝑛 𝑡𝑖𝑚𝑒 𝑠𝑡𝑒𝑝𝜎=𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑟𝑟𝑖𝑣𝑎𝑙 𝑝𝑎𝑐𝑘𝑒𝑡𝑠
𝑡𝑖𝑚𝑒 𝑠𝑡𝑒𝑝

IV.

,.

V.

(1)

In this study, we calculated the synchronization rates between
each node using network model composed of oscillators andinvestigated the performance of information transmission when
the synchronization rate was taken into account in the numberof transmitting packets 𝑆. As a result, it was confirmed that theaverage drop rate 𝜌̃ is the small when the number of generating
packets 𝑅 is small (the range of 𝑅 from 2 to 6) in the both
network models. Further it was confirmed that the averagearrival rate 𝜎̃ is large.

(2)

RESULTSIn this study, the simulation parameters the buffer size for eachnode 𝐵 = 100 and the number of transmitting packets 𝑆 is set
in three patterns. The first sets the 𝑆 of each node randomly.
The second sets the 𝑆 of all nodes constant. The third is to set 𝑆
based on the synchronization rate. Here, based on thesynchronization rate means to increase the value of 𝑆 when thesynchronization rate between nodes is high, and to decrease thevalue of 𝑆 when the synchronization rate is low. Further thetotal number of time steps 𝑇 = 2000 and the relationshipbetween the average drop rate 𝜌̃ and the number of generating
packets 𝑅 is shown in Figure 3 and the relationship between theaverage arrival rate 𝜎̃ and the number of generating packets 𝑅is shown in Figure 4.

REFERENCES[1]

Jiajing Wu, Chi K. Tse, Francis C. M. Lau and Ivan W. H. Ho, “Complex
Network Approach to Communication Network Performance Analysis”,
2012 IEEE International Symposium on Circuits and Systems (ISCAS)
May 2012.

[2]

Jiajing Wu, Chi K. Tse, Francis C. M. Lau and Ivan W. H. Ho “Analysisof Communication Network Performance From a Complex NetworkPerspective”, IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSⅠ: REGULAR PAPERS, vol. 60, NO. 12, DECEMBER 2013.

[3]

Yuuki Morita and Takayuki Kimura, “An improved routing algorithmusing chaotic neurodynamics for packet routing problems”, Nonlinear
Theory and Its Applications, IEICE, vol. 9, no. 1, pp. 95-106, 2018.
Kenta Ago, Yoko Uwate and Yoshifumi Nishio “Synchronization ofCoupled Circuits with Parameter Dispersion in Small-World Network”,
2015 International Symposium on Nonlinear Theory and Its Applications
NOLTA2015, Kowloon, Hong Kong, China, December 1-4, 2015
Shuhei Hashimoto, Takahiro Chikazawa, Yoko Uwate, and Yoshifumi
Nishio “Synchronization Phenomena in Complex Networks of Coupled
Chaotic Circuits with Different Degree Distribution”, 2017 RISPInternational Workshop on Nonlinear Circuits, Communications andSignal Processing (NCSP'17) Guam, USA, February 28th to March 3rd,2017

𝜌̃𝜌̃

[4][5]
𝑅𝑅(a) ER random.

(b)BA scale-free.

CONCLUSION𝜎̃𝜎̃

Figure 3: The relationship between the average drop rate𝜌̃ and the number of generating packets 𝑅.

𝑅(a) ER random.

𝑅(b)BA scale-free.

Figure 4: The relationship between the average arrival rate𝜎̃ and the number of generating packets 𝑅.
From Fig. 3, we see that 𝜌̃ is small when 𝑆 is set based on thesynchronization rate in the range of 𝑅 from 2 to 6 in the both
network models. Further 𝜌̃ is small when 𝑆 is set to constant in
the range of 𝑅 from 8 to 20 in the ER random while 𝜌̃ is almost
the same in the range of 𝑅 from 8 to 20 in the BA scale-free.
From Fig. 4, it is confirmed that 𝜎̃ is large when 𝑆 is set based

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