2017 IEEE Asia Pacific Conference on Postgraduate Research in Microelectronics and Electronics (PrimeAsia)

Spread of Chaotic Behavior
in Scale-free and Random Networks
Takahiro Chikazawa, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {chikazawa,uwate,nishio}@ee.tokushima-u.ac.jp

Abstract—In this study, we investigate the spread of chaotic
behavior to complex network. We propose three types random
network models with different network topology. In each networkmodel, the proposed network is composed of coupled chaoticcircuits when one circuit is set to generate chaotic attractor andthe other circuits are set to generate three-periodic attractors. By
using computer simulations, we investigate ratio of propagationand spread of chaotic behavior by changing the initial chaos
position. We compare their differences of simulation results inscale-free and random networks. From the simulation result,in scale-free network, the ratio of propagation is effected byclustering coefﬁcient and path length.

attractor and the other circuits are set to generate threeperiodic attractors. First, in each model, we investigate ratio ofpropagation by changing the initial chaos position according todegree distribution. Next, we investigate ratio of propagationby comparing scale-free and random networks. Furthermore,
we investigate the spread of chaotic behavior in three types
random networks.
II. C IRCUIT MODEL
The chaotic circuit is shown in Fig. 1. This circuit consistsof a negative resistor, two inductors, a capacitor and dualdirectional diodes. This chaotic circuit is called Nishio-Inabacircuit.

I. I NTRODUCTION
In the complex social network, various type of propagation
have attracted a great deal of attention from various ﬁelds.
The pandemic outbreak of viral infection and the trafﬁc jamof the transportation network are mentioned as an example ofpropagation in the real network. It is important to investigate
the chaos propagation and the spread of chaotic behavior undersome difﬁcult situations for the circuits. For example of some
difﬁcult situations, network is brieﬂy given external stimulation and frustration is occurred in the network. Therefore weconsider that it is necessary to investigate that behavior ofunlike the others inﬂuence the whole system. In the biology,
mechanism of the communication in nerve cells and viral
infection have not ﬁgure out yet. Furthermore, it is difﬁcult
to analyze propagation mechanism. So, we can prevent theunknown virus spreading if we comprehend the mechanismof the communication in nerve cells and viral infection. Additionally, it is important to investigate propagation phenomena
observed from coupled chaotic circuits for future engineering
applications. However, there are not many studies of largescale network of continuous-time real physical systems suchas electrical circuits [1]-[3].
In our research group, the chaos propagation is already studied in simple networks such as ring topology [4]. Moreover, weinvestigate the spread of chaotic behavior in scale-free network[5]. In these previous studies, the spread of chaotic behavior
becomes weak when the initial chaos position is set to thehigh degree node.
In this study, we propose random networks using of chaotic
circuits coupled by the resistors as different topology network. In these model, one circuit is set to generate chaotic



Fig. 1. Chaotic circuit.

The circuit equations of this circuit are described as follows:⎧di⎪⎪L1 = v + ri⎪
⎪dt
⎪⎪
⎪⎪
⎨
diL2 = v − vddt⎪
⎪⎪
⎪
⎪
dv⎪
⎪⎪
⎩ C dt = −i1 − i2 ,(1)The characteristic of nonlinear resistance is described asfollows: rd V  V (2)i2 +  − i2 −  .
vd =2 rdrd

978-1-5386-0524-0/17/$31.00 ©2017 IEEE21

2017 IEEE Asia Pacific Conference on Postgraduate Research in Microelectronics and Electronics (PrimeAsia)

By changing the variables and parameters,⎧√
CL1 C⎪⎪
V xn , i2 =V yn , v = V zn⎪ i1 =⎪⎪
L
L
12⎪
⎪
⎪
√
⎨
C
L1 C
L1(3)
α=r, β=, δ = rd,
⎪LLL
⎪
122⎪
⎪
⎪
⎪
⎪
⎪
⎩ γ = 1 L1 , t = L1 C2 τ,R C
In the proposed system, each circuit is connected to only adjacent circuits by the resistors.The normalized circuit equationsof the system are given as follows:

number of possible links between those neighbors. Third, pathlength(L) shows the shortest path in the network between twonodes. These are given as follows;

⎧dxi
⎪
⎪
= αxi + zi⎪
⎪dτ⎪
⎪⎪
⎪
⎨ dyi = zi − f (yi )dτ⎪dzi⎪⎪= −xi − βyi −γ(zi − zj )
⎪⎪dτ
⎪
⎪
j∈Sn⎪
⎩
(i, j = 1, 2, · · ·, N ).

Figures 3 and 4 show the proposed different type networkmodels. Model-A is already proposed in previous study [5].
The characteristic of Model-A is the scale-free network topology. The characteristic of Model-B is the random network
topology. In each model, each chaotic circuit is coupled by
one resistor R. We use 25 coupled chaotic circuits and 34
resistors in each network model.
Figure 5 shows degree distribution of each network. In thisgraph, the vertical axis denotes the number of nodes and thehorizontal axis denotes the value of degree. Moreover, the feature quantities of the proposed each network are summarizedin Table I.

NC=N −1(4)  
yi + 1  − yi − 1  .
δδ() ( ) (7)zxN(5)In this chaotic circuit, we deﬁne αc to generate the chaotic
attractor (see Fig. 2(a)) and αp is deﬁned to generate thethree-periodic attractors (see Fig. 2(b)). For the computer
simulations, we calculate Eq. (4) using the fourth-order RungeKutta method with the step size h = 0.01. In this study, weset the parameters of the system as αc = 0.460, αp = 0.412,
β = 3.0 and δ = 470.0.

z(6)2L=l(m, n).
N (N − 1) m=1 n=m+1

In Eq. (4), N is the number of coupled chaotic circuits andγ is the coupling strength. f (yi ) is described as follows:
1f (yi ) =
2

N

1
2En
1.
Cn =
N n=1
N n=1 kn (kn − 1)

















Fig. 3. Model-A (Scale-free network). Fig. 4. Model-B (Random network).

x Fig. 2. Attractors of chaotic circuit.

III. C OMPARE DIFFERENT TYPE NETWORK MODEL
A. Proposed system model
Topological structures in complex networks of N nodes andE edges can be evaluated by the typical three types structuralmetrics (degree, clustering coefﬁcient and path length). First,degree (k) is the number of edges which is connected on anode. Second, clustering coefﬁcient (C) shows the numberof actual links between neighbors of a node divided by theFig. 5. Degree distribution of each network..

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2017 IEEE Asia Pacific Conference on Postgraduate Research in Microelectronics and Electronics (PrimeAsia)

TABLE IFEATURE QUANTITIES OF PROPOSED EACH NETWORK .

FeatureThe number of nodes
The number of edges
Avg. degree
Avg. path length
Avg. clustering coefﬁcient

Model-A
2534
2.720
3.503
0.112

Model-B
2534
2.720
3.110
0.092

B. Ratio of propagation
We investigate the ratio of propagation when we change thevalue of degree in the initial chaos position according to degree
distribution. Moreover, in scale-free and random networks, wecompare their differences of simulation results. For example,in model-A, when the value of degree k in the initial chaosposition is 4, we set the chaotic attractor in 9th or 13th node.
Also, in model-B, when the value of degree k in the initialchaos position is 4, we set the chaotic attractor in 9th or 17thnode.

Fig. 7. Ratio of propagation according to degree distribution in Model-

B (Random network).

IV. S PREAD OF CHAOTIC BEHAVIOR IN RANDOM NETWORK
A. Proposed system model
In this research, we propose two types random networkmodels different from Model-B (see Figs. 8 and 9). Model-B,
C and D is used same coupled chaotic circuits. However, wechange the number of edge compare to Model-B. Therefore,
average path length become shorter and average clustering
coefﬁcient become larger than model-B. The feature quantitiesof the proposed each proposed network are summarized inTable II.

The simulation results of ratio of propagation according todegree distribution are shown in Figs. 6 and 7. In addition,
we investigate the ratio of propagation in the static state when
we ﬁx coupling strength as γ = 0.001, 0.005. Here, we deﬁne
the ratio of propagation as number of chaotic circuits of wholenetwork at steady state. Further, the simulation result of Fig. 6
has been reported [5].



Fig. 8. Model-C (Random network). Fig. 9. Model-D (Random network).

TABLE IIFEATURE QUANTITIES OF PROPOSED EACH NETWORK .

Fig. 6. Ratio of propagation according to degree distribution in Model-Feature
The number of nodes
The number of edges
Avg. degree
Avg. path length
Avg. clustering coefﬁcientA (scale-free network).

From the result, in scale-free network, chaos propagation
become to more difﬁcult, when we increase the value ofdegree in initial chaos position. On the other hand, in random
network, even though we increase the value of degree in initialchaos position, the ratio of propagation changes only a little.

Model-C
2550
4.000
2.617
0,234

Model-D
2575
6.000
2,173
0.509

B. Simulation result
We investigate the spread of chaotic behavior in each random network by increasing the coupling strength. Moreover,23

2017 IEEE Asia Pacific Conference on Postgraduate Research in Microelectronics and Electronics (PrimeAsia)

when we set the initial chaos position in same condition,
we compare each random network for spreading of chaotic
behavior. We summarize the each simulation result in Figs. 10and 11.
Figure 10 shows the spread of chaotic behavior when we weset the initial chaos position in minimum degree. In ModelB, the minimum degree is 1; hence, we set the initial chaos
position in 5th, 13th, 18th, 24th or 25th node. In Model-C, theminimum degree is 2; hence, we set the initial chaos positionin 16th or 23th node. In Model-D, the minimum degree is 4;
hence, we set the initial chaos position in 11th, 20th or 22thnode. Moreover, we increase the coupling strength when we
set the step size to γ = 0.0001.
From the result, when we increasing the coupling strength,each three periodic attractor are affected from the chaotic
attractors. Furthermore, the ratio of propagation in Model-D
reach 90% earlier than other model.

Fig. 11. Spread of chaotic behavior in random network when we set

the initial chaos position in maximum degree.

V. C ONCLUSIONSIn this study, we have investigated the spread of chaotic
behavior in coupled chaotic circuits of our proposed models.
By the computer simulations, we conﬁrmed that the threeperiodic attractors are affected from the chaotic attractors when
the coupling strength increases.
First, we compare scale-free network and random networks.
In scale-free network, chaos propagation is more difﬁcult when
we increasing the value of degree in the initial chaos position.
By contrast, in random network, even though we change theinitial chaos position, the ratio of propagation changes only alittle.
Next, we investigate the spread of chaotic behavior in eachrandom network. From the above results, we conﬁrm thelargest clustering coefﬁcient network is easily affected fromthe chaotic behavior than other random network. We consider
that the spread of chaotic behavior depends on average clustering coefﬁcient in random network.
For the future works, we will focus on clustering coefﬁcientand path length in each node. Furthermore, we compare scalefree and random networks in large scale complex networks.

Fig. 10. Spread of chaotic behavior in random network when we set

the initial chaos position in minimum degree.

Figure 11 shows the spread of chaotic behavior when we weset the initial chaos position in maximum degree. In ModelB, the maximum degree is 6; hence, we set the initial chaos
position in 1st node. In Model-C, the maximum degree is7; hence, we set the initial chaos position in 1st node. InModel-D, the maximum degree is 9; hence, we set the initialchaos position in 1st node. Moreover, we increase the couplingstrength when we set the step size to γ = 0.0001.
From the result, when we increasing the coupling strength,each three periodic attractor are affected from the chaotic
attractors. Maximum node as well as minimum node, the ratioof propagation in Model-D reacesh 90% earlier than othermodel.
From each result, as clustering coefﬁcient become larger, thethree-periodic attractors are easily affected from the chaotic
attractor in random network. Additionally, there is no changein the ratio of propagation when the coupling strength exceed
the ﬁxed threshold.

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[2] L. Jinhu, Y. Xinghuo and C. Guanrong, ”Chaos synchronization ofgeneral complex dynamical networks” Physica A: Statistical Mechanics
and its Applications, vol. 334, pp.281-302, 2004
[3] A.L. Barabasi and Z. N. oltvai, ”Network biology: understanding thecell’s functional organization”Nature Reviews Genetics 5, pp.101-113,
2004[4] Y. Uwate and Y. Nishio, “Chaos Propagation in a Ring of Coupled Circuits Generating Chaotic and Three-Periodic Attractors” Proceedings ofIEEE Asia Paciﬁc Conference on Circuits and Systems (APCCAS’12),
pp. 643-646, Dec. 2012.
[5] T. Chikazawa, Y. Uwate and Y. Nishio,“Spread of Chaotic Behavior inComplex Network with Coupled Chaotic Circuits” IEICE Proceeding
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