Diffusion Analysis of Direction-Preserving Small-World CNN
Kazuya Tsuruta, Zonghuang Yang, Yoshifumi Nishio∗ , Akio Ushida∗∗∗∗∗Tokushima University, JapanTokushima Bunri University, Japan

ABSTRACT: Recently, the authors have proposed Small-World Cellular Neural Network (SWCNN), which is constructed by introducing somerandom couplings between cells of the original Chua-Yang CNN. In thisarticle, the Direction-Preserving SWCNN is proposed and the features areinvestigated.

1. Introduction
Studies of network map are very important, because they help us to understand the basic featuresand requirements of various systems. So far many connection topologies of network assumed
to be either completely regular or completely random have been studied in the past. Cellular
Neural Network (CNN) model invented by Chua and Yang in 1988 [1] is a typical of thosecompletely local connectivities, which is presented as a preferred implementation of locallyand regularly coupled neural networks. The CNN has been successfully used for various highspeed parallel signals processing applications such as image processing, pattern recognition as
well as modeling of various phenomena in nonlinear systems [1]-[4]. However, in many casesin real life, many network topologies such as biological, technological and social networks are
known to be not completely random nor completely local but somewhere in between. This wasmodeled in an interesting work by Watts and Strogatz in 1998 [5] as the small-world model. The
model is a network consisting of many local links and fewer long range ‘short cuts’. Therefore,
it has a high clustering coefficient like regular lattices and a short characteristic path lengthof typical random networks. Interesting examples are shown by collaboration of movie stars,
connectivity of internet web pages or neural nets, etc.
Recently, the authors have proposed Small-World Cellular Neural Network (SWCNN) [6],which is constructed by introducing some random couplings between cells of the original ChuaYang CNN. In [6] we have reported some basic results using the concept of SWCNN. In thisarticle, we propose a Direction-Preserving SWCNN and investigate the features.

2. Network Topologies of the Direction-Preserving SWCNN
In this section, we describe the connection topology of the Direction-Preserving SWCNN composed of a two-dimensional M by N array structure.
The Direction-Preserving SWCNN is obtained by rewiring some couplings with direction
keeped between cells in the original Chua-Yang CNN. Namely, we choose a cell and a coupling that connects it to its nearest neighbour. We reconnect this coupling to the another same
direction cell chosen at random over the network. We repeat this process for all cells with theprobability pc . As a result, we get a map which is more similar to the small-world network inthe Ref. [5]. Figure 1 shows a sketch map of the Direction-Preserving SWCNN consisting of
4 × 4 cells. Obviously, when pc = 0, the SWCNN is completely the same with the original
CNNs, and the maps correponding to 0 < pc < 1 and pc = 1 are respectively shown in themiddle part and the right hand side of Fig. 1.

pc=0

pc=1

Increasing randomness

Figure 1: Direction Preserved SWCNN architecture.
The state equation of each cell c(i, j) of the SWCNN is formulated by Eq. (1), and theoutput equation is defined by Eq. (2).
ẋij (t) = −xij (t) + I +X
c(k,l)∈Nr (i,j)

XA(i, j; k, l)ykl (t) +c(k,l)∈Nr (i,j)

B(i, j; k, l)ukl (t)(1)+wc M (i, j; p, q)ypq (t)1yij (t) = (|xij (t) + 1| − |xij (t) − 1|)2(2)

i = 1, 2, ..., M, j = 1, 2, ..., N.
where Nr (i, j) denotes the neighbor cells of radius r of a cell c(i, j); A, B, and I are realconstants called as feedback template, control template and bias current, respectively; xij , yij ,uij denotes the state, input and output of the cell, respectively; M (i, j; p, q) describes the smallworld map that is randomly created by program with indicating the probability pc in advance, if
there is a coupling between one cell c(i, j) and another cell c(p, q), then the M (i, j; p, q) is equal
to 1, otherwise is zero; and wc stands for the coupling weight between the randomly coupledcells.
In order to investigate the features of the network, we calculated the characteristic path
length L(pc ) and the clustering coefficient C(pc ) as varying pc . The characteristic path lengthL(pc ) is defined as the number of edges in the shortest path between two vertices, averaged overall pairs of vertices [5]. The clustering coefficient C(pc ) is defined as follows; Suppose that avertex v has kv neighbours; then at most kv (kv − 1)/2 edges can exist between them. Let Cv
denote that fraction of these allowable edges that actually exist. Define C(pc ) as the average ofCv over all v [5].
The results are shown in Fig. 2. Because we make a restriction such that one cell has at mostone random coupling, the clustering coefficient does not approach zero even if pc becomes 1.0.
However, the characteristic path length becomes shorter and the network possesses the featureof the small-world networks.

1.2

C(pc)/C(0)L(pc)/L(0)
1.0

0.8

0.6

0.4

0.2

00.0001

0.001

0.01

0.1

1pc

Figure 2: Characteristic path length and clustering coefficient of Direction-Preserving
SWCNN.

3. Diffusion analysis
Image processing is one of important applications of the CNN. As well known, many templates
and algorithms used for this purpose have been developed. In this section, we investigate theeffect of applying the Direction-Preserving SWCNN to diffusion analysis.
The diffusion template is shown as follows.

0.1 0.15 0.1
0 0 00.15  , B =  0 0 0  , I = 0.
A =  0.15 00.1 0.15 0.1
0 0 0(3)This is a propagating template, the computing time is proportional to the length of the image.
In order to evaluate the ability of the SWCNN, we investigated the convergence speed of theSWCNN for this task with various pc . The result is shown in following Figures.

(a)

(b)

(c)

Figure 3: Diffusion simulation by the original CNN. (a) Initial state image. (b) Output imageat 5 times. (c) Steady state output image.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 4: Diffusion simulation of Direction-Preserving SWCNN. (a),(b),(c) Output image at 5,30 and 50 times with pc = 0.1. (d) Steady state output image with pc = 0.1. (e),(f),(g) Output
image at 5, 30 and 50 times with pc = 1.0. (h) Steady state output image with pc = 1.0.
1
original CNN
pc=0.1
pc=0.3pc=0.5pc=1.0transient

0.1

0.01

0.001

0100

200

300

400

500

600

700

time

Figure 5: Transient with various pc .
From the fig. 5, we can say that the Direction-Preserving SWCNN converges more quicker
than the original CNN. Moreover, in lower pc , we can confirm that the change of convergence
speed is greater, in fig.6. However, in fig. 4, the convergence speed become higher with the pc
increases, but the noise has occurred, so it may not work well as diffusion task. So we adjustthe random coupling weight wc . Basically, the Direction-Preserved SWCNN is obtained byreconnect the couplings, here, we don’t reconnect but add the couplings, and adjust the rate of
the parameter between the parameter of A-template and wc in a shortcut cell. The results areshown in fig. 7 and fig. 8. In ifg. 7, the parameter is set of 0.8(the parameter of A-T emplate)
to 0.2(the random coupling parameter wc ).

1000
original CNN
pc=0.05
pc=0.1
pc=0.5pc=1.0

900
800
700

time

600
500
400
300
200
100
012

3

456789 10 11 12 13 14 15 16 17 18 19 20network maps

Figure 6: Convergence speed in various network maps.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 7: Diffusion simulation by improved method. (a)-(e) Output images at 5, 30, 50, 100
and 150 times with pc = 1.0, respectively. (f) Steady state output image.

4. Conclusions
In this article, we have proposed Direction-Preserving network topologies of the SWCNN andinvestigate the features of the SWCNN. We have applied the Direction-Preserving SWCNN todiffusion. Because we used the templates designed for the original CNN, the obtained outputs
were not sufficient for the tasks. However, we confirmed that the SWCNN could improve theconvergence speed.
Our important future researchs are design of the template for the SWCNN and analysis of
the stability of the SWCNN.

1original CNN
pc=0.1
pc=0.3pc=0.5pc=1.0transient

0.1

0.01

0.001

0100

200

300

400

500

600

700

time

Figure 8: Improved results of transient with various pc .

References[1] L.O. Chua and L. Yang, “Cellular neural networks: theory and applications,” IEEE Trans. Circuits& Syst., vol.35 pp.1257-1290, Oct. 1988.
[2] L.O. Chua and T. Roska, “The CNN paradigm,” IEEE Trans. Circuits & Syst., vol.40, pp.147-156,Mar. 1993.
[3] T. Roska, L. Kék, L. Nemes, A. Zarandy and P. Szolgay “CNN software LIBRARY: templates andalgorithms”, Version7.3, Analogical and Neural Computing Laboratory, Computer and Automation
Institute, Hungarian Academy of Sciences, 1999.
[4] Z. Yang, Y. Nishio and A. Ushida, “Image processing of two-layer CNNs – applications and theirstability –,” IEICE Trans. Fundamentals, vol.E85-A, no.9, pp.2052-2060, Sep. 2002.
[5] D.J. Watts and S.H. Strogatz, “Collective dynamics of small-world networks”, Nature 393, pp.440442, 1998.
[6] K. Tsuruta, Z. Yang, Y. Nishio and A. Ushida, “Small-world cellular neural networks for image
processing applications,” Proceedings of ECCTD’03, vol.1, pp.225-228, Sep. 2003.