DYNAMICAL HYSTERESIS NEURAL NETWORKS FOR GRAPH COLORING PROBLEM Kenya Jin'no, Hiroshi Taguchi, Taka0 Yamamoto, and Haruo Hirose Nippon Institute of Technology 4-1 Gakuendai, Miyashiro, Minami-Saitama, Saitama, 345-8501 Japan
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[email protected] ABSTRACT We analyze syncronization phenomena in a dynamical hysteresis neural network. The dynamical hysteresis neural network accounts a phase differece in each neuron to be an information. Based on this result, we propose an application for solving 2-colorable graph coloring problems.
will analyze synchronization phenomena in a coupled hysteresis neuron. And, we will propose an appIication which can dye 2-colorable graph. 2. HYSTERESIS NEURON The objective hysteresis neuron[6][7] is described as
1. INTRODUCTION Recently, a system that can treat dynamical information, is attracted to great attention[l]-[3]. The reason why such system receives great attention, is that dynamical information processing functions can be found in biological neural networks. Especially, we think that a synchronization phenomenon plays an important role for signal processing in the brain. In our previous works, we proposed and analyzed a hysteresis neural network[4][5]. Our system consists of relaxation oscillators in which contain hysteresis elements. The relaxation oscillator is regarded as a multi-vibrator, namely, the system takes bistable, monostable, and astable state. By using stable and monostable state, we proposed a combinatorial optimization solver that its stable equilibrium point is regarded as an information[4]. On the other hand, various kinds of attractors exist in the hysteresis neural network. We consider that periodic and aperiodic attractors have rich information. For exploiting such attractors, we proposed a novel dynamical associative memory whose information is represented by phase difference.[5] This behavior relates synchronization phenomena. In this article, we
Figure 1: A normalized bipolar hysteresis.
0-7803-7761-3/03/$l7.00 02003 IEEE
wherex(t) ~%isastatevariable,y(r)E {+l,-I} isanoutput, and e(y(t)) E 93 denotes a feedback parameter. h ( x ( t ) ) is a bipolar piecewise linear hysteresis as shown in Fig.1.
Figure 2: These waveforms are from a hyteresis neuron. The upper waveform is observed in the case where the feedback parameter is given as e(y(t)) = -Zy(t). The lower waveform is observed in the case where the feedback parameter is given as e ( y ( t ) )= -4y(t). The state variable of (1) vanes toward an equilibrium point that denotes e ( y ( f ) ) . Since the equilibrium point depends on its output, the equilibrium point becomes a constant while the output does not change. If the equilibrium point does not exist on the hysteresis branch, the trajectory reaches the threshold of the hysteresis before it converges to the equilibrium point. When the trajectory hits the threshold
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of the hysteresis, the output changes its sign, and the corresponding equilibrium point changes too. If the equilibrium point e(y(t)) satisfies the following condition, the trajectory can not reach the equilibrium point[7]. e(y(t))y(O 5 - 1
(2)
In this case, the system must exhibit an oscillatory state[7]. The oscillation period is given as
T =log{
1-e(l) -l-e(-l) }+log{ l p e ( - l ) -I-e(1)
1.
Namely, the connection coefficient matrix has a diagonally dominant. If the system adopts such assumption (8), the condition (7) is satisfied. Therefore, this system must exhibit oscillatory state. Figure 3 shows typical waveforms from two neurons system. In this figure, these waveforms correspond to the state variables, XI and x2. respectively. Both results are observed into the following configurations
(3)
Since the equilibrium point depends on the self feedback, the oscillation frequency can be controlled by the feedback parameter. Figure 2 shows examples of waveforms of the state variable in a hysteresis neuron. The different point between two waveforms is only the self feedback parameter: The feedback parameter is given as e(y(t)) = -2y(r) in Fig.Z(a), and it is given as e(y(r)) = -4y(t) in Fig.Z(b). The oscillation frequencies of Fig.Z(a) and (b) are Fa = 1/21og3 and Fb = 1/210g respectively. Next, we consider the case where some hysteresis neurons are coupled as
5,
(a) in-ohase svnchronization
(b) opposite-phase synchronization Figure 3: The waveform from two coupled hysteresis neurons. The waveform of (a) is observed in the case where the system has an excitatory connection, and (b) is observed in the case where the system has an inhibitory connection. of the initial values
where T;> 0 is a time constant, N is the number of hysteresis neurons, w;, E %(i # j) is a coupling coefficient, and wjj is a self feedback parameter. For simplicity, we assume all time constants have the identical value, hereafter. The equilibrium point ej(y(t)) is given from Eq.(4) by N
ei(y(t)) =
WijYj(f).
(5)
j= I
If the following condition is satisfied, the system exhibits oscillatory state[5].
ei(y(t))yi(r) 5 -1,3i.
i -1,Vi.
(7)
Then, the system has no stable output state completely. We suppose that the connection coefficients satisfy the following relation:
(9)
In the case of Fig.3(a), the configuration of parameters is
The cross-connection coefficient has a positive value. Namely, the coefficient can be regarded as an excitatory connection. In this case, the system exhibits an in-phase synchronization. On the other hand, the parameters configuration of Fig.3(b) is
(6)
Especially, if the following condition is satisfied, all neurons keep oscillatory state. ei(y(O)y;(t)
(Xl(O),Yl(O)) = ( + L + 1 ) , (XZ(O),Y2(0)) = @ , + I ) .
( ;;; ;; ) ( =
-4 -1
-4
).
(11)
The cross-connection coefficient has a negative value, that is an inhibitory connection. In this case, the system exhibits an opposite-phase synchronization. For what has been discussed above, the fundamental frequency is affected by the self feedback parameter. The coupling parameter plays a role as frequency modulation. As the self feedback parameter W;is -4, its fundamental frequency FJ is Ff = 1/210g representing in Fig.2(h). The synchronized oscillation frequency F, is F, = 1/210g2 which is affected by the interaction with another neuron. We define a retum map for analyzing these synchronization phenomena. Since this system always satisfies (7), a
3
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F'
range of the state variable is [- 1, + I ] . Let @ be a one dimensional domain such as @=
{x2(T)llnz(T)I c
I,Xl(T)
= I}.
2
(12)
4
We can define a retum map from Q to itself
9:CJ-a
(13)
This mapping can be easily calculate by using our proposed algorithm[4]. Figure 4 shows the return map 9. The parameter configuration of Figure 4(a) obeys Eq.(lO), and (b) obeys Eq.(l I). These return maps contain two fixed points
Figure 4: Return map 9of the two neurons system. (a) and (h) correspond to the case of Figures 3(a) and (b), respectively. "A" denotes an attractor, and " R 3denotes a repeller. that are indicated as " R and "A', respectively. The point "R" denotes a repellent fixed point, and " A denotes an attractive fixed point. All initial values converge to an attractive fixed point. In the case of Fig.4(a), the attractive fixed point is x2 = I . Based on the definition of this return map, this attractor represents an in-phase synchronization. When the parameters set as (IO), all initial values converge to an in-phase synchronized attractor such as Fig.3(a). On the other hand, in the case of Fig.4(b), the attractive fixed point is xz = - I .O. Based on the definition of this return map, this attractor represents an opposite-phase synchronization such as Fig.3@). It follows from what has been said thus far that the excitatory connection leads to in-phase synchronization, and the inhibitory connection leads to opposite-phase phase synchronization.
Figure 5: Example of 2-colorable graph with 5 venices[8]. studied the possibility of coloring graphs by means of synchronized coupled relaxation oscillators[8]. However, this system has some problems that there is a case which the system has a stable equilibrium point, and so on. Also, it is difficult to set the parameters. On the other hand, the behavior of our hysteresis neuron can be easily controlled by its equilibrium point. If the connected neurons have an inhibitory connection, they do not share the same phase. By using this ability, we apply an oscillatory hysteresis neural network for solving graph coloring problems. Each hysteresis neuron is assigned to each vertex. Figure 5 shows a simplest example of 2-colorable graph with 5 vertices which is introduced in Ref.[@. From Fig.5, an adjacent matrixA of the graph is given as
A=
01 10 10 01 01
(14)
.
where a;, E {0, I} is a component element of an adjacent matrix A . ajj = 1 represents connected between i-th vertex and j-th vettex, and ai, = 0 represents disconnected edge. Based on this adjacent matrix, we determine a connection matrix for the oscillatory hysteresis neural network.
where a and E are positive constants. For example, an connection matrix W for solving a graph such as Fig.5, is given as
-5
3. GRAPH COLORING PROBLEMS I n this section, we investigate a dynamical hysteresis neural network for solving graph coloring problems. The description of graph coloring problems is that a vertex coloring of an undirected graph is an assignment of a label to each node. It is required that a minimum coloring of a graph is a culoring that uses as few different labels as possible. Dr.Wu
[:;;;8) 0 -1.2
0
0
0 -5
where a = I .2 and E = 3.8. In this case, the system exhibits a periodic attractor as shown in Fig.6. Figure 6 shows waveforms of the state of each neuron, which corresponds to each
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vertex. These waveforms can be classified into two clus-
Figure 6: The oscillatory state corresponds to the solution of 2-colorable graph with 5 vertices. The horizontal axis corresponds to the time evolution, and the vertical axis corresponds to the neuron. ters by using its phase. One cluster consists of XI, x2, and q,these exhibit in-phase synchronization. Another cluster contains 3rd and 4th neuron, therefore, these vertices are assigned the same color. Namely, a phase difference in each neuron corresponds to a solution of graph coloring problems. We show another example of 2-colorable graph with 16 vertices as shown in Fig.7. Figure 8 shows a result which indicates a solution of Fig.7.
Figure 7: Example of 2-colorahle graph with 16 vertices[S].
4. CONCLUSIONS In this article. we analyzed the synchronization phenomena in an oscillatory hysteresis neural network. We clarified that this system exhibits an in-phase synchronization when the cross-connection coefficient denotes an excitatory connection. Also, the system exhibits an opposite-phase synchronization when the cross-connection coefficient denotes an inhibitory connection. Based on these results, we proposed an application which can dye 2-colorable graphs. For solving k-colorable graph by proposed dynamical hysteresis neural network, is one of our future problems.
Figure 8: Waveforms of the inner states of 16 neurons. These waveform can be classified into two kinds which corresponds to coloring graph of Fig.7.
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