心身科学 第 11 巻第 1 号（ 1 7 ）（2019）

An elementary theory of a dynamic weighted digraph（2）Naohito Chino＊１）
Abstract
This paper is the second part of the revised version of my handout presented elsewhere（Chino,
2018a）. We examine the trajectories of（ 1 ）one of our nonlinear complex diﬀerence equation
models which describe changes in a dyadic relation between two objects over time,（ 2 ）random
walk which was introduced by Pearson（1905）, and（ 3 ）Brownian motion which was initiatedby Brown（1827）. We revisit some of the results in Chino（2018b）for these trajectories, ﬁrst,
and notice that some of the qualitative behaviors of our nonlinear dyadic system are reminiscentof random walk and Brownian motion. Then, we examine their mathematical properties further.
We conclude that the three types of trajectories generated by diﬀerent processes are diﬀerent
from each other in some respects.
Key words: complex difference equation, Hilbert space, Chino-Shiraiwa theorem, dynamic
weighted digraph, chaos, random walk, Brownian motion.

１．Revisit of some of the results in Chino（2）（2018b）
In Chino（2018b）, we discussed a bit about ourwheredifference equation model described by a set of
non-linear dyadic difference equations describedandbelow,The simplest system is such that

= 0 inEq.（ 2 ）. To attain this value, we set the α s as
=0.9564 ,andwhich is denoted as Eq.（11）in that paper.

=­ 1 ­0.9564 ,=0.01,=1.5375­0.9564 .

As discussed there, this type of system has a

It is well known that the Julia set of this systemvery desirable property in that we can utilize theis the unit circle with origin 0 in C（equivalently,

heritage of the theory of the complex dynamicalH）, where C is the complex plane.

system developed in mathematics directly in

it is easy to show that this system has two fixedclassifying its trajectories. In fact, deﬁning a new

points, 1 and 0 （the origin）, and these points

variable,are repelling and superattracting, respectively.

and transforming it

Moreover,As a result, when we start from an arbitrary point

linearly, we have a new systemon the unit circle except for the fixed point 1 ,＊ 1 ）Department of Psychology, Faculty of Psychological & Physical Science, Aichi Gakuin University.
Requests for reprints should be sent to : chino@dpc.agu.ac.jp
This is the second part of the revised version of my handout titled An elementary theory of a dynamic weighted digraphpresented at the 46th annual meeting of the Behaviormet-ric Society of Japan , Tokyo, Japan.

―1―Naohito CHINO

the trajectory moves chaotically on the circle, atby Eq.（ 1 ）. Fig. 1 shows trajectories of the four

least theoretically. （Notice that this circle is not a

indices of the original dyadic system.

fractal since the unit circle is not self-similar.） AsFirst, we computed the largest Lyapunova trial, we set the initial value of this system as ion the unit circle in C, since this point is not theIt converges to 0.693147148…, which is very close

exponent of this dyadic system. Fig. 2 shows it.

ﬁxed point of the system. As pointed out in Chino

to that of the linearized system（2018b）, the strict Lyapunov exponent of this one-

discussed above. This result clearly shows that thedimensional system,

dyadic system has chaos.

is λ= ln ２

Second, we shall zoom up trajectories shown= 0.69314718…, thus the system has chaos.

in Fig. 1 to examine further the features of these

Next, we proceeded to examine qualitative

trajectories.

behaviors of the original dyadic system described

For example, Fig. 3 shows the

expanded trajectories of the third time series in Fig.
1 from iteration 45,000 to 46,000. This trajectory
is reminiscent of one-dimensional random walk or
Brownian motion（Brown, 1828）.

Fig. 1. Trajectories of the four indices of the original
dyadic system. 10-a and 10-b are those of node j（member A）and node k（memberB）, respectively.
10-c is the trajectory of the proximity from node j tonode k. 10-d is the trajectory of the angle from node
j to node k. （This ﬁgure is reproduced from Fig. 10 in
Chino（2018b）.

Fig. 3. Expanded trajectories of Fig. 1 -c from iteration
45,000 to 46,000（This ﬁgure is reproduced from Fig.
12 in Chino, 2018b）.

Fig. 2. The largest Lyapunov exponent of the original
two dimensional system（This figure is reproduced
from Fig. 11 in Chino（2018b）.

Fig. 4. Simultaneous plot of the two trajectories shownin Figs. 13 and 14 in Chino（2018b）.

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An elementary theory of

20,000 iterations.

Third, we show trajectories of the dyadic
system after 50,000 iterations. Fig. 4 shows the

It is interesting to note that trajectories of oursimultaneous plot of the trajectories of the two

special dyadic system are reminiscent of theobjects（nodes）using diﬀerent colors, yellow and black.

random walk or the Brownian motion.
In the following sections we abbreviate theFig. 4 shows that the trajectories of two nodes
are very close to each other. This point may befractal dimension as D F , the Hausdorﬀ-Besicovitch

contrasted with those of the usual random work as

dimension as D HB , the topological dimension as

well as the Brownian motion.

D T , and the Euclidean dimension as D E .

To compare the trajectories of our dyadic systemwith those of the random walk and the Brownian

２．Mathematical properties of random walkmotion, we draw Figs. 5 and 6 , which are

Random walk was introduced by Pearson（1905）.

trajectories of a simple random walk on the x-axis

A special random walk in the one-dimensional caseis called the simple random walk . Assuming the nand a planar Brownian motion on the y-axis.
Figs. 7 and 8 are trajectories of a two-dimensional

independent random variables,,,…,, having

random walk and a planar Brownian motion aftervalues - 1 or 1 （step size equals 1 ）, the simple

Fig. 5. Trajectories of a simple random walk on thex-axis.

Fig. 7. Two-dimensional random walk after 20000
iterations.

Fig. 6. Trajectory of a planar Brownian Motion in
y-axis after 20000 iterations.

Fig. 8. Planar Brownian motion after 20000 iterations.

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Naohito CHINO

random walk on Z is deﬁned as,
（5）

（3）

The random variable
expectations:

Mathematically, physical Brownian motion is a

has the following

Gaussian process as one of the stochastic process= 0 , and（e.g., Hansen, 2018）, and historically it is also the

If the step size approaches to zero in the simple
random walk, we have the one-dimensional Wiener

process ,Wiener process introduced by Wiener（1923）.
M o re g e n e ral l y, a pro ce ss i s Gaussian if, and the density function is written as

all the finite dimensional marginal distribu-（4）

tions（ fidi）are Gaussian function,

This process is equivalent to the physicalas knowledge of these func

Brownian motion in the one-dimensional casetions allows us to write up every fidi.

which will be discussed in the next section.

the covariance matrices constructed from theHere,

Random walk has no scale invariance, and thus

covariance functions have for instance to be

is not self-similar . It is sometimes called astatistically self-similar （e.g., Mandelbrot, 1977）.

symmetric and positive semi-deﬁnite（p.s.d.）（e.g.,
Hunsen, 2018）.
For example, the two-dimensional GaussianMoreover, in one-dimensional random walk（i.e.,

process,

the simple random walk）, both the fractal
dimension （D F ）and the topological dimension（6）（D T ）are 1 ʼs（De Gennes, 1979）. Therefore, it is nota fractal （i.e., D F =D T ）. It is sometimes called thediscrete fractal . In the two-dimensional random
walk, D F is 4 / 3 （de Gennes, 1979）, and D T is 1 ,thus it is a fractal （i.e., D F > D T ）. Since theis a Brownian motion for any ０ 㱡 s 㱡 t. If weconsider the increment

over the intervalfrom s to t , we havevariance of the random walk diverges, it is not a（7）chaos .

Eq.（ 7 ）is equivalent to Eq.（ 5 ）which Einstein（1905）deduced.

３．Mathematical properties of Brownian

In the one-dimensional physical Brownian

motionThe so-called Brownian motion can be classiﬁed

motion, the Euclidian dimension（D E ）is 1 , D T is

into two categories. One is the physical Brownian

1 , and the Hausdorﬀ-Besicovitch dimension（D HB ）motion （e.g., Mandelbrot, 1977）, and the otherthe fractional Brownian motion（fBm）. Physical（i.e., D F ）is 1.5（Sprott, 2003）. Therefore, theBrownian motion is named after Brown（1827）,
while fBm was proposed by Mandelbrot and Van

fractal（D HB > D T ）. In the two-dimensional physicalBrownian motion D E is 2 , D T is 1 , and D HB is 2

Ness（1968）. According to them, the notion of（Mandelbrot, 1977）. Therefore, the two-dimen-

one-dimensional Physical Brownian motion is a

fBm has already been considered elsewhere（Hunt,sional physical Brownian motion is not a fractal

1951; Kolmogorov, 1940; Lamperiti, 1962; Yaglom,（D HB =D T ）. The physical Brownian motion is selfsimilar, and is sometimes called, exactly self-similar

1958）
.

（e.g., Sprott, 2003）. Moreover, it is not a chaos,
３．１ Physical Brownian motion

because it is a diﬀusion process.

This is the random motion of tiny particles
suspended in a fluid. Physically, it is thought of

３．２ Fractional Brownian motion（fBm）

as a diﬀusion process, one-dimensional Brownian

According to Khoshnevisian（2018）, fBm is amotion which Einstein（1905）described as

generalization of the physical Brownian motion,―4―

An elementary theory ofat any instant of time. As a result, changes inand has a special covariance function,

digraphs over time are considered as changes in（8）

conﬁgurations of nodes inover time.

Here, α is the Hurst parameter of X, and α > 0.

Our elementary theory of dynamic digraph then

The Hurst exponent H , which is introduced inassumes that these changes in the configurations

Mandelbrot （1977）is equivalent to the Hurstof nodes can be described by a set of complex

parameter α , in which H is deﬁned as

nonlinear diﬀerence equations ingeneral case.

（9）in the mostThe purpose of our theory is toclassify elementary patterns of changes in digraphs

where R（d）is the sample average river dischargeover time, by assuming that there exists a latent

between years 0 and d, while S（d）is a scaling

process which governs these changes in digraph,factor. According to Mandelbrot（1977）, D HB is

which can be described by a set of complex

equal to the inverse of H, i.e., D HB =1 /H. If H isnonlinear diﬀerence equations in.

In this paper, we restrict the dimension p of the/2 , then it is nothing but a（two-dimensional）

1

physical Brownian motion , and thus. D HB =２ .

state space to one, and conducted some simulation

Moreover, fBm is a special case of the Gaussianstudies in order to classify elementary patterns ofprocess. Falconer（2003）calls α the index of fBm.

changes in digraphs over time. It is easy to show

That is, he deﬁnes fBm of index-α（ 0 <α< 1 ）as athat such patterns can be enumerated simply in theGaussian process X: [ 0 ,㱣] → R on some probability

case when the latent dynamics are linear（Chino,

space such that2017a, b; Chino, 2018a, b）. Our major results inthis paper are concerned with the patterns in the（ 1 ）With probability 1 , X（t）is continuous andcase when the latent dynamics have quadratic

X（0 ）
=0;
（ 2 ）For every t 㱢 0 and h> 0 the increment

terms especially in the case where N= 2 andX（t+h）− X（t）has the normal distribution

q= 2 in Eq.（ 1 ）. As discussed elsewhere（Chino,with mean zero and variance2017a, 2018a, 2018b）, this type of system has a, so thatvery desirable property in that we can utilize the（10）heritage of the theory of the complex dynamicalsystem developed in mathematics directly inclassifying its trajectories.

４．Conclusion

In fact, the transformed new system of Eq.（ 1 ）,In this paper we propose an elementary theoryof a dynamical weighted digraph. In this theoryi.e., Eq.（ 2 ）becomes the Mandelbrot set

we ﬁrst assume that the weight matrix associated（Mandelbrot, 1977）in some cases.

with a weighted digraph denotes the proximity

companion paper（Chino, 2018b）, we have foundstrengths among nodes at any instant of time, andthat the original quadratic system Eq.（ 1 ）also

In thethat it varies as time proceeds. Then we apply

exhibits chaotic behaviors which are very similar to

HFM （the Hermitian Form Model）（Chino &random walks and Brownian motions. Therefore,

Shiraiwa, 1993）to the weight matrix, and obtainin this paper we examine the similarities as well

the conﬁguration of objects（nodes）at any instantas differences among （ 1 ）random walks, （ 2 ）of time in a p-dimensional Hilbert space,

Brownian motions, and（ 3 ）our original quadraticorsystem.

an indeﬁnite metric space. Here, we assume thatthe Hermitian matrix corresponding to the weight

As regards the fractal property, one-dimensional

matrix is positive-semideﬁnite , in the most general（thus the simple）random walk has

case and its weights are measured at a ratio level.

D（= 1 and D T = 1 （De Gennes,1979）, and isF D HB ）Then, we have the correspondence among digraph,

not a fractal.

weight matrix, and conﬁguration of objects（nodes）

random walk has D（= 4 / 3 and D T = 1 （DeF D HB ）―5―

However, the two-dimensional

Naohito CHINO

Sprott, 2003）. Sometimes, it is called exactly self-

Gennes,1979）, and is a fractal.

similar.

One-dimensional physical Brownian motion has

D（=1.5 and D T = 1（e.g., Sprott, 2003）, and isF D HB ）Our quadratic system described by Eq. （ 2 ）But, the two-dimensional physicalmight probably be self-similar. In fact, its computer

Brownian motion（i.e., ordinary Brownian motion）

simulation shows that all the four trajectories

has D F（D HB ）= 2 and D T = 2 （Mandelbrot, 1977）,of this system shown in Fig. 1 are almost self-

and is not a fractal.

similar, since all the log L（k）
-log k plots of thesea fractal.

The one-dimensional fBm has D F（D HB ）= 2 ­ α

trajectories by Higuchiʼs method（Higuchi, 1988）

and D T = 1 （Mandelbrot, 1977）, and is a fractal

show almost linear in the whole range of abscissa（Falconer, 2003）, where α is the index of fBm, or（e.g., Fig. 9 , Fig. 10）.

is also called the Hurst parameter （e.g.,

As for the chaos property, not only random

Khoshnevisian, 2018）. It is sometimes called thewalks but also Brownian motions （which in-

random fractal（Sprott, 2003）. In contrast, theclude physical Brownian motions as well as

two-dimensional fBmʼs has D F（D HB ）= 2 and D T = 2

fractal Brownian motions, i.e., fBm）might not（Mandelbrot, 1977）, and is not a fractal.

be chaos, since all of these trajectories diverge
as time proceeds.

The boundary of the trajectory of the transformed

In contrast, the Lyapunov

one-dimensional complex system（i.e., Eq.（2））ofexponent（LE）of the trajectory of the transformed

our quadratic system（i.e., Eq.（1））, which belongs to

one-dimensional complex system of our original

the Julia set in the unit circle with origin 0 in C

quadratic system computed numerically by a chaos（equivalently, one-dimensional Hilbert space）, is

software（i.e., Takahashi & Yamada, 2000）was

not a fractal geometrically .

very close to the theoretical Lyapunov exponents

As for the self-similarity property, random（i.e., ln 2 ）. Moreover, the theoretical Lyapunovwalk has no scale invariance（Wikipedia, 2018）

exponent of the original quadratic system wasand thus is not self-similar. In fact, a computer

0.693147148276891, which is equivalent to thesimulation of simple random walk by Higuchiʼstheoretical LE of the transformed one-dimensional

method（Higuchi, 1988）shows that the log L（k）

system up to seven decimal places. These resultsplotted against log k with base 2 is nonlinear inindicate that our original quadratic system has

the whole range of abscissa.

chaos.

Acknowledgements

Physical Brownian motion is self-similar（e.g.,

Fig. 9. The log L（k）against k plot of the trajectories ofmember A（node j） In Figure 1 -a by Higuchiʼs method（Higuchi, 1988）.

Fig. 10. The log L（k）against k plot of the trajectories of theproximity from member A（node j）to member B（node k）
in Figure 1 -c by Higuchiʼs method（Higuchi, 1988）.

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An elementary theory of

D , 31, 277-283.
Hunt, G. A.（1951）. Random Fourier transforms,

The author is indebted to Gregory L. Rohe forTransactions of the American Mathe-maticalSociety , 71, 38-69.

proofreading of an earlier version of this paper.

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