Journal of the Institute for Psychological and Physical Science, Vol. 7 , No. 1 , 31┉47, 2015

A simulation study of a Hilbert state space model forchanges in afﬁnities among members in informal groups
Naohito Chino＊1）

This is the revised version of the supplement of the paper which appeared in the proceedings of the workshop
on Problem Solving through the Applications of Mathematics to Human Behaviors. In that supplement we
added its motivation, tables and ﬁgures, and so on to the paper in order to complete it. Various solution curvesin the ﬁgures suggest that our complex difference equation model covers enormous possible scenarios for theformation and dissolution of afﬁnities among members of informal groups. Since the original model (Chino,2002) assumes not only the ﬁnite-dimensional Hilbert space but also the indeﬁnite metric space as the state
space, this model might have wide applicability to all sorts of phenomena in which asymmetric relationships
among objects are essential. An extended version of our model is also proposed in that supplement, in whicha constant disturbance term is added. It is evident that this term enriches possible scenarios curiously anddrastically. Differences between our model and the extant complex difference equation models are also
discussed. Finally, we shall consider some open problems to be solved.
Keywords: finite-dimensional Hilbert space, indefinite metric space, complex difference equation model,
longitudinal asymmetric relational data matrices, n-body problem, Chino and Shiraiwaʼs theorem,
tripartite deadlock, bifurcation theory, stability problem, holomorphic function

1 Motivation
In any branch of science, we ﬁrst start with the careful observations of the phenomenon under study, and establish the socalled empirical law which governs the phenomenon. For example, Keplerʼs law was established by Kepler using a body ofastronomical observations of planetary motions which had been gathered by Tycho Brahe. According to this law, especially theﬁrst law, the earth goes round the sun on the elliptical orbit.
However, the Keplerʼs law will never teach us why and how the earth moves around the sun. Furthermore, we may ask
whether the elliptical orbit is the only possible orbit or not. To answer these questions, we need two theoretical laws. One is
Newtonʼs second law, and the other Newtonʼs law of gravitation.
On the one hand, Newtonʼs second law, F = ma, is written as a second order differential equation, if we denote x(t) as the
position vector of the particle at time t:＊ 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 revised version of the supplement of the paper presented at the workshop on Problem Solving through the Applications of Mathematics
to Human Behaviors by the aid of The Ministry of Education, Culture, Sports, Science and Technology in Japan, February 15, 2014.

─ 31 ─Naohito Chino

where m denotes the mass of the particle.
On the other hand, Newtonʼs law of gravitation states that a body of mass m 1 exerts a force F on a body of mass m 2 suchthat F =, where r denotes the distance between their centers of gravity and g a constant. Thus, if m 1 lies at the origin of

R and m 2 at x ∈ R 3 , the force on m 2 can be written as
3

By solving the above differential equation, we can obtain possible three types of orbits, i.e., a hyperbola, parabola, andellipse according to whether the total energy of the system E > 0 , E = 0 , or E < 0 (e.g., Hirsch & Smale, 1974, p.26). As isapparent from this example, an important role of any theoretical law is that it enables us to predict the phenomena which havenever been observed empirically.
The above problem can easily be extended to the so-called“n-body problem”. The Newtonian n-body problem, which isthe prototype of other n-body problems, can generally be written as,

where V (x) is a C 2 function, andHowever, the n-body problem seems to be not easy to solve, and no clear picture has emerged on these equations (Hirsch &Smale, 1974).
In the next section we shall brieﬂy discuss n-body problems mainly in the social and behavioral sciences, and consider somebasic problems associated with the n-body problems in these areas of research.

2 n-body problems mainly in the social and behavioral sciences
Let us now ask some questions about n-body problems of phenomena in the social and behavioral sciences as well as the
biological sciences. The following list might be considered typical examples of the phenomena observed in these sciences:
relationships among members of any informal groups such as classmates at school or college and changes in these relationships
over time; relationship called pecking order among a group of hens and cocks in a chicken house and changes in these
relationships over time; amounts of trade among nations and their changes over time; strength of connections among neurons
and changes in it over time. In all of these examples, members of the group can be thought of as constituents of the n bodies in
the n-body problem.
What are the state spaces in which these objects are embedded? Are they embedded rationally in the Euclidean space? Have
we already gathered ample observations necessary to construct some empirical laws? Can we construct some theoretical laws
based on a few principles or statements about the phenomenon under study?
The ﬁrst question is concerned with the basic problem in asymmetric multidimensional scaling (abbreviated hereafter, asasymmetric MDS) in psychometrics (e.g., Chino, 2012), and the answer to the second question is negative in the strict sense,
because the distances from member j to member k as well as the one from member k to member j deﬁned in the Euclidean
space are the same and this relation contradicts the asymmetric similarities between members usually observed in thephenomena in the social and behavioral sciences. To cope with this problem, various augmented distance models have been
proposed which assume the Euclidean space or Minkowskiʼs l-metric space and augment the distance by adding some factors
on asymmetry (e.g., Okada & Imaizumi, 1984, 1987).
In contrast, Chino and Shiraiwa (1993) have shown that objects can be embedded in a ﬁnite dimensional Hilbert space or an─ 32 ─

A simulation study of a Hilbert state space model

indeﬁnite metric space, given an asymmetric similarity matrix S whose (j, k) element is the observed similarity from object j toobject k. Let us here construct a Hermitian matrix H as follows:

where i 2 = － 1. Chino and Shiraiwa have proven that a necessary and sufﬁcient condition for a set of distances djk = dkj to be
the true interpoint distances in a (complex) Hilbert space is the positive semi-definiteness of H. This is an extension of the
Schoenberg-Young-Householder theorem (Chino et al., 2012) on MDS to the case of the complex space.
Chino and Shiraiwa have also shown that objects are embedded in an appropriate space by solving the eigenvalue problemof H. In fact we havewhereand Λ is a diagonal matrix of order n which is the number of non-zero eigenvalues of H, i.e., Λ = diag(λ1;λ2;…; λn). The
matrix X is the special real N× 2 n coordinate matrix of objects, i.e., X = (Ur ,Uc), where U 1 = Ur +iUc and U 1 are composedof the complex eigenvectors of H corresponding to its non-zero eigenvalues. It should be noticed that all the eigenvalues of H
is real.
It is easy to show that Eq. ( 6 ) can be rewritten as

This equation shows the relation between observed similarities among objects and the coordinates of objects in an
appropriate complex space. For further detail, see Chino and Shiraiwa (1993) and Chino (2012).
If S are measured at a ratio scale level, we can embed objects in an appropriate complex space. Chino and Shiraiwa (1993)
call the above method for asymmetric MDS Hermitian Form Model (abbreviated as HFM). In application, however, it is often
the case that S are not measured at a ratio scale level. For example, Table 1 shows a sociometric data, in which each element
designates the afﬁnity of a member toward another member in a class of a senior-high school.

and Figures
It is measured at an interval level. In such a case we Tables
must estimate
the coordinates of objects by a suitable method. Saburi

and Chino (2008) show one such method. In their method S may be measured at either the ordinal, interval, or ratio level.
As regards the third question, we have not had ample
observations regarding many of the problems listed above,
especially in cases in the social and behavioral sciences. In
such a situation, a set of longitudinal asymmetric relational
data matrices (to be precise, longitudinal attraction data
matrices) among members of a dormitory in The Universityof Michigan is a rare example, which was gathered by
Newcomb (1961). In general, it is laborious to gather such alongitudinal asymmetric data matrices in phenomena in thesocial and behavioral sciences. Thus, it will be appropriate
and natural at present to simulate such phenomena using
some mathematical models.

Table 1:1010
students
in ainhighschoolTable
1 : SociometricSociometricdata
dataforfor
students
a highschoolrater \ ratee

1

2

3

4

5

6

7

8

9

10

1

4

3

4

3

5

5

6

4

7

7

2

2

3

4

6

7

6

5

5

4

5

3

4

4

3

5

5

4

4

3

4

5

4

4

7

6

3

7

7

4

6

4

5

5

1

7

6

7

4

7

6

6

6

5

6

4

5

4

6

5

7

4

4

4

4

7

4

5

4

4

3

3

6

6

4

6

8

2

4

4

4

5

4

5

2

4

4

9

6

5

5

5

5

6

5

5

4

6

10

4

4

4

3

4

4

4

4

4

4

(reproduced from Chino (1978))─ 33 ─Naohito Chino

One such candidate may be a difference equation model. Considering the Chino and Shiraiwaʼs theorem, we assume that thestate space in which objects or members are embedded is a Hilbert space or an indeﬁnite metric space. Furthermore, we assume
that members obey the following basic principles of interpersonal behaviors:
1. The asymmetric sentiment relationships among members make their afﬁnities change.
2. If a member has a positive sentiment toward another member, then he or she approaches to the target member.
3. If a member has a negative sentiment toward another member, then he or she parts from the target member.
There exist two minor principles in this difference equation model, as listed below:
1. The magnitude of change in coordinate of members is proportional to the sine of the difference in angles (arguments)
between two members in a complex plane.
2. The magnitude of change in coordinate of members is proportional to the norm of the coordinate in a complex plane.
The ﬁrst one is concerned with our asymmetric MDS, HFM, and the magnitudes of change in coordinates of members takethe maximum value when the angle is ±π=2. The second one is associated with the self-similarity of each member, because
the norm represents the magnitude of similarity to oneself. Recently, we have proposed a more general model which drops
these two assumptions. We shall later discuss it in the discussion section.
One of the models which fulﬁll these requirements is the ﬁnite-dimensional complex difference equation model proposed by
Chino (2002)：

Here,

Moreover,In the above model, N denotes the number of members in informal groups, and p and q represent appropriate constants.
Moreover, rj;n and θj;n denote, respectively, the norm and the argument of member jʼs position vector, zj;n, at time n. Of course,
the vector is a p-dimensional vector in a ﬁnite-dimensional complex space. The space may be either a Hilbert space or an
indeﬁnite metric space. In this paper we assume the former space.
On the one hand, it is assumed in Eq. ( 9 ) that the positive direction of the configuration of members in each of the complex
planes associated with the complex dimensions is counterclockwise (Chino, 1978, 1990). On the other hand, it is assumed thatthe positive direction is clockwise in HFM (Chino & Shiraiwa, 1993). In this paper we choose the clockwise direction in eachof the complex plane as the positive direction. As a result, Eq. ( 9 ) must be rewritten as,

Here, it is necessary to change the signs of each of the elements in Eq. (10) by exchanging j and k. Moreover, we simpliﬁed
the factors in the sine function of Eq. (12) a bit, and added new parameters b and c, as follows:─ 34 ─

A simulation study of a Hilbert state space model

In this supplement, we also discuss a more general model, in which a small complex constant(vector) is added to Eq. (13), i.e.,

In the appendix, we show some restricted simulation results of the above general difference equation model described by Eq.
(15). It will be seen that the above constant enriches possible scenarios for the formation and dissolution of afﬁnities among
members of (informal) groups.

3 Restricted simulation results of a general complex difference equation model
In this section and the appendix, several simulations are conducted to examine our complex difference equation models
under different conditions. In all of the simulations, we ﬁrst show the initial conﬁguration of objects (members) which we
assumed. In this paper, we assume the unidimensional Hilbert space to be the state space, i.e., the usual complex plane,
although our model assumes a ﬁnite-dimensional Hilbert space or an indeﬁnite metric space in general.
In looking at conﬁgurations of objects in these simulations, some caution must be exercised. In psychometrics it is usual to
depict the positions of objects as dots. However, in this section we draw them as position vectors in order to show the change
in these positions over time (iteration) as clearly as possible. Another reason is that in HFM similarity between two objects isrepresented as an inner product (to be precise, Hermitian inner product), and therefore the origin is crucial. It should be noticed
that the positive direction of the conﬁguration is clockwise since we utilize HFM in this simulation.
Simulation 1 : dyad relation, N= 2 , n=200, p=q= 1 , b=c= 1 /50
In Sim.1, we examined change in a dyad relation during 200 iterations, where the number of members equals 2 , p=q= 1 ,and b=c= 1 /50 in Eq. (13). Fig. 1 illustrates the change in conﬁgurations of two members in the complex plane. It is apparent
that the skewness of afﬁnity between the two gradually decreases, as both of them rotate in the clockwise direction, as timeproceeds. Here, it should be noticed from Fig.2a, 2 b that self-similarities of both members, i.e., norms of their position vectors,
increase monotonically after earlier iterations. In contrast, the skewness of afﬁnity between them decreases monotonically
immediately after the ﬁrst iteration, as is apparent from Fig.2c.

Figure 1 : Locomotion of a dyad relation in a unidimensional Hilbelt space

Figure 1: Locomotion of a dyad relation in a unidimensional Hilbelt space─ 35 ─Naohito Chino

self-sim.

2.5
2
1.5
1

0

20

40

60

0

20

40

60

0

20

40

60

80

100

120

140

160

180

200

80

100

120

140

160

180

200

80

100

120

140

160

180

200

a. Self-sims. of member A (b=c=1/50).

1.4

self sim.

1.2
1
0.8
0.6
0.4

b. Self-sims. of member B (b=c=1/50).

2

angle

1.5
1
0.5
0

c. Angles from A to B (b=c=1/50).

Figure 2 : Changes in self-similarities of the dyad ( 2 a, b) and change in the angle ( 2 c)

Figure2:orbits
Changes
self-similaritiesof thedyadb) andchangethe angle
Finally, Fig.3
depictsof the indyad
during 200 iterationsin thesame(2a,complexplane.
In thisinﬁgure,
a black (2c)dot represents
the origin. Moreover, A 1 and B 1 in this ﬁgure denote the initial points of the two members, respectively. It is apparent from thisﬁgure that both of the two members part from the origin as iteration proceeds, moving on the same line while preserving the
distance between them.
As can be seen from Sim.1, locomotion of a dyad in the complex plane is simple and linear, and the skewness of afﬁnity
between the dyad diminishes asymptotically. However, in the case of triad relation, their locomotions become nonlinear andshow curious scenarios. However, in many cases the skewness of afﬁnity among members diminishes asymptotically after
fairly long iterations. We shall show such scenarios in the several subsections that follow.

•

Figure 3 : Orbits of the dyad in simulation 1

Figure 3: Orbits of the dyad in simulation 1
─ 36 ─

A simulation study of a Hilbert state space model

4 Discussion
In this paper, we have first made a brief review of the asymmetric MDS methods which have been developed in
psychometrics, and have pointed out that those methods which deal with longitudinal asymmetric relational data are rare.
Next, we have made a restricted simulation study, in which a revised version of a complex difference equation model proposed
ﬁrst by Chino (2002) is examined. In this paper, we have examined a special difference equation model whose state space isassumed to be a unidimensional Hilbert space.
As shown in the simulation, it was found that this model includes a body of curious scenarios concerning the change in
afﬁnities of members. Furthermore, we have proposed a more general model which adds a complex constant (vector) in theabove model. This model extends the original one in such a way that it introduces interesting oscillation phenomena in theprocess of change in afﬁnities over time (iteration).
There are a lot of open problems to be discussed. Some of them are ( 1 ) the problem of estimating model parameters, given
empirical data, ( 2 ) identiﬁcation of the bifurcation parameters by for example utilizing SEM, ( 3 ) stability analysis of ourmodel, ( 4 ) the problem of deﬁning energy in our model.
In the appendix, we have shown major results on several simulation studies other than Sim.1. These simulations suggest thatthe difference equation models proposed in this paper can predict a body of curious possible scenarios for the formation anddissolution of afﬁnities among members of informal groups.
However, there exists a fundamental defect in our models, the reason being that our two minor principles in these models
include the complex conjugate, z̄jn, of the coordinate vectors, zjn, of each member in a complex space. It is well known inelementary complex analysis that the complex conjugate of any coordinate in the complex space is not differentiable (Bak &Newman, 1982). This means that zj,n+ 1 in Eqs. (13) and (15) is not a holomorphic function (Chino, 2014; Shiraiwa, K., personalcommunication, March 3 , 2014). As a result, our differential equation models proposed in this paper cannot be called thecomplex dynamical system in the traditional sense.
Of course there exists a way to avoid this defect. To do so, we may assume that the state space in our differential equation
models described by Eqs. (13) and (15) is the 2 p-dimensional Euclidean space, where p is the number of dimensions of theHilbert space or the indeﬁnite metric space. Another way is to drop Eq. (12) of our models. This means that we shall consider
a more general model without discarding the assumption that the state space of our models is the complex space (Chino, 2014).
We have now been conducting a new simulation study based on this new general model. The important point of this model isthat we can utilize the theory and methods of the complex dynamical system in the traditional mathematical sense in this case.

Acknowledgements
The author is indebted to Gregory L. Rohe for proofreading of an earlier version of this paper.

(Eds.) Measurement and Multivariate Analysis, Tokyo: Springer.

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A simulation study of a Hilbert state space model

Appendix
Simulation 2 : triad relation, N= 3 , n=7000, p=q= 1 , b=c= 1 /50
In Sim.2 we examined change in a triad relation during 7000 iterations, where the number of members equals 3 , p=q= 1 ,b=c= 1 /50 in Eq.13. Fig.4 suggests that as time proceeds the triad relation which exhibits asymmetric sentiment structures
among members at the initial coordinates converges to an almost symmetric relation after the lengthy iterations.
Such a tendency can be ascertained by the fact that angles among the triad seem to converge to zero or 2π in Fig.6.
This tendency can also be conﬁrmed by the fact that the three orbits of the triad seem to converge to a line in Fig.7, as timeproceeds.

Figure 4 : Locomotion of a triad relation in a unidimensional

Figure 5 : Changes in self-similarities of the triad

Figure 5: Changes in self-similarities of the triad
space relation in a unidimensional Hilbelt space
Figure 4: Locomotion Hilbelt
of a triad
2

1
B

1

•

0

A

1

C

1

imaginary axis

-1

-2

-3

-4

-5

-6
-15

Figure 6 : Changes in angles among the triad

-10

-5

0

5

10

15

20

Orbits of members, A, B, and C in Simu.2

25

30

Figure 7 : Orbits of the triad in simulation 2

Figure 6: Changes in angles among the triad

Figure 7: Orbits of the triad in simulation 2

─ 39 ─

35

Naohito Chino

Simulation 3 : tripartite deadlock, N= 3 , n=600, p=q= 1 , b=c= 1 /50
In Sim.3 we examined change in the so-called tripartite deadlock relation during 600 iterations. It is apparent from Fig.8 thatthe deadlock relation remains unchanged with regard to angles among three members even if time proceeds. This feature can bedirectly conﬁrmed from Fig.10. It is also apparent from Fig.9 that the self-similarities of members, i.e., the lengths of members
from the origin increase gradually as time passes, such features being ascertained by drawing Fig.9 more clearly. Fig.11 depicts
orbits of the tripartite in Sim.3, which illustrates these features simultaneously in a ﬁgure.

Figure 8 : Locomotion of a tripartite in a unidimensional

Figure 9 : Changes in self-similarities of the tripartite

Figure 9: Changes in self-similarities of the tripartite
Hilbelt
Figure 8: Locomotionof aspacetripartite in a unidimensional Hilbelt space

1
B1

imaginary axis

0.5

A

1

0

•

-0.5

C

1

-1

-1

Figure 10: Changes in angles among the tripartite

-0.5

0

real axis

0.5

1

Figure 11: Orbits of the tripartite in simulation 3

Figure 10: Changes in angles among the tripartite

Figure 11: Orbits of the tripartite in simulation 3

─ 40 ─

A simulation study of a Hilbert state space model

Simulation 4 : linear triad relation, N= 3 , n=200, p=q= 1 , b=c= 1 /50
In Sim.4 we examined change in a linear triad relation over time, where the initial coordinates of the three members arelocated in line, as can be seen in Fig.12a. It is apparent from Fig.12 that asymmetric relations among members graduallydisappear as time proceeds, which can be directly veriﬁed by looking at Fig.14. In Fig.14a and 14b, angles from A to B as wellas B to C converge to zero, while in Fig.14c, the angle from C to A converges to 2π, as time passes. It is apparent from Fig.15
that orbits of the triad in Sim.4 may form in line, as time passes.

Figure 12: Locomotion of a triad in a unidimensional Hilbelt

Figure 13: Changes in self-similarities of the triad

space of a triad in a unidimensional Hilbelt spaceFigure 13: Changes in self-similarities of the triad
Figure 12: Locomotion

4

2
C

1

B

imaginary axis

1

•

0

A

1

-2

-4

-6

-6

Figure 14: Changes in angles among the triad-4-2

real axis

0

2

4

Figure 15: Orbits of the triad in simulation 4

Figure 14: Changes in angles among the triad

Figure 15: Orbits of the triad in simulation 4

─ 41 ─Naohito Chino

Simulation 5 : dyadic relation, N= 2 , n=200, p= 1 , q= 2 , b=c= 1 /50
In Sim.5, we examined change in a dyadic relation during 200 iterations. Results of the simulation are illustrated in Figs.16
through 18. Initial coordinates of the dyad are the same as those in Sim.1. In this case, however, the system of difference
equations is quadratic, while it is linear in Sim.1. It is apparent from these results that the nonlinearity of the system makes
much difference to the scenarios of change in the dyadic relation between the two members.
First, the conﬁguration of members at iteration 100 (i.e., Fig.1d) in Sim.1 shows that members A and B become a symmetric
relation in which they like one another, while in Sim.5 they become the other symmetric relation in which they dislike one
another as can be seen in Fig.16d. Second, the self-similarities of the dyad no longer increase in Sim.5 as shown in Figs.17a
to 17b, while they increase monotonically or increase after some point in time at an initial stage. It is interesting to note thatthe angle between the two members increase monotonically at the initial stage but that it converges to a value at which two
members stand in line and cease to move. Such motions can be inferred by looking closely at Fig.18.

Figure 16: Locomotion of a dyad in a unidimensional Hilbelt

Figure 17: Changes in self-similarities of the dyad (17a, b)

FigureHilbelt
17: Changes
space of a dyad in a unidimensionaland changeofin thethe dyadangle (17a,(17c) b) and change in the angle (17c)Figure 16: Locomotion
spacein self-similarities
2

1.5

imaginary axis

1

0.5

B

1

A

•

0

1

-0.5

-1

-1.5
-1.5

-1

-0.5

0

real axis

0.5

1

1.5

2

Figure 18: Orbits of the dyad in simulation 5

Figure 18: Orbits of the dyad in simulation 5

─ 42 ─

A simulation study of a Hilbert state space model

Simulation 6 : tripartite deadlock, N= 3 , n=600, p= 1 , q= 2 , b=c= 1 /50
In Sim.6, we examined change in a tripartite deadlock relation, whose relation is the same as that in Sim.3. However, theorder of Eq.(13) in Sim.3 is linear (i.e., q = 1 ), whereas it is quadratic in Sim.6. In contrast with the results of Sim.3, thistripartite deadlock relation at an initial point in time, dissolves as time proceeds, as can be seen in Fig.19. On the one hand,self-similarities of members no longer increase as time proceeds in marked contrast to those shown in Fig.9 of Sim.3. On theother hand, angles from A to B as well as B to C converge to π, while the angle from C to A converges to 0. Such features canbe seen in the orbits of these members illustrated in Fig.22. It will be interesting to note that in this ﬁgure locations of the threemembers seem to approach to a line.
It should be noticed that the initial configuration in this simulation is the same as that in Sim.3. Nevertheless, theconﬁguration at iteration 100 is completely different from that, as shown in Fig.19. Moreover, self-similarities of the triad
no longer increase monotonically, and after earlier iterations they begin to decrease and then approach to some constants
asymptotically, as depicted in Fig.20. Furthermore, orbits of the triad are no longer linear, as shown in Fig.22.

Figure 19: Locomotion of a tripartite in a unidimensional

Figure 20: Changes in self-similarities of the tripartite

Figure 20: Changes in self-similarities of the tripartite
Hilbelt space
Figure 19: Locomotionof a tripartite in a unidimensional Hilbelt space

1
B1

imaginary axis

0.5

A

1

0

•

-0.5

C

1

-1

-1

Figure 21: Changes in angles among the tripartite

-0.5

0

real axis

0.5

1

Figure 22: Orbits of the tripartite in simulation 6

Figure 21: Changes in angles among the tripartite─ 43 ─

Figure 22: Orbits of the tripartite in simulation 6

Naohito Chino

Simulation 7 : dyad relation, const=0.01i, N= 2 , n=1000, p=q= 1 , b=c= 1 /50
In Sim.7, we examined change in a dyadic relation, whose initial conﬁguration is the same as that shown in Sim.1 and 5 ,and whose system is linear. However, this system is completely different from those in these simulations in that it includes acomplex constant term, 0.01i, as described in Eq. (15). In this case, a striking feature emerges in change in the angle between
the two members at around 100th iteration, as can be seen in Fig.24c. That is, the angle begins to oscillate for a whole, and thenit behaves like a damped oscillator (see, for exmaple, Thompson & Stewart, 1986). Such oscillations can also be observed in
Fig.25, which shows the orbits of the dyad in Sim.7.

Figure 23: Locomotion of a dyad relation in a unidimensional

Figure 24: Changes in self-similarities of the dyad (24a, b)

Figure 24:Hilbelt
Changesspacein self-similarities
space relation in a unidimensionaland changeofin thethe dyadangle (24a,(24c) b) and change in the angle (24c)Figure 23: Locomotion Hilbelt
of a dyad
60

50

40

imaginary axis

30

20

10

0

B
• 1

A

0

1

1

-10

-20

-30

-40

-4

-3

-2

-1

real axis

2

3

4

5

Figure 25: Orbits of the dyad in simulation 7

Figure 25: Orbits of the dyad in simulation 7

─ 44 ─

A simulation study of a Hilbert state space model

Simulation 8 : triad relation, const=0.0001i, N= 3 , n=20,000, p=q= 1 , b=c= 1 /50
In Sim.8, we examined change in a triad relation which starts with the same conﬁguration in Sim.2. Although the system
is linear as in Sim.2, it has a complex constant, 0.0001i, in Eq. (15). In this case, self-similarities increase after some point in
time, and angles among three members exhibit very curious behaviors, as can be inferred from Fig.28. In all of the three ﬁgures
in Fig.28, there seems to be complex oscillations in angle. Moreover, the orbits of the triad depicted in Fig.29 show interestingand striking behaviors during 20,000 iterations.

Figure 26: Locomotion of a triad relation in a unidimensional

Figure 27: Changes in self-similarities of the triad

Figure 27: Changes in self-similarities of the triad
space relation in a unidimensional Hilbelt space
Figure 26: Locomotion Hilbelt
of a triad
8

7

6

imaginary axis

5

4

3

2

1
B

1

•

0

A

1

C

1

-1
-0.6

Figure 28: Changes in angles among the triad

-0.4

-0.2

0

0.2

0.4

real axis

0.6

0.8

1

Figure 29: Orbits of the triad in simulation 8

Figure 28: Changes in angles among the triad

Figure 29: Orbits of the triad in simulation 8

─ 45 ─

1.2

Naohito Chino

Simulation 9 : triad relation, const=0.0001i, N= 3 , n=8600, p= 1 , q= 2 , b=c= 1 /50
In Sim.9, we examined change in a triad relation in which two of them (members, B and C) like one another but a third(member A) dislikes them both at the initial conﬁguration. Such interpersonal relationships is symmetric. This system includes
a complex constant term (0.0001i) and is quadratic. Self-similarities of this system exhibit interesting behaviors as timeproceeds, and after long iterations they seem to diverge. As in Sim.8, angles among the three members oscillate in various
manners, as shown in Fig.32. Orbits shown in Fig.33 are also reminiscent of such oscillations.

Figure 30: Locomotion of a triad relation in a unidimensional

Figure 31: Changes in self-similarities of the triad

Figure 31: Changes in self-similarities of the triad
space relation in a unidimensional Hilbelt space
Figure 30: Locomotion Hilbelt
of a triad
20

15

10

imaginary axis

5

B
1
C •

0

1

A

1

-5

-10

-15
-10

Figure 32: Changes in angles among the triad

-8

-6

-4

-2

0

real axis

2

4

6

8

Figure 33: Orbits of the triad in simulation 9

Figure 32: Changes in angles among the triad

Figure 33: Orbits of the triad in simulation 9

─ 46 ─

10心身科学 第 7 巻第 1 号（31┉47）（2015）

A simulation study of a Hilbert state space model forchanges in afﬁnities among members in informal groups
千野

直仁

要約：
この論文は，人間行動への数理の応用による課題解決に関するワークショップの予稿集論文の補遺についての修正版である．その予稿集論文の補遺において，我々は予稿集論文を完全なものにするために，論文の動機，表と図，などを追加した．図に描かれたいろいろな解曲線は，我々の複素差分方
程式モデルが非公式集団の成員間の親近度関係の形成や崩壊に対する膨大な可能なシナリオをカバーすることを示唆している．もとのモデル（Chino, 2002）は有限次元ヒルベルト空間のみならず不定計量空間を状態空間として仮定しているので，このモデルは対象間の非対称な関係が本質的であるところのすべての種類の現象に対する幅広い適用可能性を持つであろう．補遺においては，われわれのモデルに定数の攪乱項が加わった拡張版も提案されている．この攪乱項が可能なシナリオを興味深くかつ劇的に豊かにすることは明白である．我々のモデルと現存する複素差分方程式モデルとの差異についても議論される．最後に，我々は未だ解明されていない幾つかの問題について考察する．
キーワード：有限次元複素ヒルベルト空間，不定計量空間，複素差分方程式モデル，縦断的非対称関係データ行列，n- 体問題，千野・白岩の定理，三すくみ，分岐理論，安定性問題，正則関数

─ 47 ─