A dynamical model of social interaction

The

_JbPa"ese

_Jb"rnat

of

Rsychonomic Science

1987,

Vol.

6,No. 1,11-oo

A

dynamical

model

of

social

interaction

Masanori

NAKAGAWA

Hbkkaido

Ubeiversitv

and

Naohito

CHiNo

Aichigakuin

l:1)tiversity

The

present

study proposes a mathematical model

for

the

dynamic

change of social

inter

action

between

two

persons.

Even

though

social

interaction

has

essentially a

dynamic

aspect which varies through time and acress situations,

the

traditionat

_psychology

has

never

succeeded

in

treating

this

dynamic

change.

First,

we assume sorne

hypotheses

on

basic

rules

for

the

dynamic

change of social

interaction.

Then

those

hypotheses

are

integrated

into

a

simple system of

differential

equations; a

dynamical

system,

Each

coeficient of terms

in

the system represents

personalities

of two

_persons,

ln

accordance with various combinations

of values of

the

coeMcients, social

intera

¢

tion

between

the

two

persons

brings

out a variety

oi

dynarnic

_phases

which are represented

in

the

phase space of

the

dynamical

system.

thermore, a concept of

dynamic

personalities,

which vary according to situations,

is

defined

using a non-linear

dynamical

system.

Finally,

the

relation

between

the

present

model･and

the method which constructs a

dynamical

systern using an

MDS,

is

cliscussed.

Key

words: social

interactions,

dynarnic

rnodels,

dynarnical

systems,

differential

equations,

dynamic

personalities.

I.

Introduction

and

basic

hypotheses

'

Social

interaction

has

essentially

a

dynamic

aspect which

varies

through

time

and across

multifarious situations.

We

often see or

hear

that

a

_quarrel

arises

suddenly

among

friendly

fellows,

or

that

a

pair

who

have

been

living

cat-and-dog

life

falls

in

love

with each other unexpectedly.

Such

stories

are

too

trite

to

be

told

in

a

fan-ciful novel

if

not

for

a

suitable'adaptation.

In

the

traditional

_psychology,

however,

there

has

been

lacking

a suitable model which can

tell

such

dynamic

stories

of

social'

interactionl

The

present

study

_proposes

a

model which

describe's

these

dynamic

changes

of

social

interaction.

Now

let

us

formulate

some

hypetheses

below,

in

which

dynamic

changes

of

social

interactions

between

two

_persons

could

be

explicated,

at

least,

theoretically.

First

we

define

some

basic

functions

each

of

which

denotes

an

intensity

of

attitude

of

one

person

toward

the

other;

Hypothesis

I

Let

the

symbols

X

and

Y

denote

two

per-sons,

and

let

the

function

x(t) represent

the

intensity

of

attitude

of

X

toward

Y

at

time

t.

Thus,

_y(t)

stands

for

that

of

Y

toward

X

at,

time

t.

Here,

each of

the

functions

x

and

y

is

continuous,

having

values which are

pos-itive

or

negative.

If

the

value of x or

_y

iS

pes-itive,

the

attitude

of

X

toward

Y

at

time

t,

or

that

of

Ytoward

Xat

t

is

_positive.

Conversely

negative

values of

these

functions

mean a

negative

attitude

of

X

toward

Y,

and

that

of

Y

toward

X

at

time

t.

Furtherrnore,

the

greater

the

absolute

value

of

each

function

becomes,

the

stronger

the

positive

or

negative

attitude

of

one

person

toward

the

other

at

the

time.

Next

we

introduce

a

_natural

hypothesis

on

the

interaction

between

the

two

_persons

in

terms

of

the

change

of

their

attitudes

toward

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_Journal

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1

Hypothesis

II

The

attitude

of

X

toward

Y

at

time

t

in-fluences

the

change of attitude

_of

Y

_toward

X

at

time

t.

The

attitude

of

Y

toward

X

at

t

also simultaneously

influences

_the

change

of

attitude of

X

toward

Y

at

the

time.

Further

we a$sume

_that

_the

_attitude

of one

person

at

time

t

influences

the

change of

his

own attitude,

This

leads

_to

_the

_third

hypoth-esis.

Hypothesis

III

The

attitude

of

X

toward

Y

at

t

infiuences

the

change of attitude of

X

toward

Y

at

time

t.

In

the

same way,

_the

_attitude

of

Y

toward

X

at

t

influences

the

change ef attitude

_of

Y

toward

X

at

t.

Hypothesis

III

may

_require

further

explana-tions.

Our

emotions are

_sometimes

accel-erated

by

_our

_own

behavior.

_It

causes

lovers

even

more

_pain

to

say

_good-bye

on

_the

morn-ing

after.

In

the

same

way,

our

_passions

are

sometimes reduced

by

our affective reactions

as

is

asserted

in

the

two

factors

theory

of

the

emotional systems.

Hypothesis

III

_generalizes

those

self-reaction system within

_{psychological}

functions

using

the

convenient

_term,

`

attitude '.

A

further

implication

of

Hypothesis

III

_will

be

detailed

in

the

following

section.

Now

we

come

to

integrate]

_the

_preceding

three'

hypotheses

into

one

sirnple

system,

Hypothesis

IV

Hypotheses

II

and

III

can

be

_represented

by

a

system of

differential

equations,

i.e.,

a

dynamical

system as shown

below,

using

the

functions

defined

in

Hypothesis

I.

li

'

iil;

'

:,

i

l.

"

:,

'

g.

'

:;}

(1,)

Each

term

in

the

system

has

a

_{psychological}

meaning

as

is

explained

in

the

next

section.

II.

The

meaaning of

the

model

Since

the

left-hand

terms

in

the

system

represent

change rates

of

attitudes

at

time

t,

each

differential

equation means

that

both

one's own attitude and

the

attitude

of

the

other at

time

_t

simultaneously

infiuence

the

change rate of one's own attitude.

The

term

aix

in

equation

(1)

denotes

the

influence

of

X's

attitude

toward

YL

If

the

infiuence

of

the

term

biy

is

negligible

(that

is,

bi==O),

and ai

is

_positive,

then

the

term

aix

has

certain

influences

_on

_the

change rate

of

X's

attitude.

That

is,

if

x

is

_positive,

aix

becomes

positive,

and

_therefore

_dxtdt

becomes

positive.

This

means x

has

a

_tendency

_to

increase,

Then,

X's

attitude

becomes

more

and more

_positive,

with a snowballing effect.

On

the

contrary, a negative x

makes

aix and

dx!dt

negative, and as a result, x

tends

to

decrease.

In

this

case,

_a

negative attitude of

X

accelerates

itself.

Summing

up, a

_positive

ai

has

_the

_effect

of self-acceleration of

X's

attitude.

Conversely,

a negative ai

has

an

_effect

_of

self-inhibition of

X's

_attitude.

_With

_a

_negative

ai, negative x makes

_aix

_positive

_and

a

posi-tive

x makes

it

negative.

Then

_a

_positive

_x

yields

negative

dxldt

and a negative x a

pos-itive

dx!dt.

That

is,

with a

_positive

_x,

_it

_is

inhibited

to

decrease,

while

with

_a

_negative

x,

it

is

inhibited

to

increase.

Considering

those

characteristics, we can

lnterpret ai

as

a representation

of

one aspect of

X's

character,

i.e.,

the

self-oriented

_property.

From

this

interpretation,

_positive

ai means

X

has

a self-accelerating character.

This

could

be

_paraphrased

as

follows:

when

he

behaves

kindly

to

Y

at

the

beginning,

his

favor

for

Y

becomes

greater

and

_greater.

On

_the

other

hand,

his

initial

malice,

to

Y

accelerates

_the

negative

_attitude

of

X

to

YL

For

negative ai,

X's

character

is

self-inhibitory.

_If

_X

_is

self-inhibitory,

X's

extremely negative attitude

is

inhibited,

and

a

too

_great

faver

of

X

for

Y

is

likewise

inhibited.

The

parameter

a2

in

the

second equation

_of

the

system

has

the

same characteristics as

discussed

above

on

ai.

The

a2 stands

for

a

self-oriented

trait

of

Y's

_personality.

The

pararneter

bi

in

the

first

equation

also

represents a

kind

of

X's

character.

The

term,

biy

in

the

equation

denotes

the

influence

of

Y's

attitude

on

the

rate of

the

change

_of

X's

M.

Nakagawa

and

N.

Chino:

A

dynamical

model of social

interaction

Table

1.

Four

basic

personalities

constru ¢ted

by

the combination of

self-oriented

_properties

(self-inhibitory

or self-accelerating), and

others-oriented

_properties

(normal

or

_perverse).

I " t

M_xx. I l t

self- l

I

I

HLx oriented: seLf-inhibitory

l

seLf-aceelerating :

others-X's. I (a<O) l (a>O) I

Oriented Ssxl l :

--"--t---)NTx-xl--tttt-te---t---T:--J---f--t--J---1

1 1 1 1 1 1

l the one vho

Lnhiblts

l

the

one who is emotiona] l

nermal l one's own emotion and

I

and reaets normally

to

l

(b>O) :reacts normalty

to

theIthe

other

I

Iother

:

:

1 d b

T---1-"---"---"f-r---"----4m---tT----t---

---

)

1 t t

S t 1

l the one vho inhibits I

the

one whe

is

emotionat

1

perverse l one'$ ovn emotion and

t

and reacts adverseLy to : Cb<O) Ireaets adverse]y

to

thelthe

ether t

lother

l l 1 1 d

-m----m---"-T--tTl

"---"- ---v---1- ----"-L--JtJ-J----J--Tt--ml

13

;.:

:

:

l

l

1

1

, I :

l

l

: : l

I

l

i,e.,

ai equals

O,

with a

_positive

b,.

If

_y

is

pos-itive,

x

increases

because

dx!dt

gets

positive,

and

negative

y

makes x

decrease.

In

contrast,

it

follows

from

a negative

bi

that

a negative

_y

causes an

increase

in

x, and

_positive

_y

brings

a

decrease

in

x.

From

the

above

interpretation

of

the

para-meter

bi,

it

follows

that.

under

the

_positive

bi,

X's

attitude

becomes

better

when

Y's

attitude

toward

X

is

_positive,

and

X's

attitude

_gets

worse

when

Y's

attitude

toward

Xis

also

bad.

Positive

br

indicates

that

X

has

an

ordinal

character with which

he

reacts straightly

ac-cording

_to

Y's

attitude.

Conversely,

for

the

negative

bi,

X

reacts

adversely.

His

attitude

gets

worse when

Y's

attitude

is

_positive,

and

gets

better

when

Y

reacts negatively.

He

answers

favor

with malice, and malice

with

favor,

It

sounds rather unnatural,

but

might

be

more comprehensible

by

interpreting

that

X

is

a

mean

_person

who

_gets

arrogant

when

the

partner

behaves

gently,

but

becomes

servile

when

his

_partner

is

haughty.

Sumrning

up

bi's

characteristics,

it

could

be

concluded

that

this

parameter

shows an

others-oriented

_part

of

X's

_personality.

The

para-meter

b2

in

the

second equation

is

also

thought

to

be

the

others-oriented

facet

of

Y's

person-ality.

Thus

far,

we

have

interp'reted

two

kinds

of

parameters,

one

denoting

a self-oriented

_part

of a

_person's

_personality

with

respect

to

human

relation, and

the

other representing an

others-oriented

character.

Combining

those

two

_parameters

that

can

take

various

values,

a

variety

of

_per$onality

styles can

be

obtained,

some of which

are

shown

in

Table

1.

Finally,

the

constants of

the

system, ci and

c2, are

interpreted

as

_expressiens

_of

_one's

basic

attitudes, which

are

independent

of

parameters,

ai, a2,

bi

and

b2

stated above.

The

next

paragraph

will

discuss

what

changes of

interaction

are caused

by

the

combination of

two

_persons

who

have

their

own original

personalities

respectively

defined

with

_parameter

values

in

the

model.

IIL

What

will

happen

between

a

_pair

?

a.

Phase

of a

dynamical

system

Before

treating

changes of

interactions

ac-cording

to

the

model,

_phases

of

the

dynamical

system representing

the

model must

be

dis-cussed.

Let

(x(t),

y(t))

denote

the

coordinates of a

point

on

a

_plane

at

time

t,

which

obey

the

dynamical

law

described

by

equation

(1).

The

orbit

which

the

_point

draws

along

the

time

formsacurve

on

the

plane

as a

solution

of

the

system.

Generally

this

curve

is

called a solutlon curve, and

the

_plane

is

called

the

_phase

space,

Depending

on

initial

_positions

from,

which

these

curves

start at

tirne

to,

there

are

infinite

solutien

curves

in

the

_phase

space.

The

forms

of

solution curves of

the

system

are

determined

only

by,parameters

of

the

system,

In

other

words,

_phases

of

the

solutions

can

be

classi-fied

by

_parameter

values of

the

system,

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dynamical

system

like

the

_present

model,

forms

of solution curves can

be

classified

without

diraculty

using

the

simple

formulas

stated

below,

where

di,

a2,

bi,

and

b2

are

the

parameters

in

the

model system:

P

=

-(ai+aa)

q=aiaa-bib2

'

D=P2

-4q

_.

When

p,

q>O

and

D<O,

the

solution curves

form

a spiral which converges asymptotically

to

a central

_point

as shown

in

Fig,

1-A.

On

the

other

hand,

when

_q>O

and

P,

D<O,

the

solution curves

draw

an outgoing spiral which

starts

from

a central

_point

and

_goes

far

away

as

shown

in

Fig.

1-B.

In

each case,

the

' '

A

I

B

' x

D

'

Fig.

1,

Various

patterns

of orbits

in

the

phase space of

the

dependi,ng

upon values ofthe parameters,

Fig.

A

corresponds to

spirals,

C

stable nodes,

D

unstable nodes,

E

t972).

F

s

linear

dynamical

system,

stable spirals,

B

unstable

M.

Nakagawa

and

N.

Chino:

A

central

_point

is

called

the

singurar

_point,

at

which

both

derivatives

in

_the

system vanish.

Thus,

classification of

forms

of solution curves

is

usually called

the

classification of singular

points.

Coordinates

of a singtdar

_point

are

given

by

the

formulas

below.

(:iCa2,EIi/ab2,Cbr,,

22,Cai,-rabi,Cb2,)

(2)

(For'convenience

all

singular

_peints

are

reset

to

the

origin

in

Figs.

1-A,

-B,

-C,

-D,

-E,

and

-F.)

If

D>O

and

_a>O,

the

singular

_point

is

called

the

node

because

solution curves

look

tied

together

at

this

_point

(see

Fig.

1-C

and

-D).

Moreover,

in

this

case, with a

_positive

P,

solution

curves

are

converging on

the

node

as

shown

in

Fig.

1-C,

while with a negative

P

these

curves are

diverging

from

the

node

(see

Fig.

1-D).

For

both

spirals

and

nodes,

singular

_points

are called sinks since solution curves are

verging

thereon.

On

the

other

hand,

singular

points

of

diverging

curves

are

called sources

because

the

curves are reminiscent of

hot

sprmgs.

For

any sinks, even

though

the

starting

point

is

drifting

from

the

singular

_point

by

some

noises,

the

curve comes

back

into

the

sink afVer all.

In

this

case,

the

singular

_point

Table

2.

The

classification of

system

in

accordance with

meters

in

the system.

T---"---L""-"-"--L---T

dynamical

rnodel of social

interaction

15

is

described

as

being

stable.

Conversely,

in

the

case of sources,

the

singular

_point

is

scribed as

being

unstable

because

the

system

is

so unstable

as

to

_get

down

from

the

source

point

with a small noise.

Unstable

cases are

not restricted

_to

sources,

but

arise also

in

the

case of negative

_q.

Singular

_points

of

this

kind

are called saddle

_points,

because

those

solution curves

form

lines

just

as

little

balls

draw

when

they

roll

down

the

saddle

of

a

horse

(see

Fig.

1-E).

There

is,

however,

a special case

in

which

q

just

equals

O;

then

solution curves

draw

'

Fig.

2.

The

classification of $lngular

points

of

the

11'near

dynarnical

system, according to the

relation of

_the

_parameter$

in

the system,

where

P

stands

for

-(ai+a2),

q

denotes

aia2

-bib2･

(From

Simmons,

1972).

'

dynamic

status of

the

model

the

relation

between

para-:::}:::::::::::t::]::::::::

-

x

.-

i s.tt-q : p

'h"

:

sxs.Ml

:

al + a2 <Ol : Sink :

i

l

el + a2 =Di

u"-u---1

I I : ai + a2 >O!

:

Sour:e

I

l

l

ala2 ) blb2 : l SLable 2,

nblb2 > -tal - a2)

:

d Node

ur---H--UJ-urU-UU"L..:tT---TT

: : Stable 2,

4blb2 <

-(al

-

a2) , : Spiral :

Centers (O > ala2 > blb2) :::l-1::::::::::lt:::::::lll nla2 < blb2 :l Unstable :l Node ::l Unstabte :: Splral ]m---i---Unstable SadtiLe 4blb2 )

-cal

-

a2)2 4blb2 <

rcal

-

a2)2 ::::::::It:::::::::::::::::

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VoL

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1

ellipses,

the

singular

_point

being

a

center, as

shown

in

Fig.

1-F.

The

singular

_points

are

naturally

called centers.

Generally

both

of

sinks and centers are stable, while

sinks

are

specifically called asymptotically stable.

All

are

summed

up

in

Fig.

2.

In

Table2

every case

is

classified using

formulas

consist-lng

of

_parameters

in

the

model system,

i.e.,

ai, a2,

bi,

and

ba,

according

to

the

signs of

_p,

q

and

D.

b.

[IIhe

dynamic

phase

of social

interaction

Social

interaction

between

two

persons

can

be

classlfied

theoretically

in

terms

of

the

para-meters

_of

_the

_system,

using

the

model

re-presented

by

the

dynamical

system

and

the

classification

of

singular

_points

of

the

system

discussed

so

far.

In

addition, remembering

that

those

_parameters

of

the

system

imply

the

_{personalities}

of

each

_person,

social

inter-action

between

the

two

can

be

classified

locally

according

to

the

combination of

their

person-alities.

For

example,

suppose

that

X

has

a

self-inhibitery

_{personality(i.e.,}

a,<O), and

that

his

others-oriented

personality

is

ordinal

(b,>O).

Suppose

further

that

he

is

more

influenced

by

his

own

attitude

than

by

the

attitude of others

(lail>bi),

and

iurthermore

that

Y

has

the

same

_personality

as

X

(a2<O,

b2>O,

and

la21>

b2).

Then

the

system

has

the

_phase

of

a

stable

node

because

aiae>btb2, at+a2<O, and

4bib2>O>-(ai-at)2

(see

Table

2).

This

means

that

the

interactlon

between

the

two

_person

in

question

gradually

stabilizes along

the

time

axis, whatever relation

they

may

have

had

at

first.

This

is

true

especially,

if

both

of

them

have

positive

basic

attitudes

(ci,c2>O),

i.e.,

if

both

are

_gentle

basicalry,

they

will

have

a

good

relationship

after

all,

even

if

they

once

opposed

each

other

due

to

somebody's

slander-ous

activities.

If

one, not

both

of

them,

is

eontrary-mincled

(i.e.

bi<e

or

b2<O

and

btb2<O)

without

changing

the

other

conditions assumed above,

the

change

of

the

interactien

between

them

is

not

so slmple

though

they

finally

reach a

good

relation.

In

this

case,

even

if

they

favored

each other at one

time,

a

quarrel

arises

between

them

at a

later

tirne.

For

the

orbit

implying

the

change

of

the

interaction

draws

a spiral converging

to

a

$ingular

_point.

When

they

have

a self-accelerating character

(i.e.,

ai>O

and

a2>O),

the

interaction

between

them

is

unstable and

_gets

into

infinite

diver-gences,

whether

for

_good

or

for

evil.

This

is

true

even

though

both

of

X

and

Yare

self-dependent

(fail,/a2f>Xbil,/b2D.

Assuming

that

either an extremely negative or extremely

positive

attitude

results

in

the

ruination

of

both

of

them,

those

divergences

of

the

inter-action may

be

interpreted

as

the

fate

of

persons.

In

our

daily

lives,

the

break-up

ef

relationships

is

brought

on

not only

by

malice

but

also

by

too

immoderate

kindness.

Under

the

above conditions

(that

is,

se!f-dependent

and

self-accelerating),

and

it

both

of

their

others-oriented

_{personalities}

are

nega-tive

or

_positive

_(b,b2>O),

the

interaction

di-verges

diametrically.

That

is

to

say,

they

head

straight

for

the

ruin

(because

biba>O>

-(at-a2)2).

_But,

when only one of

them

has

a contrary-minded character

(bib2<O)

and

their

self-oriented

_{personalities}

are equal

to

each

other

(ai=at),

the

fate

comes

after

some

twists

and

turns

(because

bib2<O=-(a!-a2)Z).

In

the

same way, we can

imagine

variety'

of

the

fate

and

the

fortune

of

_persons

assum-ed

in

the

_present

model.

Some

of

them

would

have

happyendings

and some

tragic,

depending

upon

_{personalities}

of

_persons

represented

by

the

_parameters

in

the

model system,

Typical

combinations of

_{personalities}

and results are

summarized

in

Table

3.

IV.

Further

development

of

the

model a.

Nonlinear

model

Although

the

results

discussed

in

the

pre-ceding chapter seem somewhat

interesting,

some

_problems

remain

to

be

considered

in

the

model

from

an experiential

_point

of

view.

For

example,

it

is

not

natural

to

suppose,

whether

for

a self-inhibited

_personality

or a

self-accelerating

one,

that

those

_{personalities}

are

invariant

along

time

or

across

situations

within a

_person.

It

should

be

noted

that

in-hibition

becomes

strong

for

too

immoderate

kindness

or malice, and

that

acceleration comes

up

for

obscure attitudes.

In

other words,

parameters

ar

and

a2, which represent

self-oriented

_{personalities,}

are not constants

but

M,

Nakagawa

and

N.

Chino:

A

dynamical

model of social

interaction

Table

3.

0ne

example of

the

classification of

dynamic

status ofsocial

interaction

predicted

by

the present model, according

to

a combination of per$onalities

definecl

in

Table

1.

1:::I:l:tt:::`::::･::::::r'

17

b i

1 t : : self-oriented persenalities r l are sperior

-

/

1 :

1 1 -b L ; l other,s-oriented : Stable l both are : persenatjties are equaa l Nede

1 1 1

t 1 l

: self-inh ±bitory : : : : others-oriented : Stable

: apersenalities are net equat: Spirai

t i

i 1

t d

i t 1

- i

i one self-inhibttery. ± cyclic variation

I other self-accelerated l

1

₁ L L

b

1

b : : : : : others-oriented SUnstab!e

: both are : persenalittes are equal t Node

I : r

l : t

t self-accelerated : others-briented rUnstable

! :personalities are not equal: Sptral

Itt----J--'t---t--ttt--rT"-tL't--uL-LL---+'tJ-t----+---of attitudes, x and

_y,

respectively.

For

a

similar

reason,

_parameters

bi

and

b2

should

not

be

constants

but

functions

of

the

intensity

of each attitude respectively.

Thus

a system

of

differential

equations

is

formulated

as

be-low

: :::::::::::t'ts::t'1I::::: others-ertented are superier Unstable Saddle

dx

:

d

l:ll[Iii;,

g

j,

(

gi.

v

::i

,,,

where

fi,

f2

and

_gr,

gz

are mathematical

func-tions

that

represent

the

change of

_{personalities}

'in

_accordance

_{with attitudes of each}

persons.

Here,

_fi

and

f2

denote

the

changes

of

self-briented

_{personalities}

of

Xand

Y;

respectively,

while

_gi

and

_g2

stand

for

changes of

others-oriented

_{personalities.}

For

example,

let

us suppose

that

fi

has

the

character shown

in

Fig.

3.

Then,

thefunction

fi

has

a negative value

against

a

large

ab-solute

value

of

X's

attitude, and a

_positive

value against

a

relatively

small value of

the

X's

attitude.

That

is,

Xbecomes

self-inhibitory

when

his

attitude

.gets

either

too

negative or

too

_positive.

Such

an

assumption

represented

by

the

function

shown

in

Fig.

3

seerns

con-vincing when

we

remember

the

hypothesis

that

too

strong

_positive

or negative

attitude

results

in

the

fatal

ruin

ef

_persons.

The

system

described

by

equation

(3)

is

_general!y

tl

x

Fig.

3.

The

graph

of

the

iunction

fl

which

.

presents

the

non-linear relation

between

the

self-oriented

_property

of

X's

_personality

and

X's

attitude

toward

Y.

nonlinear.

It

will

be

natural

to

suppose

a

situation

in

which

the

self-oriented

_{personalities}

are

in-fltienced

not

only

by

the

self attitude

but

also

by

the

_partner's

attitude.

Such

an

influence

may

have

an

interactive

effect

on

the

attitude

of

both

_persons

toward

each other.

Then

we

must

represent

the

influence

of

the

attitudes

on

the

change of each

_persQn's

a･ttitude

as

general

nonlinear

'functions

with

two

terms

The Japanese Psychonomic Society

NII-Electronic Library Service

The JapanesePsychonomic Society

18

The

_Japanese

_Journal

of

_.

/ii:g,:lzl:i1

(4,

where

functions

F

and

G

denote

the

total

in-fluences

of

the

attitude of

_two

_persons

on

the

changes

_of

_each

_person's

_attitude,

_{respectively,}

while a and

b

indicate

_parameter

_vectors.

Generally

system

(4)

has

_plural

_singular

_points,

which

_are

_given

as

_points

at

which

both

F

and

G

are

equal

to

O.

The

initial

attitudes

of

both

_persons

will

determine

the

singular

polnt

at which

_the

sy$tem

will

converge

(where

at

least

some

_of

_singular

_points

are

assumed

to

be

stable).

In

the

neighborhood of each singular

_point,

the

system

can

be

approximated

by

_the

for-mula

below:

dx

0F

OF

dt

= .ax

(Xo,Yo)x+

oy

(xoTyo)y+hi(x,y)

1

ddyt

=

gGx

(x,,y,)x+

ti.Gy

(.,,y,)y+h,(x,y)

J

(5)

where

(xo,yo)

denotes

_the

coordinates of a

singular

_point,

The

matrix consisting of

the

_parameters

of

the

linear

part

in

the

formula

is

_generally

called

the

Jacobian

matrix

as shown

below:

(;

･

E

･

[::::i

Z.i

･

[i:

･

i:])

,,,

From

the

above

formula,

it

follows

_that

_each

singular

_point

is

characterized

locally

as

if

it

were

the

singular

_point

_of

_a

linear

system

represented

as

follows,

when any real

_parts

of

eigenvalues

are not equal

to

O.

if-

li

F

pt

" .]::i::

'

l.;

'

ii:l;:]:

i

(7)

That

is

to

say,

each

singular

_point

can

be

classified using

the

Jacobian

matrix under

the

above condition.

In

the

neighborhood of each singular

_point,

each value of

Jacobian

terms

in

the

formula

Psvchonomic

Science

Vol,

6,

No.

1

<6)

is

interpreted

as

deneting

_each

_personality

of each

_person

discussed

earlier

in

the

linear

model.

However,

_the

values of

Jacobian

terms

are

determined

individually

by

the

_position

of

each

singular

_point,

Furtherrnore,

even

though

the

system

is

approximated

by

_the

linear

system

whose

_parameters

are

Jacebian

_terms

in

the

neighborhood

of an

ordinal

_point,

the

Jacobian

matrix

changes

in

accordance with

the

change

of

coordinates of

_points,

i.e.,

per-sons'

_attitudes.

In

other words,

in

the

case

of

the

_present

model represented

by

system

(3),

the

tendency

of

each

_person's

_personality

depends

on

his

own attitude change

and

that

of

the

_partner's,

_while

_the

_tendency

_in

_the

linear

model

is

independent

of changes of

attitudes.

For

_the

_present

model,

at

least,

personalitie$

each

have

individual

values at

each

individual

_position

in

the

_phase

space.

This

means,

_again

_in

_the

_present

medel, we

assume

a

dynarnic

_personality

which

has

_the

possibility

of change according

to

the

situation

in

which

the

_person

is

involved.

It

follows

that

interactions

between

the

･personalities

of

persons

and

the

situations around

them

are

modeled at

Ieast

theoretically

in

the

non-linear

system.

Generally

those

situations

involving

_persens

do

not consist only of

_persons'

attitudes,

btit

of other

factors

arso,

_such

_as

_their

social or

physical

environment and

their

own

psycho-logical

state.

Taking

account of such

factors,

we will substitute constant

_parameter

vectors

a,

b

in

the

system

by

_general

vector

fupctions

of

time,

a(t),

b(t)

which represent changes

of

situations along

time.

Thissubstitution

gener-ates

the

formula

below:

ll

X

l;g,zx

･

,::;::}

(s)

where at

least

_a

_term

of vectors a(t) or

b(t)

can

be

t

itself.

The

system represented

by

the

formula

₍₈₎

is

the

most

_general

dynamic

model of social

interactions

in

the

case of

two

_persons.

For

the

system,

the

_positions

and

the

nurnber of

singular

_points

in

the

_phase

space are not

constant,

but

they

change

in

accordance

_with

M.

Nakagawa

and

N.

Chino:

A

Changes

ef a(t) or

b(t)

sometimes cause

drastic

changes of

the

_positions

and

_the

number of

singular

_points

(these

drastic

changes are

called

bifurcations).

This

means

that

some

factors

cause

drastic

changes of

the

interaction

between

two

_persons,

even

if

the

interaction

was

stable

in

the

first

stage.

b.

N

persons

case

Here,

we

will

discuss

a

more

_general

case

in

which

the

number of

_persons

is

_given

by

n(>2).

Construction

of

the

model

in

_this

case

is

not very

dithcult,

at

least,

theoretically.

Now,

let

Xi,

Xh,

X3+･･Xh

denote

n

_persons

and

xiy(t)

denote

values of

the

attitude

_of

Xl

toward

.X]

at

time

t(i,J'--1,2･･･n).

Then

a

general

dynamic

model of social

interactions

among n

_persons

is

_given

by

the

system

be-low

:

dxie

=fl(V12t _X13]

'

'

't

_{Mn,n-l, al(t))}

dt,

)

dcXiti3

=f2(xiz, _xi3, ''', xn,n-i, a2(t))

1

(g)

I

I

T

dxnn-i

J

=fN(X12, _Xle,

'

'

',Xn,

n-1, aN(t))

dt

where

functions

fte

and vector

function

aic(t)

(fe=1,2,･･-,M=n(n-1)})

are

_generally

non-linear.

However,

it

is,not

as easy

to

analyze

_the

present

dynarnical

system as

it

is

to

merely

construct

the

modeL

It

is

very

dithcult

even

to

imagine

the

_general

variations of solution

curves and singular

_points

in

the

_general

p-dimensional

_phase

space.

'In

_the

present

case,

if

we can

deduce

the

system

into

a more simple

form,

for

example

a

system with a smaller

dimension,

without

missing

the

mathematical

features

of

the

orig-inal

system,

it

will

then

be

_possible

to

do

a

fairly

strict

analysis of

the

system.

Indeed,

for

the

linear

systern,

the

deduction

of

the

dimension

was reported

in

a

study on a

dy-namic

model of social

development,

in

_which

the

_principal

component

analysis

was

applied

to

social

indices

for

countries

of

the

world

(Naka-gawa

&

Ohsawa,

1980).

Ii

another method

can

be

found

for

the

deduction

of

dimension,

multidimensional scaling could

be

used

effec-tively.

dynamical

rnodel of social

interaction

19

V.

Reconstruction

of

the

model

based

upon

_MDS

If

xi,･(t)

in

the

system

(9)

can

be

measured

by

a

relative and nonnegative

_scale,

_such

as

the

ordinary scale

in

sociometric,

the

observed

value of xij(t) on

the

scale can

be

assumed

as an asymmetric relation

between

_persons

XL

and

XY

at

time

t.

Then

these

asymmetric

relations can

be

analyzed

by

asymmetric

MDS

to

construct

the

space

in

which

_every

_person

is

represented

as

a

_point

in

accordance with

the

relationships at

the

time.

If

xij(t) can

be

measured at any

_given

_time,

_the

_points

ing

persons

draw

curves along

time

in

the

space constructed

by

MDS,

using

time

series

observations

of xid(t),

Therefore,

assuming

certain coordinates

in

the

space,

it

is

_possible

to

construct a

dynamical

system

in

which

the

phase

space

is

_the

space constructed

by

MDS

and

the

above

mentioned

curves

are

ed

as

solution

curves of

the

system,

Since

the

dimension

of

the

space

constructed

by

MDS

using observed

data

is

often

not

as

many as

the

sample number,

the

dynamical

system

is

expected

to

have

a smaller

sion

than

the

number

of

_persons,

and

would

be

represented

by

the

following

_general

form.

dy,

(lt

==

Fi

(yl

ya,

''',

Yr,

t)

i

(lo)

dit'

;pi.(yi,

y2,

･･･,

yr,

t)

J

Here

r

is

the

dimension

of

the

system.

_yi(t)

are solution

functions

corresponding

_to

_an

arbitrary coordinates, and

FL

(i=1,･`･,r)

are

nonlinear

functions

of r

terms.

Generally

i;}

have

some

_parameters,

which may

be

functions

of

tirne

in

themselves.

If

the

forms

of

the

functions

are

known

already,

it

is

not so

ficult

to

reduce such

_parameters

_or

_parameter

functions

from

the

given

data

in

such a way

that

the

system satisfies

the

data.

However,

forms

of

_the

functions

_are

usually

not

known

_prior

to

the

analysis.

Even

in

such cases,

if

the

dimension

of

the

system

is

small enough and suthcient

data

are

_given,

it

is

possible

to

reconstruct

the

functions

selves.

Indeed,

using

Newcomb's

sociometric

data

over

16

weeks with

17

students

The Japanese Psychonomic Society

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20

The

Japanese

Journal

of

Psychonomic

systern

is

now

being

researched upon

to

ex-plain

changes

of

the

relationships

_among

persons

along

time

(Chino,

1984;

Chino

&

Nakagawa,

1983).

In

these

researches,

drastic

changes

of

singular

_points

_were

6bserved

as

discussed

in

the

last

chapter,

and

the

changes

of social

interactions

among

_persons

were

analyzed

_{qualitatively}

through

the

_phase

of

bifurcations.

'

VL

Discussion

There

are some

difi

¢ulties

in

_practical

ap-plications

of

the

first

model

which

may

be

too

large

to

allow analysis,

in

_accordance

with

the

nurnber of

_persons

as$umed,

even

though

the

model

can represent

the

concrete relations

between

the

changes

_oi

social

interaction

and

dynamic

personalities

of each

_person.

Con-versely

in

the

last

model,

it

is

_possible

to

re-duce

the

dimension

of

the

system such

that

the

functions

assumed

in

the

system can

be

reconstructed

through

the

estimation

of

the

given

data

using some smoothing methods

for

general

nonlinear

functions,

However,

it

is

not so clear what

_{psychological}

meanings

the

iunctions

or

the

_parameters

have,

though

the

model

has

a

_good

_possibility

of application

to

the

_practical

data

for

the

changes

of

social

interaction,

Taking

account

of

th.e

advantages

and

dis-advantages of each model,

it

is

appropriate

to

make clear

the

relation

between

two

models,

as

well

as

to

deduce

the

_{psychological}

mean-ing

of

inner

mechanism

in

the

last

model,

in

terms

of

the

first

model.

To

our

regret,

it

is

not easy

to

realize

the

above

_goal

_generally,

However,

in

the

last

model,

if

each

solution

function

yi(t)

can

be

represented

as

a vector

function

of

_given

re-Iatienships

between

_persons,

like

_{ut(t)==gt(x),}

where

yi=(yii,yi2,･･･,yiic,･-i,yir),

i

denotes

ith

person,

k

signifies

'kth

coordinate

in

the

last

model, x=(xii,xi2,

･.･,xn-i,n),

then

that

model can

be

translated

into

the

system

below,

which

is

similar

to

the

formula

of

the

first

moclel.

Science

Vol.

6,

No,

1

dxtj

of

dt

=

Eig,

(gi(x)7

gj(x))jF<gi(x))

+

_QElfg,.

(gi(x),

_{gs(x))iFKg,･(x))}

where

tt'

,

'

Y'

=

-'

k

(

,

"

lgY

"

(

'

,,i=,,...,.,.I

(ii)

.

The

_present

translation

_provides

a

suthcient

condition

for

the

description

of

the

last

model

by

the

formula

of

the

first

model,

but

it

does

not

_provide

necessary conditions.

Also,

there

is

no absolute

_proof

for

_the

existence of

func-tions

between

ui(t)

and x(t),

that

is

_gt(x),

as-sumed

there.

Even

if

the

functions

can

be

supposed,

it

is

not

_yet

clear what conditions

are necessary

for

functions

of

the

system

in

the

Iast

model

in

order

to

represent

it

in

the

form

of

the

first

medel.

The

research on

those

conditions

is

one of

the

important

prob-lems

left

for

the

future

as

is

the

theoretical

development

of

the

model and

the

method

_of

its

application

to

the

_practical

data.

Furthermore,

if

persons

in

the

present

model

are substituted

by

nations,

it

is

_possible

to

introduce

a

dynamic

model of social

interaction

between

nations,

like

Richardson's

model of

arms

expenditures

(Richardson,

1939).

Th･en,

we may

be

able

to

discuss

the

various

prob-lems

or

troubles

among

nations

in

the

world

in

terms

of

the

model.

References

Chino,

N.

1984

Toward

a theory of

dynamical

system

in

_group

dynamics.

Working

_paper

in

education and

_psychology

at

Nagoya

University.

Chino,

N.,

&

Nakagawa,

M.

1983

A

vector

field

model

for

sociometric

data.

Paper

presented

at

the

11th

annual meeting of the

Behaviormetr,ic

Society

of

Japan,

Kyoto,

September.

Nakagawa,

M.,

&

Ohsawa,

S.

1980

Construction

of a system

dynamics

model

by

_principal

ponent analysis,

Behattiometriha,

8,

57-73.

Newcomb,

T.M.

1961

The

acquaintance

Process.

New

York

:

Holt,

Rinehart

and

Winston.

Richardson,

L.

F.

1939

Generalized

foreign

_politics.

The

British

lbesrnat

oj"

Ilsycholqg:vT,

Mbnogrmph

StipPlements

No,

23,

Simmons,

G.F.

1972

Difiizrential

eqttations.

New

Yorl{

:

McGraw-HiLl.