Ising Models of Social Impact: the Role of Cumulative Advantage

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Ising Models of Social Impact: the Role of Cumulative Advantage

G. Kohring

To cite this version:

G. Kohring. Ising Models of Social Impact: the Role of Cumulative Advantage. Journal de Physique

I, EDP Sciences, 1996, 6 (2), pp.301-308. �10.1051/jp1:1996150�. �jpa-00247186�

Ising Models of Social Impact: the Role of Cumulative Advantage

G-A- Kohr1~lg (*)

Institute for

Applied Mathematics,

Research Center Jülich

(KFA),

D-52425 Jülich,

Germany

(Received

16 November 1995, received in final form 23 November

1995, accepted

28 November

1995)

PACS.89.90.+n Other _areas of

general

interest to

physicist PACS.05.50.+q

Lattice

theory

and statistics:

Ising problems

PACS.64.60.Cn Order-disorder and statistical mechanics of model systems

Abstract. A statistical mechanical model for Latané's

theory

of social

impact

is discussed and extended to include

learning.

When the

persuasive

and

supportive strengths

_are not

equal,

a

ferromagnetic

and _a

spin-glass phase

_emerge. When the

persuasive

and

supportive strengths

are

equal,

and _a cumulative

advantage learning

scheme _is

used,

then the model exhibits _two

ferromagnetic phases distinguishable by

the structure of the correlations. In the first

phase

the

correlations

are

normally

distributed. In tl~e second

phase

ail elements _are shown to be

highly

correlated with either _a

single

element _{or a} small number of elements. These "leaders" determine the equihbrium state of the system.

1. Introduction

Tl~e _success of reductionism in the natural sciences has led many researcl~ers to attempt ^similar

formulations _m the social sciences

[2-6, _9j.

In

particular, Nowak, Szamrej

and Latané

(NSL)

[2j

attempted

to build _a

simple

model based upon the successful

theory

of social

impact

in human societies first introduced

by

Latané _m 1981

iii.

This

theory

_was based upon substantial

empirical

evidence and has found wide

spread support

within the

psychological community.

In its current

form,

Latané's

theory _suggest

tl~at _a

group's

impact _upon an individual within the _group

depends

_upon tl~ree factors:

1)

^{tl~e attitude} or

opinion

of all members of tl~e group,

2)

the number of individuals _m tl~e group and

3)

the

strength

of the

psychological coupling

between the individuals. Tl~e

psycl~ological coupling depends

_{upon many}

factors, including:

social status,

education,

rl~etorical

abilities, pl~ysical distance,

etc. and it may vary witl~ time.

Altl~ougl~

_one

migl~t expect people's opinions

to vary

continuously

between extreme

values,

tl~ere _{is some} evidence to

suggest

^tl~at

opinion

distributions _on

"important"

issues _are

nearly

bimodal

[2j,

^witl~

peaks

at tl~e extreme values. For tl~is _reason NSL

proposed using

an

Ising type

^model in order to introduce

dynamics

into Latané's otl~erwise static

tl~eory.

In its

simplest form,

tl~e NSL model cl~aracterizes tl~e

psycl~ological strengtl~ by

two

qualities, persuasiveness, P~j,

and support,

S~j.

^{The former describes the}

degree

to wl~ich the

j-th

individual _cari

persuade

tl~e1-th individual to

change

bis opinion and the latter describes the

(* ^{Author for} correspondence: address _as of January 1, 1996: NEC Research Laboratones Europe,

Rathausalle 10, D-53757 Sankt

Augustin,

Germany

(e-mail: [email protected])

©

Les

Éditions

_de

_Physique

₁₉₉₆

302 JOURNAL DE

PHYSIQUE

I N°2

degree

to wl~ich tl~e

j-tl~

individual

supports

^{tl~e1-tl~ individual in} his

opinion.

The

impact

^of

a group witl~ N members _on the1-tl~

individual, I~,

is tl~en

given by:

~ ~

~~

_{~~ + °~°J}

É

P~j11

~~~~

~~~ J-i

ii)

where _{a~ E}

(-1,1) represents

^the

opinion

of the 1-tl~ individual and P~~, S~~ ^> 0. As _cari be seen, the ternis _m the first summation

only

contribute when the 1-th and

j-th

individuals sl~are the _same

opinion,

1-e-, _a~ = a~, ^and tl~ose _m tl~e second summation

only

contribute when a~

~

_a~.

Tl~e NSL model also admits _a

parameterization describing

tl~e distance between tl~e indi- viduals

[2j. However,

in _a world witl~ radio _or

television,

distance would _seem to be irrelevant ,vhen it _comes to

"important"

_issues. Therefore this _paper considers

only fully coupled

models.

As tl~e impact ^{of tl~e} _group on an mdividual

changes,

_so does tl~e individual's

opinion.

Tl~e

dynamics

of tl~is

change

_is

gi,~en by

a

simple

Monte Carlo

procedure

in wl~ich all the indi,~iduals evaluate the social

impact

and

adjust

their

opinions accordingly:

Prob[a~(t

+

1)

₌

a~(t)]

c~ e°~'(~~

(2)

where

fl represents

^the amount of noise in the

system.

^{Tl~is noise arises}

primarily

_as a result of

misunderstandings.

In tl~e absence of noise

equation (2)

reduces to:

a~jt

+

i)

=

a~jtj sign jI~jtjj j3j

Equations il

and

(3)

form the basis for _a

simple dynamical tl~eory

of social

impact.

In tl~eir

original

work NSL used

uniformly

distributed random numbers for tl~e

persuasive

and sup-

portive couplings

and

assigned

tl~e

spins

to _a two-dimensional

grid. Tl~ey

sl~owed tl~at in sucl~

models

equation (3)

has

ferromagnetic

fixed

points

^{witl~ tl~e final value of tl~e}

magnetization depending

_upon the initial distribution of

opinions.

Lewenstein et ai. [3]

investigated

the above

equations using

_mean field

techniques

for several diiferent distance

metrics, including

_a

fully coupled

mortel like tl~at _use

l~ere,

^{and arrived} at

essentially

tl~e _same conclusion.

Another

important aspect

^of tl~e

dynamics

of social

impact

^{is tl~e}

ability

of individuals to leam from

past

^bel~avior. Rl~etorical abilities in

particular

_are leamable. On tl~e otl~er

l~and;

if

people

are

persuaded

tl~at _a

particular

individual _was

trustwortliy

on one

issue, tl~ey

_may be

more inclined to trust that person on otl~er issues _in the future.

In tl~e

present

context,

leaming

means that the

qualities,

_P~~ and S~~ should _vary with time and tl~at tl~e variation sl~ould be correlated _to tl~e values of _a~ and _aj. Wl~at _one would

expect

is tl~at tl~ose wl~o

develop

better _persuasive skills tl~an otl~ers _may be able to influence tl~e

opinion

of tl~e

majority

^{of tl~e} _group ^members. In tl~is _sense,

tl~ey

would become tl~e ~ieaders of tl~e

group".

Tl~is paper examines tl~e conditions under wl~ich sucl~ transformations _can take

place.

^In tl~e

following

section

learning

is examined wl~en

persuasiveness

and

supportiveness

are consider _two

separately

learnable

qualities. Following

tl~at _we examine tl~e _case when these

quahties

_are considered _one and the _same.

Finally,

_some consequences are discussed.

2.

Learning

with

fl~ ~ Su

A reasonable

leaming

rule _con be found

by examining

other human activities. Price [7] bas shown tl~at for _a wide _range of human

activities,

tl~e

probability

of

marginally increasing

one's

performance

is

proportional

to the current level of

performance.

Price termed this the "cumu- lative

advantage" principle. Altl~ough

this

principle

_was first

l~ypotl~esized

in connection witl~

human

activities,

it has since been

successfully applied

to certain

types

^{of social} interactions within animal societies

[8,9j. Hence,

"cumulative

advantage"

_may well be _a universal

property

of all social

systems.

One _area, related to the

present study,

in whicl~ it _con be shown tl~at the cumulative advan-

tage principle applies,

_is that of scientific

productivity [loi.

To _a

large degree,

_success in any

scientific

activity

consists _in

persuading

other scientist of tl~e correctness of one's results. In

tl~e context of tl~e present

study,

_we would then expect

persuasiveness

to follow _a cumulative

advantage principle.

Tl~e

original

_paper of NSL considered

persuasiveness

and

supportiveness

_as two

separate

skills. In tl~e _context of

learning tl~en,

_we would

expect

that tl~e S~~ sl~ould increases

only

when the

j-th

individual has

supported

the i~tl~ individual in bis

opinion

^{and tl~at}

l~~

^sl~ould

increase

only

when the

j-th

individual has

persuaded

tl~e1-th individual to

accept

^bis

opinion.

Let tl~e initial state of tl~e1-th individual be

represented by: a~(0)

and tl~e final state

by:

a~

IF).

A mean-field

expression

^for a cumulative

advantage learning

rule _can be written _as:

si

=

Su

₊

Osso

if OEj10) = OEj

IF)

₌

OE;

IF)

=

ai10);

j~~

V S~~ otherwise.

and

~, l~~

+ ap

l~~

^if a~

(0)

₌

a~(F)

= a,

IF)

and

a~(F) # a;(0);

j~

V

l~~

^otherwise.

a~ and ap are

predetermined _parameters

wl~ich control tl~e

learning speed.

We also set tl~e self

couplings

to zero, 1-e-,

Pu

=

Su

= 0. Tl~is is _a

positive

enforcement

learning

scheme in wl~icl~ _success is

rewarded,

but failure is trot

penahzed. Introducing

_a

penalty

for failure does

not bave _a

qualitative

affect _on tl~e results. In tl~is _{paper we set a~}

= ap m order _to

simplify

tl~e

analysis.

In tl~e

learning stage,

^the

system

was

repeatedly

started in _a random

configuration

and the

learning

rules _m

equation (4)

and

là)

_were

applied.

After _{a given} number of

leaming steps,

^the

system

was tl~en started in _a number of random states and allowed to evolve to a fix

point.

At eacl~ fixed

point

tl~e

magnetization

and

spin-spin

correlations _were measured.

Typically

100

states were chosen for evaluation. Since the

leaming

rule defined above is _open

ended,

_{i e.,}

there _{is no criterion}

by

which the

leaming

sl~ould be

l~alted,

tl~e final _state of tl~e

system

is a

function of botl~ tl~e number of

learning

_steps and tl~e

learning

rate.

As _a first step, we calculated tl~e

magnetizatioii

of the

system:

m~ =

j _{(a~(F), (6)}

1=1

and

compared

it witl~ that of NSL.

Figure

1 shows the results for

systems

^{with 200} individuals and lo0

leaming

steps. ^{Eacl~ data}

point

is

averaged

_over 200 systems. ^For

leaming

rates near

unity (no leaming)

tl~e

system

shows the _same

ferromagnetic

bel~avior _seen

by

NSL.

However,

as tl~e

leaming

rate

increases,

_a

phase

transition to _a

regime

of very low

magnetization

_occurs.

In order to _more

clearly

understand tl~e

regime

of low

magnetization,

_we calculate tl~e fol-

lowing

correlation functions:

~ ^N

c(1)

₌

~j

_<

_a~(F)a;(0)

_W

₍₇₎

~ j=1 and

dit)

"<

°tif)°;1°)

>,

18)

304 JOURNAL DE

PHYSIQUE

I N°2

1 m m

0.8

o.6 a

w 0.4

o.2

m

m m

0

1 1-1 1~2 1.3 1.4 1.5

OE

Fig.

1.

lvlagnetization,

_{m~ as a} function of

learning

rate _cx with N

= 200 and 100 traming steps.

Each data point ^is

averaged

_over 100 systems.

ioooo

iooo

ioo

io

,

1

0.1

-1 -0.S 0 0.5 1

d(1)

Fig.

2.

Histogram

of the correlation

function, d(1).

The solid fines _are for

o = 1.05 and the dotted bues _are for

o = 1.30. The system ^{consisted of 200} individuals and 100

training

steps. The

histogram

is

averaged

_over 100 systems.

where tl~e average is taken _over the 100 _states chosen for tl~e evaluation.

c(1)

measures tl~e correlation between tl~e initial value of

spin

and tl~e final value of all tl~e otl~er _spins, wl~ile

d(1)

_measures tl~e correlation between tl~e initial value of

spin

and its final value.

In

Figure

2 _a

l~istogram

of

d(1)

is given for two different values of tl~e

leaming

rate. As _can be _seen, for values of tl~e

learning

rate near

unity,

^tl~e self-correlations _are

normally

distributed

<

d(1)

>- 0 _{as o ~} l. In tl~e

phase

of low

magnetization

tl~e final value of eacl~ _{spin is} correlated witl~ its initial

value,

_{i e.,}

d(1)

_~ l for all1. In otl~er

words,

tl~e

system

^is a frozen

spin-glass

and _no

dynamical

evolution _occurs. Tl~e

c(1) (not sl~own)

_are

normally

distributed in both

phases, l~owever,

tl~e width of the distribution _{goes to zero} in the frozen

phase.

It

migl~t

be

wondered,

whetl~er _or not tl~is result is related to the cumulative

advantage

ioooo

iooo

ioo

io llll

1

0.1

-1 -0.S 0 0.S 1

cri)

Fig.

3.

Histogram

of the correlation

function, c(1).

The solid fines _are for _cx

= 1.10 and the dotted hnes _are for _{o =} 2.00. The systems ^{consisted of 200} individuals and 100

training

steps. ^The

histogram

is

averaged

_over 100 systems.

(Note:

the dotted fines have been offset 0.025 units for

clarity.)

leaming

scheme. In order _to rule this

out,

we

investigated

_some

simple

additive rules and

learning

rules wl~icl~

penalized

failure. In all cases a

phase diagram

similar to tl~at sl~own in

Figure

_was obtained and tl~e

phase

of low

magnetization

l~ad tl~e _same correlation structure _as

depicted

in

Figure

2.

Hence,

_{we can} conclude tl~at tl~is

type

of

phase diagram

is cl~aracteristic of tl~e NSL model and is

independent

of

leaming

rules. Tl~e location of the

phase

transition

point,

_oc, in

Figure

does

depend

_upon tl~e number of

leaming steps,

witl~ _{oc ~} l _as tl~e number of

leaming steps

_goes to

infinity.

An

interesting question

for

psycl~ologist

is: How many

leaming

_steps and wl~at

leaming

rates are realistic?

3.

Learning

with

Pç

₌

Sç

Tl~e NSL mortel _assumes

Pç # Sq.

Tl~is amounts to

assuming

tl~at

persuasiveness

and support- iveness _{are two} separate skills. There _seems to be _no

empirical _support

for this

assumption

and

one could argue that

supportiveness

is

simply

the act of

persuading

_someone tl~at his

original opinion

is _in fact correct. In this _case

persuasiveness

and

supportiveness

could be considered

one and the _same skill and _we should take

Pç

₌

Sç

in our simulations.

With

Pç

₌

Sç

the

leaming

rule described in the

previous

section becomes:

si

=

Su

+

Osso

if _OEj10)

= OEj

IF)

= ai

IF);

j~~

V

Sç

otherwise.

Here,

_we also set tl~e self

couplings

to zero, i-e-,

Pu

=

Su

= 0. Tl~is modified

learning

rule increases

Pç

and

Sç

wl~enever the

j-th

individual has been

sùpportive

or persuasive.

Again,

the

system

_was allowed to leam for 100 time

steps.

^{Then the}

system

was started in

a number of random _states and allowed _to reach _a fixed

point.

In tl~is _case tl~e

magnetization,

m m 0.92 +

0.02,

_was

relatively independent

of tl~e

leaming

rate, for _a less tl~an 10.

However,

tl~e correlations indicate two distinct

phases.

Figure

3 shows _a

l~istogram

of tl~e correlation

function, _c(1)

for two different values of tl~e

leaming

rate: a = 1.10 and _a

= 2.00. As _can be

clearly

_seen, tl~ere _{is a}

qualitative

difference

306 JOURNAL DE

PHYSIQUE

I N°2

1

0.8

0~6

0.4

~

"

0.2

o

-0.2

-0.4

0 50 100 150 200

1

Fig.

4. Correlation function,

c(1),

for _a typical system of 200 individuals with 100 training steps and _a

learning

rate of

cx = 2.0.

in tl~e

shape

of the distribution for

larger

values of _a. Most of tl~e 100

systems

evolved states in which all

opinions

_were

l~ighly

correlated witl~ tl~ose of _a

single

individual _or witl~ tl~ose of

a small _group of individuals. These

individuals,

wl~ich determine tl~e final state of tl~e system,

can be called tl~e "leaders".

Figure

4

depicts

tl~e correlations for _a

typical system.

Here tl~e leader is

clearly

visible. All spins are

higl~ly

correlated with tl~is

single spin. By

and

large,

tl~is _spin alone determines tl~e end state of tl~e

system. However,

since tl~e

magnetization

is less tl~an

unity (m

=

0.94),

tl~ere

does exist _a small

minority

of individuals wl~o _are not

always swayed by

the leader.

Unlike tl~e _case witl~

Pç # Sç,

tl~e

leaming algorithm

does make _a difference wl~en

Pç

₌

Sç.

For _{a very}

large

^{number of}

leaming steps,

tl~e current

algorithm eventually

^reaches _a ^frozen

spin-glass phase

like that discussed in the previous section.

Hence,

the

leadersl~ip phase

_{is a}

long-lived,

meta-stable

phase. Only

tl~e cumulative

advantage approacl~ produces

tl~e

phase containing

leaders.

Simple

additive

algoritl~ms

for

example, produce

results similar to tl~ose

discussed in tl~e

previous

section.

4.

Sunlnlary

and Discussion

Tl~is paper bas extended tl~e model for social

impact proposed by Nowak, Szamrej

and Latané

(NSL)

_[2] to include

learning

^of tl~e

primary qualities: persuasiveness, Pç,

and

supportiveness, Sq. Leaming

imparts a very ricl~ structure onto tl~e

system.

^Witl~

Pç # Sç,

tl~ere exist two

phases,

_a

ferromagnetic phase

and _a frozen

spin-glass

like

phase.

In tl~e frozen

phase

all individuals retain tl~eir initial

opinions.

In tl~e

ferromagnetic phase, nearly

all individuals reacl~ tl~e _same

opinion.

Tl~is bel~avior _was found to be

independent

of tl~e

leaming

rule. Wl~en

Pç

₌

Sç,

_a tl~ird

phase emerged

in which tl~e final _{opinions were}

l~igl~ly

^{correlated witl~ those} of _a

single

individual. These leaders determined the final _state of the

system.

That

leaming plays

_an

important

role _in any

society

^is

indisputable

and it is well known tl~at individuals _vary

greatly

_in tl~eir

ability

to

persuade

and in tl~e

degree

to wl~icl~

tl~ey

_can be

persuaded by

otl~ers.

Although

_{one can}

question

whether _or not the

leaming

rules studied l~ere _are

realistic, they

do show tl~at

introducing learning

into the NSL model leads _{to new}

pl~enomena

not _{seen in} tl~e

original

model.

Tl~eir _are many directions _{one can} take to make tl~e

leaming cycle

_more realistic:

1)

Tl~e

leaming

rate could be made different for different individuals

(as

it

undoubtedly

is in

reality).

2)

Tl~e

leaming cycle

could also be different for different individuals.

3) Leaming

could be made

probabilistic. (Normally

_we do not leam

sometl~ing

from _every

success.) 4) Leaming

could be made to affect ail tl~e

couplings

associated witl~ _a

given

individual.

5)

Otl~er

qualities

coula be added _or the

quahties

of

persuasiveness

and

supportiveness

^{could be subdivided into} components.

6)

Individual resistance to the

opinions

of others could also be introduced

by letting Su

be

greater

than _zero.

Normally

_one thinks of

leaming

as

goal-orientated, however,

here _we bave _an

example

of

non-goal-orientated leaming,

_{1-e-, we}

place

_no

requirements

on tl~e social system ^{to reach} a

particular

state. The state in wl~icl~ leaders _are present, for

example,

_emerges

spontaneously.

Otl~er

examples

of

non-goal-orientated

_or

"unsupervised" leaming

_con be found in _some neural network models

il Ii.

In tl~ese models _a neural network is

given

certain

inputs

and tl~e network must

organize

these

inputs

without any outside intervention. It may be

interesting

to

study

the

extent to which such

unsupervised leaming

rules _can be

applied

to social models and vice-versa.

Of _course, the NSL model is _not tl~e

only example

of _a reductionist model in tl~e social sci-

ences. We bave concentrated upon it in tl~is paper, because it is _a model wl~icl~ _,vas

specifically

developed

for l~uman societies. As another

example

consider the

self-organizing

hierarchical model

recently

introduced

by Bonabeau,

Tl~eraulaz and

Deneubourg (BTD)

_[9]. Tl~eir model

~vas

originally designed

to

explain

l~ierarcl~ical tendencies in animal societies. In tl~eir

model,

animals _{move on a} two dimensional _square lattice and _upon

meeting

another

animal,

a non-

lethal

fight

_ensues. Tl~e

probability

of

winning

the

figl~t

is

proportional

to tl~e number of

figl~ts

tl~e animal bas

previously

_won, in otl~er words _a cumulative

advantage principle

_is assumed.

Tl~ey

find tl~at if tl~e

density

^of animals _per unit _{area is}

large enougl~,

tl~en tl~e societies

develop

a natural l~ierarcl~ical structure

(or pecking-order)

witl~ _some animals

winniiig nearly

all tl~eir

fights

and others

losing nearly

all their

fights.

From _a

modeling

_point of _view, there _are two

major

^difference between the NSL model and the BTD model. l The BTD model _assumes

only two-body interactions,

while tl~e NSL model

assumes

multi-body

interactions.

2)

Tl~e BTD model

ignores

tl~e current state of the

animal,

1-e-, whether it _{won or} lost the last

figl~t,

and

only

considers tl~e animal's

complete l~istory;

wl~ereas tl~e NSL mortel considers tl~e current state and tl~e

complete l~istory.

Of _course, from

a

sociological point

of _view, tl~e mortels _were

designed

to

represent

two

quite

^different

aspects

of societal interactions and for tl~at _reason tl~e

similarity

^of their

resulting

structures is all the

more

intriguing.

Given that tl~ere _are

striking qualitative

difference between tl~e

learning

^{rules examined}

l~ere and in otl~er

models,

it may

possible

to

design

studies

involving

real groups in order to determine l~ow

learning

affects social

impact.

In

particular,

it may be

possible

to determine such factors _as tl~e

leaming

rate and tl~e

typical

number of

leaming cycles

witl~in _a group. It may also be

possible, though

_more

diflicult,

to

distinguish

between different societal

models,

such _as the BTD model and the NSL model. If such studies could be

designed

it would show that reductionist models may

perform just

_as well _in the social sciences _as

they

have in tl~e

pl~ysical

_sciences.

308 JOURNAL DE

PHYSIQUE

I N°2

References

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J-A-, Krogl~

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