l;
l"
.l ou rn a I o.[ M a I hcnta I ic a l,loc
io lo g y
1998,Vol. 23(t),pp. I 27
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STATISTICAL MECHAI{ICS OF
OPil\ION FORMATIOI\ AI\D
COLLECTIVE BEF{AVIOR:
MICRO-SOCIOLOGY
DAVID B. BAHR ''* and EVE PASSERINI b'i
" Cooperatitte hzstitute .for Researclt in Environmertal
b Department of
Socioloql', (i771,'srsitt,o.f Colorado, Boulder,
Sciences,
CO 80309, LrSA
The process of opinion formation leading to coliective behavior in large groups is
modeled with a probabiiistrc and statistical mechanical theory of micro-sociological
behar,ior. By assuming that the probabilitl' of making a given ciecisionis proportional
to the number of peopie who have made the same decision. this tireorv of microiuteractions predicts the manner in ra,hichindividuais rvill respond ro croups, hou,
groups will respond to individuals. and hou'minorities and majorities u'ill respond to
each other. In particular. the theorv rccuratch' predicts observations of chivalrl'. tip
sizes,conformity, and sarvking in groups. Guicied by intuition of social behavior and
anaiogies with physrcai theories.social forces and sociai temperatures have also been
introduced as concepts relevant to group interactions. These paramcters signilicantll'
improve the theory's fit to empirical data.
II\TTRODUCTION
Coliectivebehar,ioris a pl-renolrenonurhiciroccurson mzrcro-sociological
scalesbut wiricir is grounded in micro-sociologicaldecisions.E,ver\/
rnember of a group lnust citooser,r,hether
to participate or not to participate in the aggresatebehavior.Explanationsfor individual decision
naking processesassume everything fr-orl pure ratiol]al choice to
altruistic choice.Aii alterlatjve approach. however.is to exalnine all
possibienticro-leveldecisions(regardlessof the reasoningu,hich leacis
*Corresponding author. Insiitute of Arctic anclAlpine Rescalch.
Campus Box 450.
U n i v c r s i t r o 1 -C o l o r a c l o .B o u i d e r .C O 8 0 - ? L ) 9 - t ) 4 _U;S
0.A.
-E.
P a s s e r i nw
i as sLrpporteb
d r N a t j o n a l S c i e n c cF o u n d a t i o n G r a n t C M S - 9 3 1 2 ( r - 1t 7c r
t h e N a t u r a l F i a z a r d sC . e n t e ra t t h e U n i v e r s i t r o f C o l o r a d o . B o u i d e r "U
. SA.
D. B. BAHR AND E. PASSERINI
to these decisions)and to then assign a probability of occurrence to
each decision.The highest probability decision can then be identified,
This is the statistical mechanical point of view used below to outiine
processes
the micro-sociological
by which an individual's decisionsare
impacted b)' the actions apd opinionsof other people.
The number of complex micro-leveiinteractionsbetweenindividuais
in a iarge group are far too numerous to detail explicitly. Imagine, for
example,a party rvith l0 members.Each individual can interact with up
to 9 other people at any given instant, or they could interact with a
subsetof 8, 7, or fewer,or evenno one at all. In total, there are 210possible different interactions. In a state legislature rrith 100 people, the
nurnber of possibleinteractionsis phenomenallylarge, on the order of
1033.Obviously,for whole communities,cities,or nations,not everyone
can interact simultaneously,but it is easy to seethat a direct enumeration of all outcomes of interactions is still irnpossiblefor any but the
srnallestgroups. Most collectivebehavior problems, however, involve
interactions within groups that are rarely as small as 10 or even 100
people. In these large group cases,the tools of probability theory and
statistical mechanicscan be used to specify the most likely micro-level
interactions.In fact, while there is never any certainty in a prediction,
thc tnathematicsdemonstrates(almost paradoxically) that as group sizes
get lar-eer,the uncertainty in collective behavior actuall)' decreases.
The mathematicalmicro-levelbasisfor large group behavior can be
formulated in tire same fundamental manner as the beiravior of large
collections of rnolecuiesin a liquid. While physicists, for example,
would never clairn to understand the exact detailed behavior of any
singlemolecuiein a cup of water, they wouid claim that the large scale
averagebehavior of the water is preciselydehned - there is a measurable volume and temperatnre,for example. Likewise, a sociologistis
not likeiy to claim that the detailed behavior of every member of a
-qroupis understandableor predictable;but for a large enough collection of people, the group dynarnicscan be predicted with well def,rned
averageproperties,even at the micro-level.For example, the decision
making processbehind any singlevoter's opinion is unknown. but large
scalepatterns in voter behavior are reproducible and are known to be
a function of large scalesocial dernograpirics.Therefore, given proper
socio-economicindicators, the "average" voter's behavior can be
loughly specified.
,I
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STATISTICAL MECHANICS
FOR MICRO-SOCIOLOGY
Tireoreticalll',the collectivebehavior of a large group could defy all
reason- all the teenagersin Neu' York City couid simultaneouslv
cirooseto rob a conveniencestore or each of the moieculesin a glassof
water could suddeniy and inexpiicably evaporate. HoweveL,tire likelihood of theseevents,while always possible,is negligibly srnall.From a
mathematical and statistical mechanical point of vieu'. every possible
collectivedecisionb,va group (or ph)'sical stateof the water) is assigned
a probability of occurrence.In a typical trvo partl, election,for example.
a group of A'people decidesbetween two candidates.If the candidates
are completelyindistinguishable"then the probabilitl' of every member
choosingthe samecandidateis smalier tiran tire probability that at least
one member wili choosea ciifierent candidate. This is becausethere is
only one way for the entire group to ciroosethe same candidate, but
there are Ar different ways (one for eacirmember for the group) to have
a sin-eledissenting vote. In fact. tire ciistribution of possible voting
outcomes is binomial, or for lar-eelii approximateiy Guassian rvith a
distinct peak representingthe most iikeil, r,oting outcome or most iikely'
collectivebehavior.For many physical and socioiogicalapplications.
the Guassiandistributionhas a verv narrou'and very high peak so that
only one outcome is probabie zr.ndall otirer outcomesare improbable.
perhaps even ludicrous (as in the conveniencestore exarnple).This
sharply defined distribution allows a mathematical formulation of the
Inost probable coliectivebehzivior.
The probabilisticand statisticalrrechanicalapproachis very general.
can be exparrdedin multipie different directions,aud has the ltotential
for applicationsin many sub-disciplinesof sociology.This paper. l-iou'ever,focusesahnostexclusivelyon au euumerationof the basicnricrosociologicaltheory and irighlightssorre of rts prirnary advantages.in
particular, u,e note that the statistic.almechanicalapproacirdoes not
assulnerationality or coinplete informarion. nol cioesit require speciiic
assutnptiotlsabout itn actor'sabiiitv to analvzecostsand benefitsassociated with their potential actions.The tireor,vrissuntesonly that indrviduals interact in a sirrple nranner consistentri,ith rnauy enrpirical
observationsand previous tireories, including one of the most bersic
p o s tu l a te so f th re s h o l dand cr-i ti caimass studi es(that deci si onsare
b a s e do n th e n u mb e ro f i ndi vi ci ual sthat ai readyhave a gi venoni ni on).
Iu fact, tire statisticalIrreciranics
predictstirresholdbehavlor.and serves.
therefore.as an alternativebasisfor tirresiroidanci crirical massn-rocieis.
D. B. BAHR AND E. PASSERiNI
ln addition to laying a foundation for studiesof collectivebehavior,
the statistical rnechairicalmicro-sociologymakes it possible to matirematically derive some of the outcornesof social interactions (this is
done, for example, in a companion paper on the nacro-sociological
implications of the micro-level thcory). The ri-qorousmathematical
foundatioll can also lead to equalll' rigorous computer rnodelsof social
interaction; for exanple, the form of the rclationships between each
variable is known. rather than hypotiresizedlike the "S" shapeddecision and production functions in coliective behavior models of
Heckathorn (1993),Macy (1991),Oiiver er al. (1985)and others.
The paper is divided into three sections.The first outlines a brief
history of simiiar approachesto specifyingthe rnicro-sociologicalprocessesof coilective behavior. The second section details the hypothesized interactions of a group, and uses quantitative analogies frotn
physics to motivate the influence of "social temperature" and "social
forces" on group interactions. This section also compares theoretical
predictions with ernpirical data setsof srnall group opinion formation.
The final section discussessome of the theory's implications.
1. HISTORY OF' COLLECTIVE
BEHAVIOR MODELS
A number of sociolo-eicalstudies have modelled large scaie collective
behavior based on the assurnption that people are influenced by the
opinions of those around them. These studieshave focused on how
consensusemersesin crowds (Johnsonand Feinberg, 1977;Feinberg
and .lohnson,1988,1990),on garue theories(Glance and Hubennan,
1993;Fleckathorn, 1993), on the thresirold percentageof others u'l.to
must act before any given person will aiso act (Granovetter, l91B;
Macy, 1990, 1991),and on critical mass theories,wlrich derive their
name from an analogy to phl,sics (Oliver et al., l9B5; Marwell and
Oliver, 1993).ln contrast to the more traditional approachesused in
these theories, the idea of assigning a different statistical mechanical
probabiiity to each possible set of social behaviors has been gaining
sr-rpportover the iast few decadesand has resultedin interestingcrossdisciplinary research on collective behavior. Sorne researchersin the
1970'sand 1980's.for exarnple,reabzedtliat many physicalphenomena
have qualitative similarities to coliectivebehavior in social s)/Stems
(e.g., Weidlicir, 1971;Callen and Shapero, 1974; Weidiich and Haag,
STATISTI CAL MECHAN I CS FO R I\,iIC]RO-SOCIOLOGY
1983). Social analo-qiesto ph,vsical concepts like lnass, eravit5'. and
temperatureare older than the 1970's(e.g.,Catton. 1965).but the first
quantitative links came with the reahzation that the sameprobabilistic
laws used to describephysical svstemscan appi.vequaliy u,ell to other'
like indiyiduals i1 largegroups(e.g.,Helbilg.
multicornponentS]/Stems
1994).The power of more recent approaches.therefore.has not been in
the analog.vto physicsbut in tire underlying probabilistic rnathernatics
whicir can be used in many different frelds of studr,. including ph,vsics
and socioiog.v.
Recent coilal-rorationsbetween social scientistsand physicists h.at'e
led to substantialrefinementsof the original sociologicaltheoriesq'hicir
rvere borrowed almost intact from the physical theory of statistical
mechanics.Bibb Latane's (1981)theor-vof socialimpact, for example,
necessarv
iras been used to explain the micro-socioiogicalir-rteractions
for the transition irom merel,vphysics-iike modeis to truiy socioiogicai
et al.,
rnodels of collective behavior (Nowal; et ai., 1990;Lern,enstern
1992^Latan6 et a!.. 1994).Most significantlr,,this neu' theory can be
straightforrvardlyirnplementedas a cornputersinrulationof the interactionsbetu,eenmany individuais. These pioneeringsimuiationsuse a
'u'ariantof the "\,oter model" (v,'hichhas a long history of stud5':e.g..
Durrett. 1988)oi "majority ru1e" inodel frorn phirsicstc erairiue the
dvnamicsof opinion forrnation.In its simplestforrn thesemodeisspecily tirat evervindividual iuteractsrvith a subsetof a group and chooses
an opinion rvhicirmatchesthe maioritl' opinion of tire subset.
Tirese and otirer cornputer simr,riationshave been successfuliu
explainingriumerousqualitative featuresof reai opinion formation in
groups (Granovetter.1978.Latane ct al., 1994).but a priman' drau'back o1'these
techniquesis the assumptionof maloritv ru.leanCrational
choice.While rational choice nrodelsmav be u'c.llsuited to sonreeconomic and poiitical decisions,there are otirer social situationsr,vhere
individuals u'ill not unfaiiingil, ciroose ti"remi4oi-it1'opinion. Macv
(1990.1991).for exanrlrie.in his iraperson threslroids,usescon-iDuter
simr,riations
to sirou'that irrational chr-tices
can significantlvchangetire
c o l i e c ti v eb e i ra v i o ro i g rcl ups.H ou,ever.u' hi i e ei vi ng i rnP orturrtneu'
results.iris modeis give no matirenraticzLl
foundation for the forr-nulati o n su s e du ,i re nd e c i d ingi f an i ndi vi dual has changedopi ni ons.W i thout this nricro-ievell'oundation.it is dil'ficult to have confidencein a
c o l x p u te rs i i n u l a ti o n ' (ormathemati caldescri pti on
) of ti re macro-l evc1
D. B. BAHR AND E. PASSERINI
behavior. Likeu,ise,although Latan6's work (1981)doesspecifya rnathematicalmicro-levelfoundation. a careful analysisshowsthat the basic
form of the micro-sociological interactions is incompatible with the
empirical clata ire conpiied. According to Latan6's social iilpact theory, the extent to which an individual's opirtion is influenccd by members of a group depelrds (among other factors) on the uumber of
individuals in the group. In particular, the probability of being co11verted from one opinion to another is Po: sAI where 1/ is the ntrmber
of individuals with the opposirrgopinion and s and 0 < I < 1 are empirically determinedconstants.Whiie this power law fits the empirical data
reasonablywell (and forms the basisfor rnicro itrteractionsin the voter
rnodel),it is immediately apparent that this formulation cannot be true
for all gtoup sizes.For large enough -A/,tire probability of changing
opiniols wiil be greater than one, which is impossible(all probabilities
must be between zeroand one (Ross,1988)).In other \t'ords,if the same
empiricai experimentsused to support the power law reiationshipwere
performed again with larger groups, the original fit would be very poor
for the new data.
Becauseit is crucial to a proper micro-leveldevelopmentbut has not
beenmentionedelsewhere,we note that for situationswhen one or a f,eiv
individuals are being influenced by a group of many, the empirical data
should satisfy three constraints. First, becausePo is a probability,
0<P^<1.
(1)
Second,the socialimpact (or probabilityof changingopiniorr)should
increasewitir increasingnumbersof group members.i.e., Po(l/) (
Po(//+ l) for all N, or equivalentlY
2D
#"0
(2)
Finally, the arnount of additional impact due to the l/th person shor-rld
be greaterthan the additional impact due to the AI* lst person'As suggestedby Latane (1981),this is analogousto money. If you lrave only
one dollar, then ono more dollar is a substantialincrease,but if you
have a rnillion doliars then an additional dollar is barely noticeable.
l.
STATISTICAL MECHANICS FOR MICRO-SOCIOLOGY
i.e., we postuiate that Po(N) - Po(N- I)> P ^Qn+ 1)- Po(N), or
equivalently,
t
A 2r pA
;;;;
E U.
(3)
U1\
All three of theseconstraints are supportedby the availabie numerical
data which is between 0 and I by definition and appearsto be monotonically increasingand roughly concavedownwards (see,for example,
data in Latan6, 1981;Tanford and Penrod,198.{).The first constraintis
essential,althou-ehthe secondand thirci constraintsshould be regarded
as strong intuitive guidelinesrather than strict requirements.It is possible,for example.to imagine that the lfth personis a conformist- i.l,hile
the Ar+ ist person is a strong leaderwith much greaterinfluence. However, over large rangesof ,&. theseconstraintsshould be approxirnatelv
correct.
While the micro-sociologicalinteractionsin many computer simulations of collective behavior can be easiiy changed. the difficulty is
finding a theoretically grounded probabiiistic description tirat aqrees
with the three constraints and fits known observationsof opinion formation. We note. for example, that in an alternative theory of social
impact. Tanford and Penrod (198a)propose that if T individuals are
the target of influencefrom l/ peopiethen Po : cre(-k"-n').This reiationship fits tire data reasonabi),well (Tanford and Penrod. 1984) and
satisfiesthe tirree constraints (except for the corrcavir)'at small lrr)
However,the form of this relationshipis not justified by anv analvtical
theory, and becauseof the paucity of ernpiricaldata (only five ciata
points in some cases).illau\i different two parameter reiationsiripsu'il1
fit the observationsu,itir iarge coelilcienrsof correlation.ln thc follou'ing section. therefore, we oritiine the foundation of a pro'babilistic
micro-sociologicaltireorl' rn'iricirastees u'ith tire constlaints and predicts a specificfit to some avaiiabiedata.
Before continuing with the theoreticalderjvations.howevet-.it is
u'orth noting that our approach is fundamentallvdifferent irom rnost
previousu,ork on collectivebehavior.Somcwcil linou'n pr-eviousstucii e s . fo r e x a tn p i e .u s e threshol d.l carni ng.and netu,orktheor-i es(e.g..
G ra n o v e tte i '1. 9 7 8 :M a c -r' 1991:
.
Marri ' el i .oi i ver and P rahl . 1988).our
a;rproach. on the other hanci" is crounded entirel\' on tire b:lsic
D. B. BAHR AND E. PASSERINI
principlesof probability theory and usesa probabiiisticdescriptionof
everyactor'sinteractions.The work most closelyrelatedto this paperis
tlre socialpsychoiogyby Latane (1981)and Lewensteinet ul. (1992),but
our analysiscoutains a different description of the micro-sociology.
Latan6's pirilosophyof social impact is similar to our philosophy of
socialinteractions,but there is little overiap betu'eentireir resultsand
our results. and our deriv:rtions are colnpietelv new and do not use
Latan6's derivations.
Threshold and critical rnassmodeis are also rela"tedto the statistical
meclranicalapproach.In thresliolds,it is hypothesized(or argued by
some means)that actors will join in a decision only after sorrrecritical
number of other actorshave rnadetire same decision(e.g.,Granovetter,
1978).In the statisticalrnechanicalmodel, this is translatedinto a probabilistic statement- the mot'epeopleu,ho have already made a decision,
then the rnore probable it is that an actor will also make that same
decision.Critical massand thresholdmodels suggestthat there is a fairly
sharp transition at the critical lnass from non-participatory to participatory behavior, but by itself, the probabilistic formulation would not
seemto lead to sharp transitions. However, without appealing to any
estimate(r'ationalor irrational) of an actor's costs and benefits(e.g.,
Macy, 1991;Oliver et a|.,1985),the statisticalmechanicspredictsthat
for large groups the probabilitiestum into a very sharp transition around
some critical number of people (who have aireadl' made a given decision). Belorvthe critical number of peopie it is highly improbable that
the actor u'illjoin their decision.Above the critical number it is highly
probable that the actor u'ill join their decision.The statisticalmechanical theory, tl-rerefore,actnally predicts the size of the threshold for
interactionsin large groups. Becauseof this, tire probabilistic approach
gives the mathematical basis for treating critical mass and thresholcl
studiesas a logicai set of consequences
of statisticalmeciranicaltheor1,.
Becausethis paper presentsa very different and new type of sociological theory, the following derivations drau' on a nulltber of analogies
from tlie physical sciences.However, the analogies are useful only
becausephysicistsuse sonre of the same mathematical language and
have aireadygiven namesto many of the variablesand equationsin the
probability theory. Thesevariables,iike "temperature",have intuitive
physical rneanin,qs;
but becausethey are only mathernaticalconstructs,
the sarnevariables apply to different situations in sociology.(No one
-:'l
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STATISTICAL IVIECHANICS FOR MICRO-SOCIOL()GY
r.vouldclaim, for example. that just becziuseit iras tire safire narrre a
"standard deviation" al$,a)'srefersto the salnedata. no matter wirat
the application. Instead,a standard deviation is useful as a variable
for many different situations.)Ali of tire quantities in this paper are
defined socioloqicalh'.are in an unambiguous rlathematical format.
and are completely independentof ph.vsicaltheories.
2. MICRO-SOCIOLOGY
OF OPINTOI*{ FORN4ATIOI{
Imagine a group of // people. and supposethat rz, have opinion one,
n rh a v e o p i n i o n tw o , a n d n, harreopi ni on three,so ti rat nr* n" * n.:
1/. If each member of tire group is equaily persuasive,then it is
reasonableto assumethat a newconer to the group (without a preformuiated opinion) will chooseopinion one. two. or three based on
the number of individuals with each opinion. This is supported by selfattention theories (Mulien. 1983) as well as aspectsof the decisiori
making and social influence theoriesof Carnilleri and Conner (1976).
Iu a generalsense,both tirresiroldand critical rnassmodels also posit
that an actor'sdecisionwill dependon tite number of other peoplewho
irave made a -eivendecision.Says Granovetter i.i978),"the costs and
benefitsto the actor of making one or the other choicedependin part
on how many others make wliich choice". So if everyonein tire group
has opinion one. for example. tiren the newconer is rnost likel1, ie
choose opinion one. In other words, the probability P, of ciroosing
opinion i is proportional to n,. So in tiris exampie,
D_
,1-
't 1' 1
,
l7r+11i-tll1
, l
^
D-:
- I
'1 - -
(4)
,rrrrttr',-li..
(,i )
and
I
D
lll
3------,
111+11
.
-li't
(6t
w i re red i v i s i o nb v th e to t al nunber of peopl e.A ' ' : / ? , * l t , + / ? : . e n S L l f e S
that the probabilities ale iess than one (as recui red) and that
P . t P . * P i : 1 ( a sr e q u i r e d ; .
10
D. B. BAHR AND E. PASSERINI
If we assumethat not all individuals have equally strong opinions,
then the probability of choosing one, two or three is proportional to
the total strengthsof each subgroup. If the subgroup of opinion orle,
for example, has a very strong and persuasiveleader while opinions
tu'o and three do not, then the probability of choosing opinion one will
be highest.Let pi.;be the strengthof opinion or the ability of individr-ral
7 to persuade(or support) individual i. It follows that the probability of
person i choosingopinion one u,ill depend on the sum of thc strengths
pii of the people7 with opinion one. Similarly, for opinions trx'o and
three, the probabilities are proportional to tire sum of the strengthsof
the individuals with opinions tu'o and tirree. i.e..
P,(i):40,,
f,t,
Pr(i):40,,
f,t,
:40,,
P.(i)
f,t,
n
n
(7)
(B)
(e)
where the nnnterator is a sum over all group members with a given
opinion (I, 2, or 3), and the denominator is a surn over all group
members. If everyone has the same strength of opinion, then tirese
reduce to equations(4), (5), and (6), as expected.
Note that p,, does not have to equal /7; i individual i could be qr-rite
good at convincing individual .j, for example, but 7 rnight be shy and
unabie to convince i. In fact, individual.i could choose (for whatever
reason)to ignore individuai i and not be any influence at all. Simiiaril',
individual.Tcould try to influencei, but i rnight chooseto ignore or only
vaguely consider the arguments of 7. In other words, actors are not
required to influence(or be influenced)if they choosenot to. The variabie
pi;, therefore,reflectsthe two-way interactionsbetweenindividuals and
not iust ti-restrerrgthof opinion of one individual; this is consistentwith
the nature of "asymmetric" interactions in many social network theories (e.g.,Maru'eil, Oiiver, and Prahl, 1988).As such, an alternative
label migirt be "interaction strenqth", rather than "opinion strength",
but we use the latter to indicate that pi.1rnay change depending on tire
issue. Normally mild-n-ranneredand uninspired peopie. for example,
l
l
i
l.
I
l,
I
t,
i,
l,
ii
li
l,
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STATISTICALMECHANICS FOR ]\4ICRO-SOCIOLOGY
II
may becorne quite impassioned and convincins u,,henthe issue and
opinions are in some u'ay personal.
The opinion of each member of a group can be representedas a
variable. For exarnple,opinion one could be representedb1,a " _ r",
opinion two by &"0", and opinion three by a "1". In qeneral.if a qroup
i s o f s i z e l y ' u ' i thrrzo p i n i onsrepresented
by the vari abl eso, ,o2,...,6m,
then the probability that an individual iras opinion t (l < k ( rr) is
P,.(i\ :
..-\
n .
+!vi!- - L'i= t Pi ' f]L
r/v
L
\-tl
Lj=lI'ij
r ,*,,((s,''j=tPtj
o 1 )l ( o , , - 6 ; ) )
(10)
where s, is the opinion of the 7th group member and the product
f]l-it,,*o just ensuresthat oniy the group mernbers q,ith opinion ft
contribute to the sum.
Examples v,ith One Opinion versusllfanl,
Many empirical observationsinvoive interactions in groups 'uvithone
person iraving a different opinion from the rest (e.g., Latane, 19g1:
Tanford and Penrod,i984). So considerthe caseu,hereolle individual
of opinion o, is being influencedb-vl/, other individuals witli opinion
61. From the theory outlined above, the probabiliti'tirat the lone
individual i will retain opinion o, is
t1
D
I
1
i r \
-----;-\ t l
T T I
/'i=1
(11)
l-'ii
u'hete Pi, is the individual's personal opinion strength (or self colfidence.abilit-vto persuadehimself/herselfaud staird.b1' his/irer beliefs.
etc.). The probabilitl, of changins opinions to o" is
P:.:1 - P1.
(lrr
If all the opinion strer-rgths
are ver-vsirnilar (no reiatjvej_r,
shrvgroup
tnembersor outspokenJeaders),
then 7r,,is the sanreconstani for all i
a n d .j , a n d e q u a ti o n s(1 1)and (l J) reduceto
(13;
and
(14t
12
D. B. BAHR AND E. PASSERINI
In other words, the probabiiity that the one individual u'ill change
strong
opiniols to pratchtha1.of the group is giyeuby Po Pr. Note the
of
shape
gener:rl
the
resernbiauceof equiitiot^r(14)(show1 in Figure t) to
(i4)
also
t5e empirical ctaia (e.g., Latan,g,1981).As desired,equation
(3)'
satisfiestiie funclamerttalcgnstt'aintsin equatiols (1), (2)' and
a
[4or-e generally, the o1$ion strengtirs will varv widely witi-rin
the
usirtg
sirnplified
(12)
be
can
equatiols (11)and
group. Irr this czLse,
u' hi ch i npl i es Il = t P i .i x
(R
oss,
1988),
n
umbers
l
a
rg
e
l
a
u
,o
f
,tro ,rg
the enclosedquantity, in
of
average
the
(
meatrs
.
.
//.
N(/,; ) for large
)
this casethe averageopinion strengthsof tire entire group' Tirerefore'
for large groups,
'p t --
(1s)
l)ii
/v(p,;)
and
D -1
t 2-t
-/ D
-1l-'
(16)
Pii
N(lr,,.)'
probabilTlre factor p il I (p,) is a constant for any given grorlp. so the
large
it1, lhot an individual will c}-rangeopinions to match tirat of a
l.l
t2
10
T4
opinion $'heu there
FIGUI{E I Plots oi'the probabilit)' that an individual will change
c depends on
The,par:rmeter
opinion.
tire
opposite
with
group
a
of
mernbers
are N
s
t
r
e
n g t h sa r e n o t
I
f
t
h
e
c
:
l
'
c
a
s
e
t
h
e
t
o
(
1
4
)
c
o
r
r
e
s
p
o
r
r
d
s
. quation
o p i n i o n s t r e r r g t h sE
the latryof large numbers'
equal (c*1) then thesecurves ate valid only for large l/due to
I
I
I
li
ii r
i:
S T . A T I S T I C A LM E C H A N I C S F O R I \ 4 T C R O - S O C I O L O G Y
i3
group is P^ - 1- clN for some constant c. Again. this gives a family of
curves (sirown in Figure i) which look like the data on opinion formawith the fundamentalconstraints(,I1,Q1. and (3).
tion and q'hich a_qree
If iryeconduct many independent experimentsdesignedto studv the
influence of lar-uegroups on individuais. then the lone indir,'idualsu,ili
have different opinion strengthsin each experiment.The averaee(or
expected)probability can then be caicuiated from tireselnanv experiments as
/n.\
,/D
\
\t
2/
_ i
\
1
*
\
tl-
\
I ' L L
c A '
\
- J :"
l).. i
L r I)/
( n,,)
16;:i-F
\
( Lt,,\
-_ rl - , - l "
/
(.'
1
'
;= t pii)
(17)
In otirer words, the probabiiitv that an individual will convert to a
g ro u p o p i n i o n d e p e n c i son the group si ze as P t:I-cl l {
w i th c= l
u,hen avera-ledover rnany different social situations.
The probabiiitiesP* appll,equaliy well to situationsu'ith an entire
group being influeircedby just one individual. In fact, in most social
settings.if one individual iras an opposine vieu,point,tiren tile group
is sirnultaneouslyinfluencing tirat individual while bein-einfluenced by
tire individual. For two opinions, the pri'ibabiiitiesare exactly the san-re
as in the previouscase.The probabiiiti' that a member of the groul)
rv i l l c h a n g eo p i n i o n sfro m o. to o, i s P t:1-P r:cl l l
i n ti re general
:l
l
l
V
c a s e :a n d P,
tv h e t tw e consi derequal opi ' i on stre' gthsor averages over lnan\/ experiments.Again these probabiiities agree rernarkabll' r,r'ellu'itir the generalshapeof the data on socialinteractions(e.g..
ti p p i ri g tre h a v i o ra n d c h i val ry (Latane. 1981)).
Exantplcs v,ith llfinori\, y,srtrrsMajoriy, Opinions
Tire probabiiistic formulation of choosing opinions can be used to
exanrinetire more generalpossibilit),tirat r?group tnembers(u'itit the
salre opinion) r.r'illchan*ceopinions to nratch thzLtof ,41others. Suppose ti-ratout o1'AItotal group rnen-ibers
^ lll have opinion one and the
remainingAt- I,I nrenrbershave opinion tu'o. Note n .-<AI-,4/. Suppose
again tl'rat we conduct many' experinrentsand corrsiderthe expected
(a v e ra -u e )p ro b a b ri i tioers.assul neti rat A ri s l areeand one of tr,Ir<A -or
D. B. BAHR AND E. PASSERINI
14
At- M<< /y',so that the lau, of large numbers applies.Then b), the sarne
arguments as before, oLIr theory shows that
D
N'-
M
( 1 8)
",-AI
and
Prxl-
ArI
(1e)
*
Therefore,the probability that any n members(out of the total 1/) rvill
chooseopinion one is
PQr:()),",,',' - P r ) ^ - "
:
N!
/U\^/.
- ai \Fl [trltoN
M\'v-'
Qa)
"i
if the lr' individuals are not allowed to changeopinions (because,for'
example they are confederatesin league with each other, or are paid
not to change),then the probability that rt of Ar- M individttals u,ith
opinion one will change to opinion two is
P'(tt):
(N - M)l
n!(N-A4-rt)l
(m\^ ( ,
M\n-M-',
'2I)
\F/ \'-Fl
Note that equations(20) and (21)are binomial distributions,rvith peak
values at n: M and r1: (lr- M)A,tll'{ respectively.Therefore, if ever)/one is allowed to cirangeopinions, then it is most probable that the
nnmber of individuals with opinion one will remain exactly the same.
If tlrere are M "confederates" who will not changeopinions, then it is
nrost likely that an additional (N- M)AflN individualswiil join them.
Critical Mass and Thresholds
Often we are not interestedin the exactnumber of individualswho u'ill
clrangeopinions but are interestedin finding the sizeof' M uecessaryto
convince /? or rrore indii'iduals to cirange opinions. This is the typc
to promote
of question asked by social movenlent organizerst11,i11o
ii
STATISTICAL MECHANICS FOR NlICRO_SOCIOLOGY
I5
collective action, or by political candidateswho need to sr.vinga key
number of votes. Let C be the number of convertsfrom opinion two to
opinion one. Then the probability that C is -ereaterthan or equal to rzis
N-M
P(Cln'):
I
(,22)
P'(i).
l=n
For large enough l{. P'(i) is approximatedby a Guassiandistribution (central limit theorem) (Ross, 1988),and the summation can be
repiacedby an integral which leads to the so-called"error function".
By standard arguments,
P(c>")= i , , r ( N - M \ - 1 " - . /
1
I
,L
"'\. .-,rao,)- lr'rr \
rt-M((N-A,I)lLl
l)
- x,r)tN))
",5u11w
(231
(Note that for iarse AI, the first term aiwavs evaluatesto l/2.) For
example. our theory shows that in a group oi i00 people u,ith 30
guaranteedto vote for a particular candidate.the probabilitS,'of 2i or
more additional individuais votins for this canciidateis
P(C>21)- ;
I
I_.1
-;Erl{
?r' -
3 0 ( ( 1 0- 03 0 )1i 0 0 )
:0.5.
2(30.1
((100- 30)/1
00),
(24)
In otlrer words, if roughly 30Toof a group^svotescan be guaranteed
for a particulai candidateand the rest of tire votesare upcomnittecl,
then the candidateltas a roughll' 50% chanceof u,inningthe election.
Note that equation (23) predicts thresholdand critical mass behavior. As r,1 increases.the likelihood of otirerstaking action lumps drauraticalh,u'hen r1'1reachesa critical iiumbcr. In particular. fol iAr.uc
Ai, P(C ) n) has an extrenreiysharo transitionft'oln a probability neai'
zero to a probabilitl'uear one. The transitionis at the critical mass ,4y'
(oliver at ul.. 198-5).
anclit is a consequcnce
of tire error function u,irich
has a classic"s"-shaped curve (Figur-e2t. Tire "S"-siralleis central tcr
nlanv thresiroidand criticalmasstireories(see.fbr example.equatrcln(l)
o f M a c r' . 1 9 9 1 ).b u t ra ti te r than berngassumedas i n nost parret' s.
ti re
e rro rfu n c ti o ni s p 1 -3 d 1 .1i"r-r
d the probabi i i strc
argurnents
o{ -ti ri spaper.
The exact value of the critical mass is given by the poiiit u,heretite
secorrdderivatir,'eof equation (23; rx,,ith
respectto AI is equal to zerct
D. B. BAHR AND E. PASSERINI
1 /
It)
r\
o- 0.
M
FIGURE 2 A plot of the probability that 50 or more people will changeopinions and
j o i n a c o l l e c t i v ed e c i s i o nu ' h e n t h e t o t a l g r o u p s i z e i s 1 0 0 0( i . e . , n : 5 0 a n d A ' : 1 0 0 0 ) .
Note the "S"-shapedtransition a1 the critical mass.
(i.e., at the inflection point). This has an exceptionally complicated
fonn, so numerical estimatesare better. Nurnerical solutionsshow that
tire critical mass is approximately the nurnber of desired cotlverts
rn i n u s o n e ,n -1 . In other w ords, a massof n-1 peopl ei s requi redto
mobilize /?lnore to a coilective action.
Beforeplacing too much emphasison this particular nuntber. however, recall that equation (23) predicts a criticai nass for situations
where the law of large numbers applies.In otirer words, it usesthe lau'
of large numbers to average the effect of all the different opinion
criticalnass pil.The results,therefore,indicatethe arrerage
strengths,
tire massaveragedover many different scenarioswith different distributicrns of p,.,. Tiris precludes the irnpact of strong leaders and easil5'
iltfluenced followers, two personaiity tlrpes u,hich q,ill clearly change
the sizeof the critical mass.
In any specificcircurnstarlces.thresholds for action will vary f rom
individual to individual (Granovetter,1978),and someu'ilI act at tlre
averagecritical mass,while others u,ill wait for an evenlarger tnass.ln
other words, titere is a distribution of responsesr.r'hichare centered
around the critical mass.This distribution could be expliciti),considered
either analyticallyor nurnericaliy.Hou'by inciudingpii in the anal1,sis,
ever, as the form of equation (10) rnight suggest,the complete analysis
I
lr
i,
lr
I
I
i
I
STATISTICALMECHANICS FOR MICRO.SOCIOLOGY
11
with p,, can be quite compiicated,while the averageresponsepresented
above gives a ciearer and more succinct picture. Rather than deriving the more complicated distribution of threshoicis,we continue the
anah'sis with several additional factors influencing mrcro-socioiogical
interactions.
2.1. Social Temperatureand External Forces
Other factors, besidesindividual opinion strength and the nurnber of
peopieu,ho have a certain opinion, wili influence decisionprocessesand
opinion formation in groups.S/ars. for exarnple.can encoura-qe
nationai
cooperation u'hen a nation might otherq,iseremain divided in opinion;
and hurricanes and earthquakescan encouragevotes for srrict building
codes whicir \^'erepreviousiy unpopular. Simiiari-y,angr1, or excited
sports fans can stampede and riot. v,'hiie the same fans might do
nothing under ciifferentcircumstances.During periods of sociai unrest,
groups of people mav be more likely to vote for cirangesthan during
periodsof socialquiescence.
Thereis alsoan eiementof unpredictabiiitl'
(or "noise") ciue to micro-levelsocial interactionsand personal decisions:miscomrnunications,for exampie.can iead to otheru,iseunlikely
decisions.
If the sarne group can behave differently when making the same
decision, then tire factors controlling this different collective behavior
need to be specified.81, using gualtitative anaiogiesu'ith ph1r5j.o1
systems.three sociai factors are jder-rtified:
social temperature,sociai
force. and noise. We discusstiresef-actorsquaiitatirei,r'hefore siring
cluantitativedefin itions.
Social Tentperuture
Considera measurelrentu'hichquantiiiesthe cxtent to u'irich a groLlp's
a v e ra g eo p i n i o n i s s u s cepti bl e
to chungi nrr.i i t the past thi s cl uanti tv
i ra sb e e n c a l i e dc i i s p o si ti ou.
sLrscepti bi i i risuugesti
,.
bi l i i l ' (Joi rnson
anci
F e i n b e rg .1 9 1 7 ).c o n ta ci on(LeB on, 196f)t.soci alfaci j i tati on(A l i port.
.
1 9 ? 4 )-a n d c i r-c u l are
r a c ti on(B l umer' 1936).
Fi ou,ever.
ti reseprcri ousl r,
defined concepts(v,'itjrthe exceptiOnof Ilir-ritier)\\iel-et.ncanito apPil'
to indirtidualsand not groi-tps.Becauseu'e ti-rinl<it is a nrore iiccLrrilte
l a b e lw h e u re i e i -ri n gto e nti resrol rps(and becausei t hasa cl oseanal ogv
to p h rrs i c su) ' e c a l l th i s measure' ' sc' rci :rl
tertrpet-al Ltrc:'
' . ti vel r' .j t rs
Intui
l8
D. B. BAHR AND E. PASSERINI
the volatiiity of each individual's decision;or a little nlore accurately,
social ternpreratureis an average(over the entire group) of each individual's volatility. However,temperaturecannot be measuredas a propert1,of'individuals, or even of Ver1,s6'1ll groups, becausedetails of
opinion strengths,personalinteractions.and other nicro-sociological
factors are too variable and unquantifiable.In other words, tire average temperaturesof a small qroup would fluctuate too dramaticallyto
be rneaningful. Tiie saue is true in ph1,5is5.Ternperatureis a c)oncept
'ur,hichoniy appiies to a lar-eeensembleof molecules,like the - 102-1
moleculesof rvater in a glass.The detailed interactions of only a few
rnoleculesare too varied to attempt delining a temperature. In both
physicsand socialsciences,
ternperatureis valid only as a statisticalor
"average" large group concept.
individuals are influenced b), the groupls temperature. Whetr a
group's socialtemperatureis high, very little provocation is necessiirl,le
induce an individual to change opinion. At low group ternperatures,
individuals appear rnore pirlegmatic or stubborn and much greater
provocatioll is required to induce a change in opinion. High sociai
temperaturesamplify tlie siightest excusefor chan-qe,whiie iow temperatures diminish the argurnentsfor change.Note that each individual's opinion strength is unaltered, but as the group's temperature
changes,so doesan individual'sdecisionmaking abilities.Although rve
are only appealingto anaiogiesat this point, as shown shortly, temperature in sociologyis a completelydefined mathematical conceptwith no
ambiguitl,.Any apparentarnbiguitycomesfrom the difficulty of describing a mathenraticalconcept with words.
Social Forces
Sirnilar to the inclusion of socialtemperature,"social forces" should be
incorporated. Floods, neu,s reports, advertisementsand other influencesexternal to a group can bias opinions, as can sanctionsancl lau,s
(asdiscussedin Heckatirorn,1993).The presenceof extemalor ent'ironmcntal social forcescan push a divided group toward a rnajorily opinion, or can forcedefectionsin a previouslyunified group. The greatera
sociarlforce. the tnore biasedin the "direction" of tire force is the collective behavior of a group. Like socialtemperature,sociaiforceshartean
analogy with forces in a physical system. Hou,e\rer,as u,ith temperature, social forceshave a precisemathematicaldefinition independent
,i
i
I
I
rT
I
'1
,l
,t
,: l I
, Ii
rl
I
S T A T I S T I C A L M E C } I . ^ ,N I C S I I O R N l 1C R O - S O C IO L O C Y
19
of the ph1,si.utanalog1,.Hebiing (1994)gives a detaiieddescriptionof
social forces and their influence.
l{oise
Finally, there needsto be some measureof miscomurunicationsand
- a near certain decisioncan be tiru,arted
potential misunderstandirrgs
by bad information. mistakes.misinterpretations and apparentl)rrandorn disruptions.This is "noise" and is added into tire decisionmaking
processas a random perturbationof the probabilitiesin equation(10).
Noise. for exampie.allor.l'sfor an individual's apparentlyinexpiicabie
cirange to a ciissentingopinion, even u'hen the entire group has previousiy settied on another opinion. The analo-e1,
to phvsical svstemsis
exact. with botli phl,sical and social noise accounting foi unkuown or
otherwise unquantifiabie perturbations to the S)/stem.
Quantif ing the Analogies
To incor-poratethe intuitive conceprsof sociai temperature( I. ), sociai
forces(/r.), and troiseinto the micro-socioiogicaltheon'. definea function AFl suchthat the probability of an individuai changingopinionsis
p
e
-od
Z
wirere Z ts a constant factor. For simplicity, we restrict tire follov,,ing
anaivsisto opinion formation in groups u,ith ouil, two choices(or:i
and o, - - i) and with equai opinion strengtirs:tire derir,'ationsare
easilygeneralized.If an individual has opinion s,. then
n r lN . i f s , : - 1 '
P\ ( i ) : f
1 , , . , n .i l ' s , : 1 .
(26)
Thet'efore.for the specialcaseof Z:I,
[ - ioe(n, //v.), i f , r - : - i '
aII1i1: i,
(-iog(n"ll{). if s,:1.
(:l \
Nor.r' briefl-v consider irou' tcmperature and iorce rvonld bc dcflned if
A11 n'ere a pirl'sicaj rather tiran a scrciological coitccl-rt(titis uill ireip
guide our deril'ations iater;. Bi' analoqv il'itir pirl,sical starisrical
n r e c h a n i c s .A 1 r ' i s t i t e c h a n g c i n t h e ' ' e t l e r s \ , ' ' o f t h e s v s l c n r a s s o c i t t e d
20
D. B. BAHR AND E. PASSERINI
with a change in opinion (Ma, 1985).The probability of change in a
physical systernis decreasedif it requireslots of energy. Physical ternperatnre, T and physical forces, /2,would be added to the probability
of change as
I
D
l;\ _
a\I/-
1
Z
-e
- ( A H 1 ' t us ; ) / T
(28)
T and /z in the Phi,sical systetn play the same role as or-tr intuitive
definitions of social temperatnre and social forces in the social system.
High (or low) I. for example, amplify (or dirninish) the change rn
energyrequired ior a changein opinion. With a high I; fewer resources
(i.e., less energy) are needed to induce a change. Likewise, a iarge
positive force, ft, makes tire probability of changing to o t:l
much
higher thaii the probabiiity of changingto or- -1. Sirnilarly, alarge
negativeforce biasesthe opinions towards 6z: -I.
To quantitatively preservethe analogiesbet'vveen
physical temperature 7'and social temperature { and physical force lt and social force
h,, Po must be modified to incorporate T, and ft, in such a way that
the form of equation (28) is unchanged.Let
]{ )1tr' ,
if
h' t' r' (n N yrt' " i f
zl
P ^ ( 1 ) : [(t l z)sh' tr' (11r1
[1tl ;;
"-
Ji:
- 1;
(2e)
Ji:1'
Tlren by taking logarithms it follou,s that T,:7, h,:h, z:2,
and
tire structureof equation (28) is presen,ed.The constant: just eusures
that pi'obabilities are conserved ( L, Pn:1) and therefore. from
equation (29),
- - eh,tr,(?)"t'* e-r,/?.
(|;)"t'
(30)
(; is called the partition function in statisticalmechanics.)
We can now step back and definesocialtemperatureand socialforces
1o be nathematically del"inedas the quantities u'hich \/ary as Z. and lr.
in equations(29) and (30). Thesedefinitionsare then completeiyindependent of the physics,u'hich has only irelpedto guide our intuition
towards mathematical forms appropriate for sociology. (It is worth
STATISTICAL MECHANICS FOR MICRO-SOCIOLOGY
2I
noting, for exampie,that the sociologicalform of A1l in equation (27)
is completely different from its form in physics). Most importantiy,
though, the quantities { and lz. defined in equations (29) and (30)
agreewith ali of the qualitative descriptjonsof social temperatureand
social forces discussedabove.
Using equations(29) and (30). the probabiiities of choosinga given
opinion can now be generalizedas
P ', : 1 e h ' r ' ( 1 ) t ' t '
:
\1vl
, r::e-
h',r'(#)ttt'
(31)
(32)
(whererz, hat,eopinion o t :7 and ti, have cpinion o 1: - 1).Note that
P, and P" nor4'have a form whicir agreesu'ith our intuitions of the
elfect that socialtemperatureand social forces have on opinion fonlation. Also note. that to inciude opinion strengths,rtrlN andn,lli can
be repiacedby the right hand side of equation (10). More than two
1e sfttlr' for opinion fr rr,'itir
opinions are alioil'ed by generalizings..rrs/rs
the constraint that !* hr:O. Noise (irom miscornmunicationsand
( i
other unknown sources)can be inciuded as random numbers 0 ( -d,u
added to P, subjectto the constraint that I^ d,,,:0.
Empiricallr,. once an issue iras been identifieci, tire social ternperature of a large grollp can be measured lr'ith a sirnpie poil. For a large
representative
subsetof a group, poll for the percentageof individuals
u ,i th e a c ho p i n i o n o ,,.Thi s gi vesnrl l ri . n.l Ir' . etc. i n equati ons(31)and
(32). (Becausetire szrmpiesizeis large. the different opinion streugtirs
of tire individuals can be treated as an averagestrength, which foiaveragesof n-ranydifferent polis rvill fzictor a\\/ay. as expiained in the
examplesabove.)Poll again at a later time and determinethe percent
of inciividuaisr','irichhave changedopinions. This grvesP1, P., etc. Ii
tirere are no exrernal social forces. equations (3i) and (32) can be
solved simultaneousll'f'or the social terrperature. lf the social forces
a re n o n z e ro .th e n a d d iti cnalpol l s cai r determi nertrl A ' ,ri .l N . P r. and
P. at 1,etanother time. if the svsternis not in equiirbrrium.then a set
of four equarionsconstructedfl'om (3i) and (32) can be used to soive
is in
for the sociaitelnperatureanclthe net socialforces Tf tire sr".<rerrr
1)
D. B. BAHR AND E. PASSERINI
or near equilibrium, then P, and P, will not changeappreciably.and
the systemof equationswill be degenerate.Note that in thesemeasurements there is no neeclto appeal to the intuition of f. and 1.. hrstead,
the matirematicsof equations(29)*(32) cornpletell,define hou, social
temperature and force shor-rldbe rneasured.
2.2. Fitting the Tlreorv to Empirical Data
To test the theory derived above, ecprations(29) and (30) are fit to
some available empirical data. We have used particularly appropriate
data culled from graphs published in Latan6 (1981).Although son-re
loss of acculacy is inevitable when transcribing information frorl
graphs, the data is still representative,and the fits will still indicate
whether the statistical mechanical theory is reasonable.While using
enrpirical data from Latan6,,we emphasizethat all of the fits are done
solely with equations fi'om our theory outiined above.
Technicaily, unless the opinion strengths p,, are included, the
probabiiitiesof choosing an opinion, Pr and Pr, appiy only to very
large groups. Unfortunately, ernpirical experiments are typicalil' on
srnall groups rvith no objectirremeasuresof opinion. Never-the-less,
the large grollp forrnulation can be fit to the small group data as an
approximation. Clonsider experimentswhere one individual is being
influenced by rnany. For large 1/, : approachesone, and by similar
argumentsto those presentedabove,
/
D _D
t L-r
_1
2-t-t
D
l ^- 1
n'\ | lT'
_l: I
\iry
(33)
(which requires the iaw of large numbers). /r. is
wlrere c:p,l(p,i)
assunredto be zero becausethe experirnentaldata was designedto hold
externaiinfluencesto a rlinimurn (although with a little extra complication /2,could be incorporatgd as a part of the factor r:). Figure 3 shou,s
equation (33) fit to trvo different data sets.VisuallS,,the flts a.regood,
and tire coefficientsof correiationare high (R2:0.88 and R2 :0.72).
The rnicro-sociologicaltireory developedin this paper, therefore,does a
good job of expiaining previously cornpiled data on the influence of
groups on individuals.
-1"*
I
I
II
I
.l
: I
II
i
I
.i
il
t:
I
li
l,
lr
l
ir
ii
il
li
li
li"
a
srATtsrrcAl MECHaNicspon MIC,RO-SOCIOLOGY
23
\4/
0.4
0.35
n1
tt
T)
16
1
0.25
/a^)
v-z
0.15
0.1
0.0s
A
=
N
I
I
i
(b)
tl
PA
{l
6
8
10 L2L4
N
FIGURE 3 (a) Equation (33) fit to data on conformitl,with r':0.31 arrd T:7.9
(R':0.72). Po is the probabiiity that irn individr-ralu'ill conform to the group opinion
tl'hen A rndividuals qive zurobviouslv incorrect answer. (b) Equation (-j3) fit tb data on
c o n f o r m i t vw i t h r ' : 0 . 3 8 a n d f : 1 . 9 ( n : : 0 . 8 8 ) . P ^ r s t h e p r o b a b i l i t yt h a t a n i n c l i v i d r - r a l
will gawk at an ol--'ject
u'hen A members of a group are aireadv gau,king. (For details
o n t l i e d a t a , s e eL a t a n e , 1 9 8 1 . )
For experimentsv,,ithman\/individuals
beinginfluenced
b1'one.the
large AI approxirnation:l b^
rP
-D
c-
I
!
I tu
tt '
/ ,'1t
tl
\F/
(341
F o r e mp i ri c a ld a ta c o mpi i edb),Latane i l 981)on ti p sj ze.chi vetl r;,for.
me n . a n d c h i v a i rv fo r \\i onen, equati on (34) pl s6i cts ti re data u' i th
R r : 0 .9 5 . R 2 : A.9 4 . and R r : 0.E 2 respecti vei \,.
A nother data ser
D. B, BAHR AND E. PASSERiNI
examining the number of people inquiring about Christ (after talks by
Billy Graham) involves lar-gegroups so that equation (34) is a nlore
reasonableapproximation. However, scatter in the data makes a good
fit to any function unlikely, inciuding equation (34),and R2:0.22. ln
general,however, the statistical mecha.nicalapproach does a good job
of explaining the data on small groups influencing individuals.
It is interestingto note that equation (34) is the same as one of the
equationspredictedby Latane (i981). However, the agreementis fortuitous and our theory matches.Latane'son orriy tiris one point. The
power law used for this casebyi La,tan6is acceptablebecause1/ is in
the denominator so that the probability remains lessthan one. On the
other hand, the power law proposed by Latane for the different case
of one individual being influenced by many, incorrectll' places1/ in thc
numerator. The correct expressionfor that situation is found in equation (33). Becausethe power laws are the same for the specificcaseof
one individual influencing man)/, the fits done by Latane matcir the
fits done in the previous paragraph. As a resuit, the R2 are also the
saine.However, in generalour probabilistic approach will predict different fits and different coefficientsof correlation.
3. CONCLUSXONS
Note that the precedinganalysislass,umes
only that actors considerthe
opinion of those individuals with whom they interact, and then weight
their decisionaccordingto the numfr of other actorsu,ith each opinion. What evaluationsgo into this considerationis not specified,nor
is the cognitive levei of consideration specified (it could range lrom
rtnconsciousto full1, sstscious). The consideration couid include a
cost/benefitanalysis(e.g.,Oliver and Marwell, i9B8)or any other decision making process.Hou'ever, the advantageof equations(10), (31)
and (32) is that they do not ha,vqto specifythe exact systemof rewards
doesnot have
and punishmentor cost and benefit.In fact, this anal.vsis
to assumerational choice and reward driven behavior at ali. These
considerations happen at a level u,hich is n-]ore detailed than is
required to specify the micro-sociologicalinteractions appropriate to
our model.
By definition, rnicro-sociology seeks to explain the interactions
betweena small nurnberof individuals.Without delvinginto the detailed
l,
I
Il r
li
l.
l,
ll
ll
l.
ll
li
ii
STATIST]CAL MECHANICS FOR MICRO-SOCIOLOGY
social processesof each individual (e.g., sociai conventions, norms!
rituals, perceived rewards, roles, etc.), the previous derivations
describethese interactions. Of course.there is another level of microsocioiogytirat explains the more detailedprocessesinvolved in maliing
the interactive decisions (conversation anal-vses,symbolic interaction
studies.ethnomethodologies,etc.). \\tith that in mind, the theory presented here might be referred to as a meso-scaiesocioiogl,.We leave
that as iargeiy a matter of semantics.However, we do note that social
temperature, social forces, and noise play important roles in each
actor's opinion formation process.and that despite their macro-level
derivations, these parameters influence decisionmaking behavior at a
level which would typicaliy be describedas micro-sociological.
One interestingaspectof the statisticalmechanicalmicro-socioiogical
theory is its prediction of a critical mass and tirreshoid behavior.
Assumptions about possiblecost and benefit structuresare unnecessar),
to show that there is a sirarp transition (threshold) from inaction to
action when some critical number (mass)of actors are already participating. The "S"-shaped functions commoniy used to produce critical
mass beiraviorin other theoriesare predictedas a direct consequenceoi
the probabiiistic structure of the starisricalmechanicaitheory.
As noted. the statistical mechanical theory could be better tested if
there was a method for quantitative lreasurementsof p;;. Collecting
larger data setsunder controlled environments(so that l. is known or
fixed) would also perrnit better tests.Botir of theseoptions. however,
could be difficult, subjective, and economicallf infeasible. A more
prornising approach. therefore, is to take advantageof tire inherent
blurring. aireadymentioned,betweenmicro- and macro-levelprocesses
in the theory. The inherent links beti,veen
micro ancintacro mai,e predictions of large scaiecoliectivebehaviorpossibieby studvins a network of
uricro-socioiogical
interactions.Tiresepreciictionscouid tiren be compared to empirical observatioirsoi macro-socioiogical
behavior. Tiie
toois and iinks needed to maiie sucir connectionsare explored in a
companionpaper.
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