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Statistical mechanics of opinion formation andcollective behavior: Micro‐sociologyDavid B. Bahr a c & Eve Passerini ba Cooperative Institute for Research in Environmental Sciencesb Department of Sociology , University of Colorado , Boulder , CO , 80309 , USAc Institute of Arctic and Alpine Research , University of Colorado , Campus Box 450, Boulder ,CO , 80309–0450 , USAPublished online: 26 Aug 2010.

To cite this article: David B. Bahr & Eve Passerini (1998) Statistical mechanics of opinion formation and collective behavior:Micro‐sociology, The Journal of Mathematical Sociology, 23:1, 1-27, DOI: 10.1080/0022250X.1998.9990210

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Journal of Mathematical Sociology © 1998 OPA (Overseas Publishers Association) N.V.1998, Vol. 23(1), pp. 1-27 Published by license underReprints available directly from the publisher the Gordon and Breach Publishers imprint.Photocopying permitted by license only Printed in India.

STATISTICAL MECHANICS OFOPINION FORMATION AND

COLLECTIVE BEHAVIOR:MICRO-SOCIOLOGY

DAVID B. BAHRa,* and EVE PASSERINIb,†

aCooperative Institute for Research in Environmental Sciences,bDepartment of Sociology, University of Colorado, Boulder, CO 80309, USA

The process of opinion formation leading to collective behavior in large groups ismodeled with a probabilistic and statistical mechanical theory of micro-sociologicalbehavior. By assuming that the probability of making a given decision is proportionalto the number of people who have made the same decision, this theory of micro-interactions predicts the manner in which individuals will respond to groups, howgroups will respond to individuals, and how minorities and majorities will respond toeach other. In particular, the theory accurately predicts observations of chivalry, tipsizes, conformity, and gawking in groups. Guided by intuition of social behavior andanalogies with physical theories, social forces and social temperatures have also beenintroduced as concepts relevant to group interactions. These parameters significantlyimprove the theory's fit to empirical data.

INTRODUCTION

Collective behavior is a phenomenon which occurs on macro-sociologicalscales but which is grounded in micro-sociological decisions. Everymember of a group must choose whether to participate or not to par-ticipate in the aggregate behavior. Explanations for individual decisionmaking processes assume everything from pure rational choice toaltruistic choice. An alternative approach, however, is to examine allpossible micro-level decisions (regardless of the reasoning which leads

* Corresponding author. Institute of Arctic and Alpine Research, Campus Box 450,University of Colorado, Boulder, CO 80309-0450, USA.

†E. Passerini was supported by National Science Foundation Grant CMS-9312647 tothe Natural Hazards Center at the University of Colorado, Boulder, USA.

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2 D. B. BAHR AND E. PASSERINI

to these decisions) and to then assign a probability of occurrence toeach decision. The highest probability decision can then be identified.This is the statistical mechanical point of view used below to outlinethe micro-sociological processes by which an individual's decisions areimpacted by the actions and opinions of other people.

The number of complex micro-level interactions between individualsin a large group are far too numerous to detail explicitly. Imagine, forexample, a party with 10 members. Each individual can interact with upto 9 other people at any given instant, or they could interact with asubset of 8, 7, or fewer, or even no one at all. In total, there are 210 pos-sible different interactions. In a state legislature with 100 people, thenumber of possible interactions is phenomenally large, on the order of1033. Obviously, for whole communities, cities, or nations, not everyonecan interact simultaneously, but it is easy to see that a direct enumera-tion of all outcomes of interactions is still impossible for any but thesmallest groups. Most collective behavior problems, however, involveinteractions within groups that are rarely as small as 10 or even 100people. In these large group cases, the tools of probability theory andstatistical mechanics can be used to specify the most likely micro-levelinteractions. In fact, while there-is never any certainty in a prediction,the mathematics demonstrates (almost paradoxically) that as group sizesget larger, the uncertainty in collective behavior actually decreases.

The mathematical micro-level basis for large group behavior can beformulated in the same fundamental manner as the behavior of largecollections of molecules in a liquid. While physicists, for example,would never claim to understand the exact detailed behavior of anysingle molecule in a cup of water, they would claim that the large scaleaverage behavior of the water is precisely defined - there is a measur-able volume and temperature, for example. Likewise, a sociologist isnot likely to claim that the detailed behavior of every member of agroup is understandable or predictable; but for a large enough collec-tion of people, the group dynamics can be predicted with well definedaverage properties, even at the micro-level. For example, the decisionmaking process behind any single voter's opinion is unknown, but largescale patterns in voter behavior are reproducible and are known to bea function of large scale social demographics. Therefore, given propersocio-economic indicators, the "average" voter's behavior can beroughly specified.

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STATISTICAL MECHANICS FOR MICRO-SOCIOLOGY 3

Theoretically, the collective behavior of a large group could defy allreason - all the teenagers in New York City could simultaneouslychoose to rob a convenience store or each of the molecules in a glass ofwater could suddenly and inexplicably evaporate. However, the likeli-hood of these events, while always possible, is negligibly small. From amathematical and statistical mechanical point of view, every possiblecollective decision by a group (or physical state of the water) is assigneda probability of occurrence. In a typical two party election, for example,a group of N people decides between two candidates. If the candidatesare completely indistinguishable, then the probability of every memberchoosing the same candidate is smaller than the probability that at leastone member will choose a different candidate. This is because there isonly one way for the entire group to choose the same candidate, butthere are N different ways (one for each member for the group) to havea single dissenting vote. In fact, the distribution of possible votingoutcomes is binomial, or for large N approximately Guassian with adistinct peak representing the most likely voting outcome or most likelycollective behavior. For many physical and sociological applications,the Guassian distribution has a very narrow and very high peak so thatonly one outcome is probable and all other outcomes are improbable,perhaps even ludicrous (as in the convenience store example). Thissharply defined distribution allows a mathematical formulation of themost probable collective behavior.

The probabilistic and statistical mechanical approach is very general,can be expanded in multiple different directions, and has the potentialfor applications in many sub-disciplines of sociology. This paper, how-ever, focuses almost exclusively on an enumeration of the basic micro-sociological theory and highlights some of its primary advantages. Inparticular, we note that the statistical mechanical approach does notassume rationality or complete information, nor does it require specificassumptions about an actor's ability to analyze costs and benefits asso-ciated with their potential actions. The theory assumes only that indi-viduals interact in a simple manner consistent with many empiricalobservations and previous theories, including one of the most basicpostulates of threshold and critical mass studies (that decisions arebased on the number of individuals that already have a given opinion).In fact, the statistical mechanics predicts threshold behavior, and serves,therefore, as an alternative basis for threshold and critical mass models.

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4 D. B. BAHR AND E. PASSERINI

In addition to laying a foundation for studies of collective behavior,the statistical mechanical micro-sociology makes it possible to math-ematically derive some of the outcomes of social interactions (this isdone, for example, in a companion paper on the macro-sociologicalimplications of the micro-level theory). The rigorous mathematicalfoundation can also lead to equally rigorous computer models of socialinteraction; for example, the form of the relationships between eachvariable is known, rather than hypothesized like the "S" shaped deci-sion and production functions in collective behavior models ofHeckathorn (1993), Macy (1991), Oliver et al. (1985) and others.

The paper is divided into three sections. The first outlines a briefhistory of similar approaches to specifying the micro-sociological pro-cesses of collective behavior. The second section details the hypothe-sized interactions of a group, and uses quantitative analogies fromphysics to motivate the influence of "social temperature" and "socialforces" on group interactions. This section also compares theoreticalpredictions with empirical data sets of small group opinion formation.The final section discusses some of the theory's implications.

1. HISTORY OF COLLECTIVE BEHAVIOR MODELS

A number of sociological studies have modelled large scale collectivebehavior based on the assumption that people are influenced by theopinions of those around them. These studies have focused on howconsensus emerges in crowds (Johnson and Feinberg, 1977; Feinbergand Johnson, 1988, 1990), on game theories (Glance and Huberman,1993; Heckathorn, 1993), on the threshold percentage of others whomust act before any given person will also act (Granovetter, 1978;Macy, 1990, 1991), and on critical mass theories, which derive theirname from an analogy to physics (Oliver et al., 1985; Marwell andOliver, 1993). In contrast to the more traditional approaches used inthese theories, the idea of assigning a different statistical mechanicalprobability to each possible set of social behaviors has been gainingsupport over the last few decades and has resulted in interesting cross-disciplinary research on collective behavior. Some researchers in the1970's and 1980's, for example, realized that many physical phenomenahave qualitative similarities to collective behavior in social systems(e.g., Weidlich, 1971; Callen and Shapero, 1974; Weidlich and Haag,

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STATISTICAL MECHANICS FOR MICRO-SOCIOLOGY 5

1983). Social analogies to physical concepts like mass, gravity, andtemperature are older than the 197O'sr(e.g., Catton, 1965), but the firstquantitative links came with the realization that the same probabilisticlaws used to describe physical systems can apply equally well to othermulticomponent systems like individuals in large groups (e.g., Helbing,1994). The power of more recent approaches, therefore, has not been inthe analogy to physics but in the underlying probabilistic mathematicswhich can be used in many different fields of study, including physicsand sociology.

Recent collaborations between social scientists and physicists haveled to substantial refinements of the original sociological theories whichwere borrowed almost intact from the physical theory of statisticalmechanics. Bibb Latané's (1981) theory of social impact, for example,has been used to explain the micro-sociological interactions necessaryfor the transition from merely physics-like models to truly sociologicalmodels of collective behavior (Nowak et al., 1990; Lewenstein et al.,1992; Latané et al., 1994). Most significantly, this new theory can bestraightforwardly implemented as a computer simulation of the inter-actions between many individuals. These pioneering simulations use avariant of the "voter model" (which has a long history of study; e.g.,Durrett, 1988) or "majority rule" model from physics to examine thedynamics of opinion formation. In its simplest form these models spec-ify that every individual interacts with a subset of a group and choosesan opinion which matches the majority opinion of the subset.

These and other computer simulations have been successful inexplaining numerous qualitative features of real opinion formation ingroups (Granovetter, 1978; Latané et al, 1994), but a primary draw-back of these techniques is the assumption of majority rule and rationalchoice. While rational choice models may be well suited to some eco-nomic and political decisions, there are other social situations whereindividuals will not unfailingly choose the majority opinion. Macy(1990, 1991), for example, in his papers on thresholds, uses computersimulations to show that irrational choices can significantly change thecollective behavior of groups. However, while giving important newresults, his models give no mathematical foundation for the formula-tions used when deciding if an individual has changed opinions. With-out this micro-level foundation, it is difficult to have confidence in acomputer simulation "(or mathematical description) of the macro-level

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6 D. B. BAHR AND E. PASSERINI

behavior. Likewise, although Latané's work (1981) does specify a math-ematical micro-level foundation, a careful analysis shows that the basicform of the micro-sociological interactions is incompatible with theempirical data he compiled. According to Latané's social impact the-ory, the extent to which an individual's opinion is influenced by mem-bers of a group depends (among other factors) on the number ofindividuals in the group. In particular, the probability of being con-verted from one opinion to another is P&=sNt where iVis the numberof individuals with the opposing opinion and s and 0 < t < 1 are empiri-cally determined constants. While this power law fits the empirical datareasonably well (and forms the basis for micro interactions in the votermodel), it is immediately apparent that this formulation cannot be truefor all group sizes. For large enough N, the probability of changingopinions will be greater than one, which is impossible (all probabilitiesmust be between zero and one (Ross, 1988)). In other words, if the sameempirical experiments used to support the power law relationship wereperformed again with larger groups, the original fit would be very poorfor the new data.

Because it is crucial to a proper micro-level development but has notbeen mentioned elsewhere, we note that for situations when one or a fewindividuals are being influenced by a group of many, the empirical datashould satisfy three constraints. First, because PA is a probability,

O ^ P A ^ I . (1)

Second, the social impact (or probability of changing opinion) shouldincrease with increasing numbers of group members, i.e., PA(N)^PA(N+1) for all N, or equivalently

%><>•

Finally, the amount of additional impact due to the Mh person shouldbe greater than the additional impact due to the N+ 1st person. As sug-gested by Latané (1981), this is analogous to money. If you have onlyone dollar, then one more dollar is a substantial increase, but if youhave a million dollars then an additional dollar is barely noticeable.

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STATISTICAL MECHANICS FOR MICRO-SOCIOLOGY 7

i.e., we postulate that P&(N)-PA(N-l)^PA(N+\)-PA(N), orequivalently,

All three of these constraints are supported by the available numericaldata which is between 0 and 1 by definition and appears to be mono-tonically increasing and roughly concave downwards (see, for example,data in Latané, 1981; Tanford and Penrod, 1984). The first constraint isessential, although the second and third constraints should be regardedas strong intuitive guidelines rather than strict requirements. It is pos-sible, for example, to imagine that the Nth person is a conformist, whilethe N+ 1st person is a strong leader with much greater influence. How-ever, over large ranges of N, these constraints should be approximatelycorrect.

While the micro-sociological interactions in many computer simula-tions of collective behavior can be easily changed, the difficulty isfinding a theoretically grounded probabilistic description that agreeswith the three constraints and fits known observations of opinion for-mation. We note, for example, that in an alternative theory of socialimpact, Tanford and Penrod (1984) propose that if T individuals arethe target of influence from N people then PA=aë ~k/ \ This relation-ship fits the data reasonably well (Tanford and Penrod, 1984) andsatisfies the three constraints (except for the concavity at small iV).However, the form of this relationship is not justified by any analyticaltheory, and because of the paucity of empirical data (only five datapoints in some cases), many different two parameter relationships willfit the observations with large coefficients of correlation. In the follow-ing section, therefore, we outline the foundation of a probabilisticmicro-sociological theory which agrees with the constraints and pre-dicts a specific fit to some available data.

Before continuing with the theoretical derivations, however, it isworth noting that our approach is fundamentally different from mostprevious work on collective behavior. Some well known previous stud-ies, for example, use threshold, learning, and network theories (e.g.,Granovetter, 1978; Macy, 1991; Marwell, Oliver and Prahl, 1988). Ourapproach, on the other hand, is grounded entirely on the basic

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8 D. B. BAHR AND E. PASSERINI

principles of probability theory and uses a probabilistic description ofevery actor's interactions. The work most closely related to this paper isthe social psychology by Latané (1981) and Lewenstein et al. (1992), butour analysis contains a different description of the micro-sociology.Latané's philosophy of social impact is similar to our philosophy ofsocial interactions, but there is little overlap between their results andour results, and our derivations are completely new and do not useLatané's derivations.

Threshold and critical mass models are also related to the statisticalmechanical approach. In thresholds, it is hypothesized (or argued bysome means) that actors will join in a decision only after some criticalnumber of other actors have made the same decision (e.g., Granovetter,1978). In the statistical mechanical model, this is translated into a proba-bilistic statement - the more people who have already made a decision,then the more probable it is that an actor will also make that samedecision. Critical mass and threshold models suggest that there is a fairlysharp transition at the critical mass from non-participatory to partici-patory behavior, but by itself, the probabilistic formulation would notseem to lead to sharp transitions. However, without appealing to anyestimate (rational or irrational) of an actor's costs and benefits (e.g.,Macy, 1991; Oliver et ah, 1985), the statistical mechanics predicts thatfor large groups the probabilities turn into a very sharp transition aroundsome critical number of people (who have already made a given deci-sion). Below the critical number of people it is highly improbable thatthe actor will join their decision. Above the critical number it is highlyprobable that the actor will join their decision. The statistical mechan-ical theory, therefore, actually predicts the size of the threshold forinteractions in large groups. Because of this, the probabilistic approachgives the mathematical basis for treating critical mass and thresholdstudies as a logical set of consequences of statistical mechanical theory.

Because this paper presents a very different and new type of socio-logical theory, the following derivations draw on a number of analogiesfrom the physical sciences. However, the analogies are useful onlybecause physicists use some of the same mathematical language andhave already given names to many of the variables and equations in theprobability theory. These variables, like "temperature", have intuitivephysical meanings; but because they are only mathematical constructs,the same variables apply to different situations in sociology. (No one

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STATISTICAL MECHANICS FOR MICRO-SOCIOLOGY 9

would claim, for example, that just because it has the same name a"standard deviation" always refers to the same data, no matter whatthe application. Instead, a standard deviation is useful as a variablefor many different situations.) All of the quantities in this paper aredefined sociologically, are in an unambiguous mathematical format,and are completely independent of physical theories.

2. MICRO-SOCIOLOGY OF OPINION FORMATION

Imagine a group of TV* people, and suppose that n l have opinion one,n2 have opinion two, and n3 have opinion three, so that nl +n2 + n3 =N. If each member of the group is equally persuasive, then it isreasonable to assume that a newcomer to the group (without a pre-formulated opinion) will choose opinion one, two, or three based onthe number of individuals with each opinion. This is supported by self-attention theories (Mullen, 1983) as well as aspects of the decisionmaking and social influence theories of Camilleri and Conner (1976).In a general sense, both threshold and critical mass models -also positthat an actor's decision will depend on the number of other people whohave made a given decision. Says Granovetter (1978), "the costs andbenefits to the actor of making one or the other choice depend in parton how many others make which choice". So if everyone in the grouphas opinion one, for example, then the newcomer is most likely tochoose opinion one. In other words, the probability P¡ of choosingopinion i is proportional to n¡. So in this example,

(4)

(5)

and

P3 = "3 (6)n1+n2 + n3

where division by the total number of people, N=nl + n2 + n3, ensuresthat the probabilities are less than one (as required) and thatP1 + P2 + P3 = 1 (as required).

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10 D. B. BAHR AND E. PASSERINI

If we assume that not all individuals have equally strong opinions,then the probability of choosing one, two or three is proportional tothe total strengths of each subgroup. If the subgroup of opinion one,for example, has a very strong and persuasive leader while opinionstwo and three do not, then the probability of choosing opinion one willbe highest. Let/?y be the strength of opinion or the ability of individualj to persuade (or support) individual /. It follows that the probability ofperson / choosing opinion one will depend on the sum of the strengthsPij of the people j with opinion one. Similarly, for opinions two andthree, the probabilities are proportional to the sum of the strengths ofthe individuals with opinions two and three, i.e.,

j=i

î Pu (8)= l

PÁÍ)=TPJ Î PiJ (9)3 / j = l

where the numerator is a sum over all group members with a givenopinion (1, 2, or 3), and the denominator is a sum over all groupmembers. If everyone has the same strength of opinion, then thesereduce to equations (4), (5), and (6), as expected.

Note that pi} does not have to equal p)t ; individual i could be quitegood at convincing individual j , for example, but j might be shy andunable to convince /. In fact, individual j could choose (for whateverreason) to ignore individual i and not be any influence at all. Similarly,individual^ could try to influence /, but i might choose to ignore or onlyvaguely consider the arguments of y. In other words, actors are notrequired to influence (or be influenced) if they choose not to. The variablePij, therefore, reflects the two-way interactions between individuals andnot just the strength of opinion of one individual; this is consistent withthe nature of "asymmetric" interactions in many social network the-ories (e.g., Marwell, Oliver, and Prahl, 1988). As such, an alternativelabel might be "interaction strength", rather than "opinion strength",but we use the latter to indicate that ptJ may change depending on theissue. Normally mild-mannered and uninspired people, for example,

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STATISTICAL MECHANICS FOR MICRO-SOCIOLOGY II

may become quite impassioned and convincing when the issue andopinions are in some way personal.

The opinion of each member of a group can be represented as avariable. For example, opinion one could be represented by a " — 1",opinion two by a "0", and opinion three by a "1". In general, if a groupis of size N with m opinions represented by the variables ax, <x2,..., <rm,then the probability that an individual has opinion k (1 < k ^ m) is

' L,j=lPij

where s¡ is the opinion of the yth group member and the productn™= 1.1 sit Jus t ensures that only the group members with opinion kcontribute to the sum.

Examples with One Opinion versus ManyMany empirical observations involve interactions in groups with oneperson having a different opinion from the rest (e.g., Latané, 1981;Tanford and Penrod, 1984). So consider the case where one individualof opinion at is being influenced by Nx other individuals with opiniona2- From the theory outlined above, the probability that the loneindividual / will retain opinion a1 is

where pa is the individual's personal opinion strength (or self confi-dence, ability to persuade himself/herself and stand by his/her beliefs,etc.). The probability of changing opinions to a2 is

P2 = l-P1. (12)

If all the opinion strengths are very similar (no relatively shy groupmembers or outspoken leaders), then pv¡ is the same constant for all iand y, and equations (11) and (12) reduce to

and

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12 D. B. BAHR AND E. PASSERINI

In other words, the probability that the one individual will changeopinions to match that of the group is given by P&=P2- Note the strongresemblance of equation (14) (shown in Figure 1) to the general shape ofthe empirical data (e.g., Latané, 1981). As desired, equation (14) alsosatisfies the fundamental constraints in equations (1), (2), and (3).

More generally, the opinion strengths will vary widely within agroup. In this case, equations (11) and (12) can be simplified using thestrong law of large numbers (Ross, 1988), which implies £7=i-P>7~N(pij} for large A . ( . . . ) means the average of the enclosed quantity, inthis case the average opinion strengths of the entire group. Therefore,for large groups,

Pii

andN<PiJ>

P —r 2 —

Pa

(15)

(16)

The factor p¡¡/(p¡j} is a constant for any given group, so the probabil-ity that an individual will change opinions to match that of a large

No

FIGURE 1 Plots of the probability that an individual will change opinion when thereare N members of a group with the opposite opinion. The parameter c depends onopinion strengths. Equation (14) corresponds to the case c=l . If the strengths are notequal (c 1) then these curves are valid only for large A due to the law of large numbers.

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STATISTICAL MECHANICS FOR MICRO-SOCIOLOGY 13

group is PA=l—c/N for some constant c. Again, this gives a family ofcurves (shown in Figure 1) which look like the data on opinion forma-tion and which agree with the fundamental constraints (1), (2), and (3).

If we conduct many independent experiments designed to study theinfluence of large groups on individuals, then the lone individuals willhave different opinion strengths in each experiment. The average (orexpected) probability can then be calculated from these many experi-ments as

Lj=iPij/ \Lj=iPij/

In other words, the probability that an individual will convert to agroup opinion depends on the group size as PA = 1—cA/V* with CK\when averaged over many different social situations.

The probabilities Pk apply equally well to situations with an entiregroup being influenced by just one individual. In fact, in most socialsettings, if one individual has an opposing viewpoint, then the groupis simultaneously influencing that individual while being influenced bythe individual. For two opinions, the probabilities are exactly the sameas in the previous case. The probability that a member of the groupwill change opinions from a2 to o is P1 = l — P2 = cIN in the generalcase; and Px = 11N when we consider equal opinion strengths or aver-ages over many experiments. Again these probabilities agree remark-ably well with the general shape of the data on social interactions (e.g.,tipping behavior and chivalry (Latané, 1981)).

Examples with Minority versus Majority OpinionsThe probabilistic formulation of choosing opinions can be used toexamine the more general possibility that n group members (with thesame opinion) will change opinions to match that of M others. Sup-pose that out of N total group members, M have opinion one and theremaining N— M members have opinion two. Note n N—M. Supposeagain that we conduct many experiments and consider the expected(average) probabilities, or assume that N is large and one of M« N or

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14 D. B. BAHR AND E. PASSERINI

N—M«N, so that the law of large numbers applies. Then by the samearguments as before, our theory shows that

P ^ (18)

and

Therefore, the probability that any n members (out of the total N) willchoose opinion one is

Nl /MV/, M\N-"

) • ( 2 0 )

If the M individuals are not allowed to change opinions (because, forexample they are confederates in league with each other, or are paidnot to change), then the probability that n of N— M individuals withopinion one will change to opinion two is

(N-M)\ /MV/. MY'"-"

Note that equations (20) and (21) are binomial distributions, with peakvalues at n = M and n = (N—M)MIN respectively. Therefore, if every-one is allowed to change opinions, then it is most probable that thenumber of individuals with opinion one will remain exactly the same.If there are M "confederates" who will not change opinions, then it ismost likely that an additional (N-M)MIN individuals will join them.

Critical Mass and ThresholdsOften we are not interested in the exact number of individuals who willchange opinions but are interested in finding the size of M necessary toconvince n or more individuals to change opinions. This is the typeof question asked by social movement organizers trying to promote

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STATISTICAL MECHANICS FOR MICRO-SOCIOLOGY 15

collective action, or by political candidates who need to swing a keynumber of votes. Let C be the number of converts from opinion two toopinion one. Then the probability that C is greater than or equal to n is

p(o«)= £ no- (22)

For large enough N, P'(i) is approximated by a Guássian distribu-tion (central limit theorem) (Ross, 1988), and the summation can bereplaced by an integral which leads to the so-called "error function".By standard arguments,

(23)IM) 2 \J2M(ÍN-M)IN))

(Note that for large N, the first term always evaluates to 1/2.) Forexample, our theory shows that in a group of 100 people with 30guaranteed to vote for a particular candidate, the probability of 21 ormore additional individuals voting for this candidate is

(24)-30)/100)/

In other words, if roughly 30% of a group's votes can be guaranteedfor a particular candidate and the rest of the votes are uncommitted,then the candidate has a roughly 50% chance of winning the election.

Note that equation (23) predicts threshold and critical mass behav-ior. As M increases, the likelihood of others taking action jumps dra-matically when M reaches a critical number. In particular, for largeN, P(C ri) has an extremely sharp transition from a probability nearzero to a probability near one. The transition is at the critical mass M(Oliver et al., 1985), and it is a consequence of the error function whichhas a classic "S"-shaped curve (Figure 2). The "S"-shape is central tomany threshold and critical mass theories (see, for example, equation (1)of Macy, 1991), but rather than being assumed as in most papers, theerror function is predicted by the probabilistic arguments of this paper.

The exact value of the critical mass is given by the point where thesecond derivative of equation (23) with respect to M is equal to zero

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16 D. B. BAHR AND E. PASSERINI

0 .8

Om 0 .6Al

O.°- 0.4

0 .2

20 40 eo 80 100

M

FIGURE 2 A plot of the probability that 50 or more people will change opinions andjoin a collective decision when the total group size is 1000 (i.e., n = 50 and //=1000).Note the "S"-shaped transition at the critical mass.

(i.e., at the inflection point). This has an exceptionally complicatedform, so numerical estimates are better. Numerical solutions show thatthe critical mass is approximately the number of desired convertsminus one, n — \. In other words, a mass of « —1 people is required tomobilize n more to a collective action.

Before placing too much emphasis on this particular number, how-ever, recall that equation (23) predicts a critical mass for situationswhere the law of large numbers applies. In other words, it uses the lawof large numbers to average the effect of all the different opinionstrengths, ptj. The results, therefore, indicate the average critical mass -the mass averaged over many different scenarios with different distribu-tions of Pij. This precludes the impact of strong leaders and easilyinfluenced followers, two personality types which will clearly changethe size of the critical mass.

In any specific circumstances, thresholds for action will vary fromindividual to individual (Granovetter, 1978), and some will act at theaverage critical mass, while others will wait for an even larger mass. Inother words, there is a distribution of responses which are centeredaround the critical mass. This distribution could be explicitly consideredby including p¡j in the analysis, either analytically or numerically. How-ever, as the form of equation (10) might suggest, the complete analysis

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STATISTICAL MECHANICS FOR MICRO-SOCIOLOGY 17

with pij can be quite complicated, while the average response presentedabove gives a clearer and more succinct picture. Rather than deriv-ing the more complicated distribution of thresholds, we continue theanalysis with several additional factors influencing micro-sociologicalinteractions.

2.1. Social Temperature and External Forces

Other factors, besides individual opinion strength and the number ofpeople who have a certain opinion, will influence decision processes andopinion formation in groups. Wars, for example, can encourage nationalcooperation when a nation might otherwise remain divided in opinion;and hurricanes and earthquakes can encourage votes for strict buildingcodes which were previously unpopular. Similarly, angry or excitedsports fans can stampede and riot, while the same fans might donothing under different circumstances. During periods of social unrest,groups of people may be more likely to vote for changes than duringperiods of social quiescence. There is also an element of unpredictability(or "noise") due to micro-level social interactions and personal deci-sions; miscommunications, for example, can lead to otherwise unlikelydecisions.

If the same group can behave differently when making the samedecision, then the factors controlling this different collective behaviorneed to be specified. By using quantitative analogies with physicalsystems, three social factors are identified: social temperature, socialforce, and noise. We discuss these factors qualitatively before givingquantitative definitions.

Social TemperatureConsider a measurement which quantifies the extent to which a group'saverage opinion is susceptible to changing. In the past this quantityhas been called disposition, susceptibility, suggestibility (Johnson andFeinberg, 1977), contagion (LeBon, 1960), social facilitation (Allport,1924), and circular reaction (Blumer, 1936). However, these previouslydefined concepts (with the exception of Blumer) were meant to applyto individuals and not groups. Because we think it is a more accuratelabel when referring to entire groups (and because it has a close analogyto physics) we call this measure "social temperature". Intuitively, it is

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18 D. B. BAHR AND E. PASSERINI

the volatility of each individual's decision; or a little more accurately,social temperature is an average (over the entire group) of each indi-vidual's volatility. However, temperature cannot be measured as a pro-perty of individuals, or even of very small groups, because details ofopinion strengths, personal interactions, and other micro-sociologicalfactors are too variable and unquantifiable. In other words, the aver-age temperatures of a small group would fluctuate too dramatically tobe meaningful. The same is true in physics. Temperature is a conceptwhich only applies to a large ensemble of molecules, like the ~ 1023

molecules of water in a glass. The detailed interactions of only a fewmolecules are too varied to attempt defining a temperature. In bothphysics and social sciences, temperature is valid only as a statistical or"average" large group concept.

Individuals are influenced by the group's temperature. When agroup's social temperature is high, very little provocation is necessary toinduce an individual to change opinion. At low group temperatures,individuals appear more phlegmatic or stubborn and much greaterprovocation is required to induce a change in opinion. High socialtemperatures amplify the slightest excuse for change, while low tem-peratures diminish the arguments for change. Note that each individ-ual's opinion strength is unaltered, but as the group's temperaturechanges, so does an individual's decision making abilities. Although weare only appealing to analogies at this point, as shown shortly, tempera-ture in sociology is a completely defined mathematical concept with noambiguity. Any apparent ambiguity comes from the difficulty of describ-ing a mathematical concept with words.

Social ForcesSimilar to the inclusion of social temperature, "social forces" should beincorporated. Floods, news reports, advertisements and other influ-ences external to a group can bias opinions, as can sanctions and laws(as discussed in Heckathorn, 1993). The presence of external or environ-mental social forces can push a divided group toward a majority opin-ion, or can force defections in a previously unified group. The greater asocial force, the more biased in the "direction" of the force is the collec-tive behavior of a group. Like social temperature, social forces have ananalogy with forces in a physical system. However, as with tempera-ture, social forces have a precise mathematical definition independent

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STATISTICAL MECHANICS FOR MICRO-SOCIOLOGY 19

of the physical analogy. Hebling (1994) gives a detailed description ofsocial forces and their influence.

NoiseFinally, there needs to be some measure of miscommunications andpotential misunderstandings - a near certain decision can be thwartedby bad information, mistakes, misinterpretations and apparently ran-dom disruptions. This is "noise" and is added into the decision makingprocess as a random perturbation of the probabilities in equation (10).Noise, for example, allows for an individual's apparently inexplicablechange to a dissenting opinion, even when the entire group has previ-ously settled on another opinion. The analogy to physical systems isexact, with both physical and social noise accounting for unknown orotherwise unquantifiable perturbations to the system.

Quantifying the AnalogiesTo incorporate the intuitive concepts of social temperature ( Ts), socialforces (/zs), and noise into the micro-sociological theory, define a func-tion A H such that the probability of an individual changing opinions is

e-AH

P * — ( 2 5 )

where Z is a constant factor. For simplicity, we restrict the followinganalysis to opinion formation in groups with only two choices (a1 = \and a2 = — 1) and with equal opinion strengths; the derivations areeasily generalized. If an individual has opinion sh then

nJN, i f j , = - l ;\J T , (26)IN, ifs,= l.

Therefore, for the special case of Z=l ,

-log(n./A0, i f i ,= - l ;

-log(«2/A0, ifs¡ = l.

Now briefly consider how temperature and force would be defined ifAH were a physical rather than a sociological concept (this will helpguide our derivations later). By analogy with physical statisticalmechanics, AH is the change in the "energy" of the system associated

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20 D. B. BAHR AND E. PASSERINI

with a change in opinion (Ma, 1985). The probability of change in aphysical system is decreased if it requires lots of energy. Physical tem-perature, T and physical forces, A, would be added to the probabilityof change as

iP¿})-JZ • (28)

T and h in the physical system play the same role as our intuitivedefinitions of social temperature and social forces in the social system.High (or low) T, for example, amplify (or diminish) the change inenergy required for a change in opinion. With a high T, fewer resources(i.e., less energy) are needed to induce a change. Likewise, a largepositive force, h, makes the probability of changing to al = 1 muchhigher than the probability of changing to <r2= — 1. Similarly, a largenegative force biases the opinions towards CT2 = — 1.

To quantitatively preserve the analogies between physical tempera-ture T and social temperature Ts and physical force h and social forcehs, P& must be modified to incorporate T s and hs in such a way thatthe form of equation (28) is unchanged. Let

J{\lz)^(nJNy, if ,,= -1;A{) lThen by taking logarithms it follows that TS = T, h%=h, z=Z, andthe structure of equation (28) is preserved. The constant z just ensuresthat probabilities are conserved (Y.s¡^&—^) a°d therefore, fromequation (29),

/ n in \1/T

(z is called the partition function in statistical mechanics.)We can now step back and define social temperature and social forces

to be mathematically defined as the quantities which vary as Ts and hs

in equations (29) and (30). These definitions are then completely inde-pendent of the physics, which has only helped to guide our intuitiontowards mathematical forms appropriate for sociology. (It is worth

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STATISTICAL MECHANICS FOR MICRO-SOCIOLOGY 21

noting, for example, that the sociological form of AH in equation (27)is completely different from its form in physics). Most importantly,though, the quantities Ts and hs defined in equations (29) and (30)agree with all of the qualitative descriptions of social temperature andsocial forces discussed above.

Using equations (29) and (30), the probabilities of choosing a givenopinion can now be generalized as

J (31)1/T*

(where nl have opinion al = 1 and n2 have opinion <x2 = — 1). Note tha tP1 and P2 now have a form which agrees with our intuitions of theeffect tha t social temperature and social forces have on opinion forma-tion. Also note, that to include opinion strengths, nJN and n2IN canbe replaced by the right hand side of equation (10). M o r e than twoopinions are allowed by generalizing e±hslT' to e**'7"* for opinion k withthe constraint tha t £ t / 2 t = 0. Noise (from miscommunicat ions andother unknown sources) can be included as r andom numbers 0 < (f>k < 1added to Pk subject to the constraint that Y.k4>k—Q-

Empirically, once an issue has been identified, the social tempera-ture of a large group can be measured with a simple poll. For a largerepresentative subset of a group, poll for the percentage of individualswith each opinion ak. This gives nJN, n2IN, etc. in equat ions (31) and(32). (Because the sample size is large, the different opinion strengthsof the individuals can be treated as an average strength, which foraverages of many different polls will factor away, as explained in theexamples above.) Poll again at a later time and determine the percentof individuals which have changed opinions. This gives Pu P2, etc. Ifthere are no external social forces, equations (31) and (32) can besolved simultaneously for the social temperature . If the social forcesare nonzero, then additional polls can determine nJN, n2IN, Plt andP2 a t yet another time. If the system is not in equil ibrium, then a setof four equations constructed from (31) and (32) can be used to solvefor the social temperature and the net social forces. If the system is in

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22 D. B. BAHR AND E. PASSERINI

or near equilibrium, then Pi and P2 will not change appreciably, andthe system of equations will be degenerate. Note that in these measure-ments there is no need to appeal to the intuition of Ts and hs. Instead,the mathematics of equations (29)-(32) completely define how socialtemperature and force should be measured.

2.2. Fitting the Theory to Empirical Data

To test the theory derived above, equations (29) and (30) are fit tosome available empirical data. We have used particularly appropriatedata culled from graphs published in Latané (1981). Although someloss of accuracy is inevitable when transcribing information fromgraphs, the data is still representative, and the fits will still indicatewhether the statistical mechanical theory is reasonable. While usingempirical data from Latané, we emphasize that all of the fits are donesolely with equations from our theory outlined above.

Technically, unless the opinion strengths p¡j are included, theprobabilities of choosing an opinion, P1 and P2, apply only to verylarge groups. Unfortunately, empirical experiments are typically onsmall groups with no objective measures of opinion. Never-the-less,the large group formulation can be fit to the small group data as anapproximation. Consider experiments where one individual is beinginfluenced by many. For large N, z approaches one, and by similararguments to those presented above,

1 (33)

where c=pit/(plJy (which requires the law of large numbers). hs isassumed to be zero because the experimental data was designed to holdexternal influences to a minimum (although with a little extra complica-tion hs could be incorporated as a part of the factor c). Figure 3 showsequation (33) fit to two different data sets. Visually, the fits are good,and the coefficients of correlation are high (/?2=0.88 and R2=0J2).The micro-sociological theory developed in this paper, therefore, does agood job of explaining previously compiled data on the influence ofgroups on individuals.

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STATISTICAL MECHANICS FOR MICRO-SOCIOLOGY 23

(a)0.4

0.35D 0 .3A 0.25

0.20.15

0.10.05

0 2 4 6 8N

(b)1

0.8

P A 0.6

0.4

0.2

0 2 4 6 8 10 12 14N

FIGURE 3 (a) Equation (33) fit to data on conformity with c=0.31 and T=7.9(R2 = 0J2). P¿ is the probability that an individual will conform to the group opinionwhen N individuals give an obviously incorrect answer, (b) Equation (33) fit to data onconformity with c = 0.38 and T-X.9 (£2 = 0.88). P& is the probability that an individualwill gawk at an object when N members of a group are already gawking. (For detailson the data, see Latané, 1981.)

For experiments with many individuals being influenced by one, thelarge N approximation is

cy/rs (34)

For empirical data compiled by Latané (1981) on tip size, chivalry formen, and chivalry for women, equation (34) predicts the data withi?2 = 0.95, J?2 = 0.94, and i?2 = 0.82 respectively. Another data set

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24 D. B. BAHR AND E. PASSERINI

examining the number of people inquiring about Christ (after talks byBilly Graham) involves large groups so that equation (34) is a morereasonable approximation. However, scatter in the data makes a goodfit to any function unlikely, including equation (34), and R2=0.22. Ingeneral, however, the statistical mechanical approach does a good jobof explaining the data on small groups influencing individuals.

It is interesting to note that equation (34) is the same as one of theequations predicted by Latané (1981). However, the agreement is for-tuitous and our theory matches Latané's on only this one point. Thepower law used for this case by Latané is acceptable because N is inthe denominator so that the probability remains less than one. On theother hand, the power law proposed by Latané for the different caseof one individual being influenced by many, incorrectly places N in thenumerator. The correct expression for that situation is found in equa-tion (33). Because the power laws are the same for the specific case ofone individual influencing many, the fits done by Latané match thefits done in the previous paragraph. As a result, the R2 are also thesame. However, in general our probabilistic approach will predict dif-ferent fits and different coefficients of correlation.

3. CONCLUSIONS

Note that the preceding analysis assumes only that actors consider theopinion of those individuals with whom they interact, and then weighttheir decision according to the number of other actors with each opin-ion. What evaluations go into this consideration is not specified, noris the cognitive level of consideration specified (it could range fromunconscious to fully conscious). The consideration could include acost/benefit analysis (e.g., Oliver and Marwell, 1988) or any other deci-sion making process. However, the advantage of equations (10), (31)and (32) is that they do not have to specify the exact system of rewardsand punishment or cost and benefit. In fact, this analysis does not haveto assume rational choice and reward driven behavior at all. Theseconsiderations happen at a level which is more detailed than isrequired to specify the micro-sociological interactions appropriate toour model.

By definition, micro-sociology seeks to explain the interactionsbetween a small number of individuals. Without delving into the detailed

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STATISTICAL MECHANICS FOR MICRO-SOCIOLOGY 25

social processes of each individual (e.g., social conventions, norms,rituals, perceived rewards, roles, etc.), the previous derivationsdescribe these interactions. Of course, there is another level of micro-sociology that explains the more detailed processes involved in makingthe interactive decisions (conversation analyses, symbolic interactionstudies, ethnomethodologies, etc.). With that in mind, the theory pre-sented here might be referred to as a meso-scale sociology. We leavethat as largely a matter of semantics. However, we do note that socialtemperature, social forces, and noise play important roles in eachactor's opinion formation process, and that despite their macro-levelderivations, these parameters influence decision making behavior at alevel which would typically be described as micro-sociological.

One interesting aspect of the statistical mechanical micro-sociologicaltheory is its prediction of a critical mass and threshold behavior.Assumptions about possible cost and benefit structures are unnecessaryto show that there is a sharp transition (threshold) from inaction toaction when some critical number (mass) of actors are already partici-pating. The "S"-shaped functions commonly used to produce criticalmass behavior in other theories are predicted as a direct consequence ofthe probabilistic structure of the statistical mechanical theory.

As noted, the statistical mechanical theory could be better tested ifthere was a method for quantitative measurements of ptj. Collectinglarger data sets under controlled environments (so that hs is known orfixed) would also permit better tests. Both of these options, however,could be difficult, subjective, and economically infeasible. A morepromising approach, therefore, is to take advantage of the inherentblurring, already mentioned, between micro- and macro-level processesin the theory. The inherent links between micro and macro make predic-tions of large scale collective behavior possible by studying a network ofmicro-sociological interactions. These predictions could then be com-pared to empirical observations of macro-sociological behavior. Thetools and links needed to make such connections are explored in acompanion paper.

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