Structural balance: A generalization of Heider's theory

VOL. 63, No. 5 SEPTEMBER, 1956

THE PSYCHOLOGICAL REVIEWSTRUCTURAL BALANCE: A GENERALIZATION

OF HEIDER'S THEORY'

DORWIN CARTWRIGHT AND FRANK HARARY

Research Center for Group Dynamics, University of Michigan

A persistent problem of psychologyhas been how to deal conceptuallywith patterns of interdependent prop-erties. This problem has been cen-tral, of course, in the theoreticaltreatment by Gestalt psychologists ofphenomenal or neural configurationsor fields (12, 13, 15). It has alsobeen of concern to social psychologistsand sociologists who attempt to em-ploy concepts referring to social sys-tems (18).

Heider (19), reflecting the generalfield-theoretical approach, has con-sidered certain aspects of cognitivefields which contain perceived peopleand impersonal objects or events.His analysis focuses upon what he callsthe P-O-X unit of a cognitive field,consisting of P (one person), 0 (an-other person), and X (an impersonalentity). Each relation among theparts of the unit is conceived as inter-dependent with each other relation.Thus, for example, if P has a relationof affection for 0 and if 0 is seen asresponsible for X, then there will bea tendency for P to like or approveof X. If the nature of X is such thatit would "normally" be evaluated asbad, the whole P-O-X unit is placedin a state of imbalance, and pressures

1 This paper was prepared as part of aproject sponsored in the Research Centerfor Group Dynamics by the RockefellerFoundation.

will arise to change it toward a stateof balance. These pressures maywork to change the relation of affec-tion between P and 0, the relation ofresponsibility between 0 and X, or therelation of evaluation between PandX.

The purpose of this paper is topresent and develop the consequencesof a formal definition of balance whichis consistent with Heider's conceptionand which may be employed in a moregeneral treatment of empirical con-figurations. The definition is statedin terms of the mathematical theoryof linear graphs (8, 14) and makes useof a distinction between a given rela-tion and its opposite relation. Someof the ramifications of this definitionare then examined by means oftheorems derivable from the definitionand from graph theory.

HEIDER'S CONCEPTION OFBALANCE

In developing his analysis of bal-anced cognitive units, Heider dis-tinguishes between two major types ofrelations. The first concerns atti-tudes, or the relation of liking orevaluating. It is represented sym-bolically as L when positive and as ~Lwhen negative. Thus, PLO means Plikes, loves, values, or approves 0, andJP~LO means P dislikes, negatively

277

278 DORWIN CARTWRIGHT AND FRANK HARARY

values, or disapproves 0. The secondtype of relation refers to cognitiveunit formation, that is, to such specificrelations as similarity, possession,causality, proximity, or belonging.It is written as U or ~U. Thus,according to Heider, PUX means thatP owns, made, is close to, or is associ-ated with X, and P~ILY means thatP does not own, did not make, or isnot associated with X.

A balanced state is then defined interms of certain combinations of theserelations. The definition is statedseparately for two and for threeentities.

In the case of two entities, a balanced stateexists if the relation between them is positive(or negative) in all respects, i.e., in regard toall meanings of L and U . . . . In the caseof three entities, a balanced state exists if allthree relations are positive in all respects, orif two are negative and one positive (9, p. 110).

These are examples of balancedstates: P likes something he made(PUX, FIX); P likes what his friendlikes (PLO, OLX, PLX); P dislikeswhat his friend dislikes (PLO, 0~LX,P~LX"); P likes what his enemy dis-likes (P~LO, 0~LX, PLX); and P'sson likes what P likes (PU0, PLX,OLX).

Heider's basic hypothesis assertsthat there is a tendency for cognitiveunits to achieve a balanced state.Pressures toward balance may pro-duce various effects.

If no balanced state exists, then forcestowards this state will arise. Either the dy-namic characters will change, or the unitrelations will be changed through action orthrough cognitive reorganization. If a changeis not possible, the state of imbalance willproduce tension (9, pp. 107-109).

The theory, stated here in sketchyoutline, has been elaborated by Heiderso as to treat a fuller richness of cog-nitive experience than would be sug-gested by our brief description. Ithas been used, too, by a number of

others as a point of departure forfurther theoretical and empirical work.We shall summarize briefly some ofthe major results of this work.

Horowitz, Lyons, and Perlmutter(10) attempted to demonstrate ten-dencies toward balance in an experi-ment employing members of a discus-sion group as subjects. At the end ofa discussion period each subject wasasked to indicate his evaluation of anevent (PLX or P~LX) which hadoccurred during the course of the dis-cussion. The event selected for evalu-ation was one which would be clearlyseen as having been produced by asingle person (OUX). The liking re-lation between each P and 0 (PLOor P~LO) had been determined bya sociometric questionnaire admin-istered before the meeting. WouldP's evaluation of the event be such asto produce a balanced P-O-X unit?If so, P's evaluation of 0 and X shouldbe of the same sign. The experi-mental data tend to support thehypothesis that a P-O-X unit tendstoward a balanced state.2

The social situation of a discussiongroup can be better analyzed, accord-ing to Horowitz, Lyons, and Perl-mutter, by considering a somewhatmore complex cognitive unit. Theevaluation of X made by P, theyargue, will be determined not only byP's evaluation of 0 but also by hisperception of the evaluation of Xgiven by others (Qs) in the group.The basic unit of such a social situa-tion, then, consists of the subject, a

2 One of the attractive features of this studyis that it was conducted in a natural "field"setting, thus avoiding the dangers of artifici-ality. At the same time the setting placedcertain restrictions on the possibility ofmanipulation and control of the variables.The data show a clear tendency for P to placea higher evaluation on Xs produced by moreattractive Os. It is not clearly demonstratedthat P likes Xs produced by liked Os anddislikes Xs produced by disliked Os.

STRUCTURAL BALANCE 279

person who is responsible for theevent, and another person who will beseen by the subject as supporting orrejecting the event. This is calleda P-O-Q-X unit. The additional dataneeded to describe these relations wereobtained from the sociometric ques-tionnaire which indicated P's evalua-tion of Q (PLQ or P~L@), and froma question designed to reveal P's per-ception of Q's support or rejection ofX, treated by the authors as a unitrelation (QUX or Q~UZ).3

Although these authors indicate thepossibility of treating the P-O-Q-Xunit in terms of balance, they do notdevelop a formal definition of a bal-anced configuration consisting of fourelements. They seem to imply thatthe P-O-Q-X unit will be balanced ifthe P-O-X and the P-Q-X units areboth balanced. They do not considerthe relation between Q and 0, nor thelogically possible components of whichit could be a part. Their analysis isconcerned primarily with the twotriangles (P-O-X and P-Q-X), whichare interdependent, since both containthe relation of P's liking of X. Wenoted above that the data tend tosupport the hypothesis that the P-0-X unit will tend toward balance.The data even more strongly supportthe hypothesis when applied to theP-Q-X unit; P's evaluation of X andhis perception of Q's attitude towardX tend to agree when P likes Q, andto disagree when P dislikes Q. Itshould be noted, however, that therewas also a clear tendency for P to seeQ's evaluation of X as agreeing withhis own whether or not he likes Q.

In a rather different approach tothe question of balanced P-O-X units,

3 Whether this relation should be treated asU or L is subject to debate. For testingHeider's theory of balance, however, theissue is irrelevant, since he holds that the tworelations are interchangeable in definingbalance.

Jordan (11) presented subjects with64 different hypothetical situations inwhich the L and U relations betweeneach pair of elements was systemati-cally varied. The subject was askedto place himself in each situation bytaking the part of P, and to indicateon a scale the degree of pleasantnessor unpleasantness he experienced.Unpleasantness was assumed to reflectthe postulated tension produced byimbalanced units. Jordan's data tendto support Heider's hypothesis thatimbalanced units produce a state oftension, but he too found that addi-tional factors need to be considered.He discovered, for example, that nega-tive relations were experienced as un-pleasant even when contained inbalanced units. This unpleasantnesswas particularly acute when P was apart of the negative relation. Jordan'sstudy permits a detailed analysis ofthese additional influences, which weshall not consider here.

Newcomb (17), in his recent theoryof interpersonal communication, hasemployed concepts rather similar tothose of Heider. He conceives of thesimplest communicative act as one inwhich one person A gives informationto another person B about somethingX. The similarity of this A-B-Xmodel to Heider's P-O-X unit, to-gether with its applicability to objec-tive interpersonal relations (ratherthan only to the cognitive structure ofa single person), may be seen in thefollowing quotations from Newcomb:

A-B-X is ... regarded as constituting asystem. That is, certain definable relation-ships between A and B, between A and X,and between B and X are all viewed as inter-dependent. . . . For some purposes the sys-tem may be regarded as a phenomenal onewithin the life space of A or B, for otherpurposes as an "objective" system includingall the possible relationships as inferred fromobservations of A's and B's behavior (17,p. 393).


Newcomb then develops the con-cept of "strain toward symmetry,"which appears to be a special instanceof Heider's more general notion of"tendency toward balance." "Straintoward symmetry" is reflected inseveral manifestations of a tendencyfor A and B to have attitudes of thesame sign toward a common X.Communication is the most commonand usually the most effective mani-festation of this tendency.

By use of this conception Newcombreinterprets several studies (1, 4, 5,16, 20) which have investigated theinterrelations among interpersonal at-traction, tendencies to communicate,pressures to uniformity of opinionamong members of a group, andtendencies to reject deviates. Theessential hypothesis in this analysis isstated thus:

If A is free either to continue or not tocontinue his association with B, one or theother of two eventual outcomes is likely:(a) he achieves an equilibrium characterizedby relatively great attraction toward B andby relatively high perceived symmetry, andthe association is continued; or (i) he achievesan equilibrium characterized by relativelylittle attraction toward B and by relativelylow perceived symmetry, and the associationis discontinued (17, p. 402).

Newcomb's outcome a is clearly abalanced state as denned by Heider.Outcome b cannot be unambiguouslytranslated into Heider's terms. If by"relatively little attraction toward B"is meant a negative L relation betweenA and B, then this outcome would alsoseem to be balanced. Newcomb's"continuation or discontinuation ofthe association between A and B"appear to correspond to Heider's Uand ~TJ relations.

STATEMENT OF THE PROBLEM

This work indicates that the ten-dency toward balance is a significantdeterminant of cognitive organization,

and that it may also be important ininterpersonal relations. The conceptof balance, however, has been definedso as to apply to a rather limited rangeof situations, and it has contained cer-tain ambiguities. We note five spe-cific problems.

1. Unsymmetric relations. Shouldall relations be conceived as sym-metric? The answer is clearly thatthey should not; it is possible for Pto like 0 while 0 dislikes P. In fact,Tagiuri, Blake, and Bruner (21) haveintensively studied dyadic relationsto discover conditions producing sym-metric relations of actual and per-ceived liking. Theoretical discussionsof balance have sometimes recognizedthis possibility—Heider, for example,states that unsymmetric liking is un-balanced—but there has been nogeneral definition of balance whichcovers unsymmetric relations. Theempirical studies of balance have as-sumed that the relations are sym-metric.

2. Units containing more than threeentities. Nearly all theorizing aboutbalance has referred to units of threeentities. While Horowitz, Lyons, andPerlmutter studied units with fourentities, they did not define balancefor such cases. It would seem desir-able to be able to speak of the balanceof even larger units.

3. Negative relations. Is the nega-tive relation the complement of therelation or its opposite! All of thediscussions of balance seem to equatethese, but they seem to us to be quitedifferent, for the complement of a rela-tion is expressed by adding the word"not" while the opposite is indicatedby the prefix "dis" or its equivalent.Thus, the complement of "liking" is"not liking"; the opposite of "liking"is "disliking." In general, it appearsthat ~L has been taken to mean"dislike" (the opposite relation) while


~U has been used to indicate "notassociated with" (the complementaryrelation). Thus, for example, Jordansays: "Specifically, '+L' symbolizesa positive attitude, '—L' symbolizesa negative attitude, '+TT symbol-izes the existence of unit formation,and ' — IT symbolizes the lack of unitformation" (11, p. 274).

4. Relations of different types. Hei-der has made a distinction betweentwo types of relations—one basedupon liking and one upon unit forma-tion. The various papers followingup Heider's work have continued touse this distinction. And it seemsreasonable to assume that still othertypes of relations might be designated.How can a definition of balance takeinto account relations of differenttypes ? Heider has suggested some ofthe ways in which liking and unitrelations may be combined, but ageneral formulation has yet to bedeveloped.

5. Cognitive fields and social systems.Heider's intention is to describe bal-ance of cognitive units in which theentities and relations enter as experi-enced by a single individual. New-comb attempts to treat social systemswhich may be described objectively.In principle, it should be possible alsoto study the balance of sociometricstructures, communication networks,patterns of power, and other aspectsof social systems.

We shall attempt to define balanceso as to overcome these limitations.Specifically, the definition should (a)encompass unsymmetric relations, (&)hold for units consisting of any finitenumber of entities, (c) preserve thedistinction between the complementand the opposite of a relation, (d)apply to relations of different types,and (e) serve to characterize cognitiveunits, social systems, or any configura-tion where both a relation and itsopposite must be specified.

THE CONCEPTS OF GRAPH, DIGRAPH,AND SIGNED GRAPH

Our approach to this problem hastwo primary antecedents: (a) Lewin'streatment (IS) of the concepts ofwhole, differentiation, and unity, to-gether with Bavelas1 extension (2) ofthis work to group structure; and (b)the mathematical theory of lineargraphs.

Many of the graph-theoretic defini-tions given in this section are con-tained in the classical reference ongraph theory, Konig (14), as well asin Harary and Norman (8). Weshall discuss, however, those conceptswhich lead up to the theory of balance.

A linear graph, or briefly a graph,consists of a finite collection of points 4

A,B, C, • • • together with a prescribedsubset of the set of all unordered pairsof distinct points. Each of these un-ordered pairs, AB, is a line of thegraph. (From the viewpoint of thetheory of binary relations,6 a graphcorresponds to an irreflexive 6 sym-metric relation on points A, B, C,Alternatively a graph may be repre-sented as a matrix.7)

Figure 1 depicts a graph of fourpoints and four lines. The pointsmight represent people, and the linessome relationship such as mutual lik-ing. With this interpretation, Fig. 1indicates that mutual liking exists be-tween those pairs of people A, B, C,and D joined by lines. Thus D is inthe relation with all other persons,while C is in the relation only with D.

'Points are often called "vertices" bymathematicians and "nodes" by electricalengineers.

5 This is the approach used by Heider."A relation is irreflexive if it contains no

ordered pairs of the form (a, a), i.e., if noelement is in this relation to itself.

7 This treatment is discussed in Festinger(3). The logical equivalence of relations,graphs, and matrices is taken up in Hararyand Norman (8).


FIG. 1. A linear graph of four points andfour lines. The presence of line AB indicatesthe existence of a specified symmetric relation-ship between the two entities A and B.

Figure 1 could be used, of course, torepresent many other kinds of rela-tionships between many other kindsof entities.

It is apparent from this definition ofgraph that relations are treated in anall-or-none manner, i.e., either a rela-tion exists between a given pair ofpoints or it does not. Obviously,however, many relationships of in-terest to psychologists (liking, forexample) exist in varying degrees.This fact means that our present useof graph theory can treat only thestructural, and not the numerical,aspects of relations. While our treat-ment is thereby an incomplete repre-sentation of the strength of relations,we believe that conceptualization ofthe structural properties of relationsis a necessary first step toward a moreadequate treatment of the more com-plex situations. Such an elaboration,however, goes beyond the scope ofthis paper.

A directed graph, or a digraph, con-sists of a finite collection of pointstogether with a prescribed subset ofthe set of all ordered pairs of distinctpoints. Each of these ordered pairsAB is called a line of the digraph.Note that the only difference betweenthe definitions of graph and digraphis that the lines of a graph are un-ordered pairs of points while the linesof a digraph are ordered pairs ofpoints, An ordered pair of points is

distinguished from an unordered pairby designating one of the points as thefirst point and the other as the second.Thus, for example, the fact that amessage can go from A to B is repre-sented by the ordered pair (A, B), orequivalently, by the line AB, as inFig. 2. Similarly, the fact that A andD choose each other is represented bythe two directed lines AD and DA.

A signed graph, or briefly an s-graph,is obtained from a graph when oneregards some of the lines as positiveand the remaining lines as negative.Considered as a geometric representa-tion of binary relations, an s-graphserves to depict situations or struc-tures in which both a relation and itsopposite may occur, e.g., like anddislike. Figure 3 depicts an s-graph,employing the convention that solidlines are positive and dashed linesnegative; thus A and B are repre-sented as liking each other while Aand C dislike each other.

Combining the concepts of digraphand s-graph, we obtain that of ans-digraph. A signed digraph, or ans-digraph, is obtained from a digraphby taking some of its lines as positiveand the rest as negative.

A graph of type 2 (8), introduced todepict structures in which two differ-

B

FIG. 2. A directed graph of four pointsand five directed lines. An AB line indicatesthe existence of a specified ordered relation-ship involving the two entities A and B.Thusjbr example, if A and B are two people,the AB line might indicate that a messagecan go from A to B or that A chooses B.


ent relations defined on the same setof elements occur, is obtained from agraph by regarding its lines as beingof two different colors (say), and bypermitting the same pair of points tobe joined by two lines if these lineshave different colors. A graph of typer, T = 1, 2, 3, • • • , is denned simi-larly. In an s-graph or s-digraph oftype 2, there may occur lines of twodifferent types in which a line of eithercolor may be positive or negative.An example of an s-graph of type 2might be one depicting for the sameP-O-X unit both U and L relationsamong the entities, where the sign ofthese relations is indicated.

A path is a collection of lines of agraph of the form AB, BC, • • • , DE,where the points A, B, C, • • • ,D,E,are distinct. A cycle consists of theabove path together with the line EA.The length of a cycle (or path) is thenumber of lines in it; an n-cycle is acycle of length n. Analogously tographs, a path of a digraph consists ofdirected lines of the form AB, BC, • • •,DE, where the points are distinct. Acycle consists of this path togetherwith the line EA. In the later discus-sion of balance of an s-digraph weshall use the concept of a semicycle.A semicycle is a collection of linesobtained by taking exactly one fromeach pair_AB or BA^BC or CB, • • • ,DE or ED, and EA or AE. Weillustrate semicycles with the digraphof Fig. 2. There are three semicyclesin_this digraph^ AD± DA; AD, DB,BA ; and AD, DB, BA. The last twoof these semicycles are not cycles.Note that every cycle is a semicycle,and a semicycle of length 2 is neces-sarily a cycle.

BALANCE

With these concepts of graphs, di-graphs, and signed graphs we may

FIG. 3. A signed graph of four points andfive lines. Solid lines have a positive signand dashed lines a negative sign. If thepoints stand for people and the lines indicatethe existence of a liking relationship, thiss-graph shows that A and B have a relation-ship of liking, A and C have one of disliking,and B and D have a relationship of indifference(neither liking or disliking).

now develop a rigorous generalizationof Heider's concept of balance.

It should be evident that Heider'sterms, entity, relation, and sign of arelation may be coordinated to thegraphic terms, point, directed line, andsign of a directed line. Thus, for ex-ample, the assertion that P likes0 (PLO) may be depicted as a directedline of positive sign PO. It shouldalso be clear that Heider's two differ-ent kinds of relations (L and U) maybe treated as lines of different type.It follows that a graphic representa-tion of a P-O-X unit having positiveor negative L and U relations will bean s-digraph of type 2.

For simplicity of discussion we firstconsider the situation containing onlysymmetric relations of a single type(i.e., an s-graph of type 1). Figure 4shows four such s-graphs. It will benoted that each of these s-graphs con-tains one cycle: AB, BC, CA. Wenow need to define the sign of a cycle.The sign of a cycle is the product ofthe signs of its lines. For conveni-ence we denote the sign of a line by+1 or — 1 when it is positive or nega-tive. With this definition we see thatthe cycle, AB, BC, CA is positive ins-graph a (-f-l '+l '+l) , positive ins-graph b (+1 • — 1 • — 1), negative


Vc

A

\\\\ /

Vc

c dFIG. 4. Four s-graphs of three points and

three lines each. Structure a and b arebalanced, but c and d are not balanced.

in s-graph c ( + 1 • +1 • —1), and nega-tive in s-graph d ( — ! • —1 • —1). Togeneralize, a cycle is positive if it con-tains an even number of negativelines, and it is negative otherwise.Thus, in particular, a cycle containingonly positive lines is positive, sincethe number of negative lines is zero,an even number.

In discussing the concept of balance,Heider states (see 9, p. 110) that whenthere are three entities a balancedstate exists if all three relations arepositive or if two are negative and onepositive. According to this defini-tion, s-graphs a and b are balancedwhile s-graphs c and d are not (Fig. 4).We note that in the examples citedHeider's balanced state is depicted asan s-graph of three points whose cycleis positive.

In generalizing Heider's concept ofbalance, we propose to employ thischaracteristic of balanced states as ageneral criterion for balance of struc-tures with any number of entities.

Thus we define an s-graph (containingany number of points) as balanced ifall of its cycles are positive.

Figure 5 illustrates this definitionfor four s-graphs containing fourpoints. In each of these s-graphsthere are seven cycles: AB, BC, CA;AB, BD, DA; BC, CD, DB; AC, CD,DA; AB, BC, CD, DA; AB, BD, DC,CA; and BC, CA, AD, DB. It willbe seen that in s-graphs a and b allseven cycles are positive, and theses-graphs are therefore balanced. Ins-graphs c and d the cycle, AB, BC,CA, is negative (as are several others),and these s-graphs are therefore notbalanced. It is obvious that thisdefinition of balance is applicable tostructures containing any number ofentities.8

A

\Y/V

D

FIG. 5. Four s-graphs containing fourpoints and six lines each. Structures a and 6are balanced, but c and d are not balanced.

8 If an s-graph contains no cycles, we saythat it is "vacuously" balanced, since all (inthis case, none) of its cycles are positive.


The extension of this definition ofbalance to s-digraphs containing anynumber of points is straightforward.Employing the same definition of signof a semicyde for an s-digraph as foran ordinary s-graph, we similarly de-fine an s-digraph as balanced if all ofits semicycles are positive.

Consider now Heider's P-O-X unit,containing two persons P and 0 andan impersonal entity X, in which weare concerned only with liking rela-tions. Figure 6 shows three of thepossible 3-point s-digraphs which mayrepresent such P-O-X units. A posi-tive PO line means that P likes 0, anegative PO line means that P dis-likes 0. We assume that a personcan like or dislike an impersonalentity but that an impersonal entitycan neither like nor dislike a person.9

We also rule out of consideration here"ambivalence," where a person maysimultaneously like and dislike an-other person or impersonal entity.

In each of these s-digraphs there arethree semicycles: PO, OP; PO, OX,XP; and PO, OX, XP. If we confineour discussion to the kind of structuresrepresented in Fig. 6 (i.e., where thereis no ambivalence and where allpossible positive or negative lines arepresent), it will be apparent that:when P and 0 like each other, thes-digraph is balanced only if bothpersons either like or dislike X (s-digraph a is not balanced); when Pand 0 dislike each other, the s-digraphis balanced only if one person likes Xand the other person dislikes X (s-digraph b is balanced); and when oneperson likes the other but the other

9 In terms of digraph theory we define anobject as a point with zero output. Thus acompletely indifferent person is an object.If, psychologically, an impersonal entity isactive and likes or dislikes a person or anotherimpersonal entity, then in terms of digraphtheory it is not an object.

FIG. 6. Three s-digraphs representing Heider'sP-O-X units. Only structure 6 is balanced.

dislikes him, the s-digraph must benot balanced (s-digraph c is not bal-anced). These conclusions are con-sistent with Heider's discussion ofP-O-X units and with Newcomb'streatment of the A-B-X model.

The further extension of the notionof balance to s-graphs of type 2 re-mains to be made. The simplest pro-cedure would be simply to ignore thetypes of lines involved. Then wewould again define an s-graph of type 2to be balanced if all of its cycles arepositive. This definition appears tobe consistent with Heider's intention,at least as it applies to a situation con-taining only two entities. For inspeaking of such situations havingboth L and U relations, he calls thembalanced if both relations between thesame pair of entities are of the samesign (see 9, p. 110). There remainssome question as to whether this defi-nition will fit empirical findings forcycles of greater length. Until furtherevidence is available, we advance theabove formulation as a tentative defi-nition. Obviously the definition of


balance can be given for s-graphs ofgeneral type r in the same way.

SOME THEOREMS ON BALANCE

By definition, an s-graph is bal-anced if and only if each of its cyclesis positive. In a given situationrepresented by an s-graph, however, itmay be impractical to single out eachcycle, determine its sign, and thendeclare that it is balanced only afterthe positivity of every cycle has beenchecked. Thus the problem arises ofderiving a criterion for determiningwhether or not a given graph isbalanced without having to revert tothe definition. This problem is thesubject matter of a separate paper (6),in which two necessary and sufficientconditions for an s-graph to be bal-anced are developed. The first ofthese is no more useful than the defini-tion in determining by inspectionwhether an s-graph is balanced, but itdoes give further insight into thenotion of balance. Since the proofsof these theorems may be found in theother paper, we shall not repeatthem here.

Theorem. An s-graph is balancedif and only if all paths joining the samepair of points have the same sign.

Thus, we can ascertain that thes-graph of Fig. 7 is balanced either bylisting each cycle separately and veri-fying that it is positive, or, using this

FIG. 7. An s-graph of eight points andthirteen lines which, by aid of the structuretheorem, can be readily seen as balanced.

theorem, by considering each pair ofpossible points and verifying that allpossible paths joining them have thesame sign. For example, all thepaths between points A and E arenegative, all paths joining A and Care positive, etc.

The following structure theorem hasthe advantage that it is useful in de-termining whether or not a givens-graph is balanced without an ex-haustive check of the sign of everycycle, or of the signs of all paths join-ing every pair of points.

Structure theorem. An s-graph isbalanced if and only if its points canbe separated into two mutually ex-clusive subsets such that each positiveline joins two points of the same sub-set and each negative line joins pointsfrom different subsets.

Using the structure theorem, onecan see at a glance that the s-graph ofFig. 7 is balanced, for A, B, C,D, andE, F, G, H are clearly two disjointsubsets of the set of all points whichsatisfy the conditions of the structuretheorem.

It is not always quite so easy todetermine balance of an s-graph byinspection, for it is not always neces-sarily true that the points of each ofthe two subsets are connected to eachother. Thus the two s-graphs of Fig.8 are balanced, even though neither ofthe two disjoint subsets is a connectedsubgraph. However, the structuretheorem still applies to both of thes-graphs of Fig. 8. In the first graphthe appropriate subsets of points areA, D, E, H and B, C, F, G; while inthe second one we take A i, Bi, As, -B3,As, B$ and A j, B%, At, B\.

In addition to providing two neces-sary and sufficient conditions for bal-ance, these theorems give us furtherinformation about the nature of bal-ance. Thus if we regard the s-graphas representing Heider's L-relation in


a group, then the structure theoremtells us that the group is necessarily de-composed into two subgroups (cliques)within which the relationships thatoccur are positive and between whichthey are negative. The structuretheorem, however, does not precludethe possibility that one of the twosubsets may be empty—as, for ex-ample, when a connected graph con-tains only positive lines.

The first theorem also leads to someinteresting consequences. Suppose itwere true, for example, that when twopeople like each other they can influ-ence each other positively (i.e., pro-duce intended changes in the other),but when two people dislike each otherthey can only influence each othernegatively (i.e., produce changes oppo-site to those intended). An s-graphdepicting the liking relations among agroup of people will, then, also depictthe potential influence structure of thegroup. Suppose that Fig. 7 repre-sents such a group. If A attempts toget H to approve of something, H willreact by disapproving. If H at-tempts, in turn, to get G to disapproveof the same thing, he will succeed.Thus A's (indirect) influence upon Gis negative. The first theorem tellsus that A's influence upon G must benegative, regardless of the path alongwhich the influence passes, since thes-graph is balanced. In general, thesign of the influence exerted by anypoint upon any other will be the same,no matter what path is followed, sincethe graph is balanced.

By use of the structure theorem itcan be shown that in a balanced groupany influence from one point toanother within the same clique mustbe positive, even if it passes throughindividuals outside of the clique, andthe influence must be negative if itgoes from a person in one clique to aperson in the other. (It should be

rD E

a

-»B,

^

48,

FIG. 8. Two s-graphs whose balance cannotbe determined easily by visual inspection.

noted that in this discussion we givethe term "clique" a special meaning,as above.) Thus, under the assumedconditions, any exerted influence re-garding opinions will tend to producehomogeneity within cliques and op-posing opinions between cliques.

Although we have illustrated thesetheorems by reference to social groups,it should be obvious that they hold forany empirical realizations of s-graphs.

FURTHER CONCEPTS IN THETHEORY OF BALANCE

The concepts of balance as de-veloped up to this point are clearlyoversimplifications of the full com-

288 DORWIN CAETWRIGHT AND FRANK HARARY

plexity of situations with which wewant to deal. To handle such com-plex situations more adequately, weneed some further concepts.

Thus far we have only consideredwhether a given s-graph is balancedor not balanced. But it is intuitivelyclear that some unbalanced s-graphsare "more balanced" than others!This suggests the introduction of somescale of balance, along which the"amount" of balance possessed by anunbalanced s-graph may be measured.Accordingly we define the degree ofbalance of an s-graph as the ratio ofthe number of positive cycles to thetotal number of cycles. In symbols,let G be an s-graph,

c(G) = the number of cycles of G,c+(G) = the number of positive

cycles of G, andb(G) = the degree of balance of G.

Thenc+(G)b(G) =c(G)

Since the number c+(G) can rangefrom zero to c(G) inclusive, it is clearthat b(G) lies between 0 and 1. Ob-viously b(G) = 1 if and only if G isbalanced. We can give the numberb(G) the following probabilistic inter-pretation : the degree of balance of ans-graph is the probability that arandomly chosen cycle is positive.

Does b(G) = 50% mean that Gis exactly one-half balanced? The

B

FIG. 9. A graph of four points which canacquire degrees of balance of only .33 and1.00 regardless of the assignment of positiveand negative signs to its five lines.

FIG. 10. An s-graph which is 3-balancedbut not 4-balanced.

answer to this question depends on thepossible values which b(G) may as-sume. This in turn depends on thestructure of the s-graph G. Thus, ifG is the complete graph of 3 pointsand G is not balanced, then the onlypossible value is b(G) — 0, since thereis only one cycle. Similarly, if thelines of G are as in Fig. 9, some ofwhich may be negative, and if G is notbalanced, then the only possible valueis b(G) = i and b(G) = 50% doesnot even occur for this structure.Thus any interpretation given to thenumerical value of b(G) must takeinto account the distribution curve forb(G), which is determined by thestructure of G.

We now consider the correspondingconcept of the degree of balance fors-diagraphs. Since an s-digraph isbalanced if all of its semicycles arepositive, the degree of balance of ans-digraph is taken as the ratio of thenumber of positive semicycles to thetotal number of semicycles.

In a given s-graph which representsthe signed structure of some psycho-logical situation, it may happen thatonly cycles of length 3 and 4 are im-portant for the purpose of determiningbalance. Thus in an s-graph repre-senting the relation L in a complexgroup, it will not matter at all to thegroup as a whole whether a cycle oflength 100, say, is positive. Tohandle this situation rigorously, wedefine an s-graph to be N-balanced ifall its cycles of length not exceeding


N are positive. Of course the degreeof ./V-balance is definable and com-putable for any s-graph. Examplescan be given of unbalanced s-graphswhich are, however, jY-balanced forsome N. Figure 10 illustrates thisphenomenon for N = 3, since all ofits 3-cycles are positive, but it has anegative 4-cycle.10

For certain problems, one may wishto concentrate only on one distin-guished point and determine whetheran s-graph is balanced there. Thiscan be accomplished by the notion oflocal balance. We say an s-graph islocally balanced at point P if all cyclesthrough P are positive. Thus thes-graph of Fig. 11 is balanced at pointsA, B, C, and not balanced at D, E, F.If this figure represents a sociometricstructure, then the concept of localbalance at A is applicable provided Ais completely unconcerned about therelations among D, E, F.

Some combinatorial problems sug-gested by the notions of local balanceand ./V-balance have been investi-gated by Harary (7). The principaltheorem on local balance, which fol-lows, uses the term "articulationpoint" which we now define. Anarticulation pointn of a connectedgraph is one whose removal12 resultsin a disconnected graph. Thus thepoint D is the only articulation pointof Fig. 11. We now state the main

10 One way of viewing the definition of N-balance is to regard cycles of length N ashaving weight 1, and all longer cycles as ofweight 0. Of course, it is possible to general-ize this idea by assigning weights to eachlength, e.g., weight 1/2" to length N.

11A characterization of the articulationpoints of a graph, or in other words the liaisonpersons in a group, is given by Ross andHarary (19), using the "structure matrix" ofthe graph. An exposition of this concept isgiven in Harary and Norman (8).

12 By the removal of a point of a graph ismeant the deletion of the point and all linesto which it is incident.

theorem on local balance, withoutproof.

Theorem. If a connected s-graph Gis balanced at P, and Q is a point ona cycle passing through P, where Qis not an articulation point, then G isalso balanced at Q.

Figure 11 serves to illustrate thistheorem, for the s-graph is balancedat A, and is also balanced at B but isnot balanced at D, which is an articu-lation point.

In actual practice, both local bal-ance and ./V-balance may be employed.This can be handled by introducingthe combined concept of local N-balance. Formally we say that ans-graph is locally N-balanced at P if allcycles of length not exceeding N andpassing through P are positive. Ob-viously the degree of local ./V-balancecan be defined analogously to the de-gree of balance.

In summary, the concept of degreeof balance removes the limitation ofdealing with only balanced or un-balanced structures, and in additionis susceptible to probabilistic andstatistical treatment. The definitionof local balance enables one to focusat any particular point of the struc-ture. The introduction of TV-balancefrees us from the necessity of treatingall cycles as equally important in de-termining structural balance. Thus,the extensions of the notion of balance

FIG. 11. An s-graph which is locallybalanced at points A, B, C but not balancedat points D, E, F.

200 DORWIN CARTWKIGHT AND FRANK HARARY

developed in this section permit astudy of more complicated situationsthan does the original definition ofHeider.

ADEQUACY OF THE GENERALTHEORY OF BALANCE

In any empirical science the evalua-tion of a formal model must be con-cerned with both its formal propertiesand its applicability to empirical data.An adequate model should account forknown findings in a rigorous fashionand lead to new research. Althoughit is not our purpose in this article topresent new data concerning ten-dencies toward balance in empiricalsystems, we may attempt to evaluatethe adequacy of the proposed generaltheory of balance in the light ofpresently available research.

Our review of Heider's theory ofbalance and of the research findingsrelated to it has revealed certain am-biguities and limitations concerning(a) the treatment of unsymmetric re-lations; (&) the generalization to sys-tems containing more than threeentities; (c) the distinction betweenthe complement and the opposite of arelation; (d) the simultaneous exis-tence of relationships of differenttypes; and (e) the applicability of theconcept of balance to empirical sys-tems other than cognitive ones. Wenow comment briefly upon the way inwhich our generalization deals witheach of these problems.

Unsymmetric relations. 11 was notedabove that, while theoretical discus-sions of balance have sometimes al-lowed for the possibility of unsym-metric relations, no rigorous definitionof balance has been developed to en-compass situations containing unsym-metric relationships. Furthermore,empirical studies have tended to as-sume that liking is reciprocated, thateach liking relation is symmetric.

By stating the definition of balance interms of s-digraphs, we are able toinclude in one conceptual scheme bothsymmetric and unsymmetric relation-ships. And it is interesting to observethat, according to this definition,whenever the lines PO and OP are ofdifferent signs, the s-digraph contain-ing them is not balanced. Thus, tothe extent that tendencies towardbalance have been effective in thesettings empirically studied, the as-sumption of symmetry has, in fact,been justified.

Situations containing any finite num-ber of entities. Heider's discussion ofbalance has been confined to struc-tures containing no more than threeentities. The definition of balanceadvanced here contains no such limita-tion; it is applicable to structurescontaining any finite number of en-tities. Whether or not empiricaltheories of balance will be confirmedby research dealing with larger struc-tures can only be determined by em-pirical work. It is clear, however,that our generalization is consistentwith the more limited definition ofHeider.

A relation, its complement, and itsopposite. Using s-graphs and s-di-graphs to depict relationships betweenentities allows us to distinguish amongthree situations: the presence of arelation (positive line), the presenceof the opposite of a relation (negativeline), and the absence of both (noline). The empirical utilization ofthis theory requires the ability todistinguish among these three situa-tions. In our earlier discussion of theliterature on balance, we noted, how-ever, a tendency to distinguish onlythe presence or absence of a relation-ship. It is not always clear, there-fore, in attempting to depict previousresearch in terms of s-graph theorywhether a given empirical relationship


should be coordinated to no line or toa negative line.

The experiment of Jordan (11)illustrates this problem quite clearly.He employed three entities and speci-fied certain U and L relations betweeneach pair of entities. The empiricalrealization of these relations was ob-tained in the following way: U wasmade into "has some sort of bond orrelationship with"; ~U into "has nosort of bond or relationship with;"L was made into "like;" and ~L into"dislike." Viewed in the light ofs-graph theory, it would appear thatJordan created s-graphs of type 2(which may contain positive andnegative lines of type U and type L).It would also appear, however, thatthe ~U relation should be depicted asthe absence of any U-line but that the~L relation should be depicted as anegative L-line. If this interpreta-tion is correct, Jordan's classificationof his situations as "balanced" and"imbalanced" will have to be revised.Instead of interpreting the ~U rela-tion as a negative line, we shall haveto view it as no U-line, with the resultthat all of his situations containing~U relations are vacuously balancedby our definition since there are nocycles.

It is interesting to examine Jordan'sdata in the light of this reinterpreta-tion. He presented subjects with 64hypothetical situations, half of whichwere "balanced" and half "imbal-anced" by his definition. He hadsubjects rate the degree of pleasant-ness or unpleasantness experienced ineach situation (a high score indicat-ing unpleasantness). For' 'balanced"situations the mean rating was 46 andfor "imbalanced" ones, 57.

If, however, we interpret Jordan's<~U relation as the absence of a line,his situations must be reclassified.Of his 32 "balanced" situations, 14

have no ~U relation and thus remainbalanced. The mean unpleasantnessscore for these is 39. The remaining18 of his "balanced" situations, hav-ing at least one <~U relation, becomevacuously balanced since no cycle re-mains. The mean unpleasantness ofthese vacuously balanced situations is51. Of Jordan's 32 "imbalanced"situations, 19 contain at least one ~Urelation, thus also becoming vacuouslybalanced, and the mean unpleasant-ness score for these is 51. The re-maining 13 situations, by having no~U relations, remain imbalanced, andtheir mean score is 66. Thus it isclear that the difference in pleasant-ness between situations classed byJordan as "balanced" and "imbal-anced" is greatly increased if thevacuously balanced situations are re-moved from both classes (balanced,39; vacuously balanced, 51; not bal-anced, 66). These findings lend sup-port to our view that the statement"has no sort of bond or relationshipwith" should be represented as theabsence of a line.13

Relations of different types. A basicfeature of Heider's theory of balanceis the designation of two types ofrelations (L and U). Our generaliza-tion of the definition of balance per-mits the inclusion of any number oftypes of relations. Heider discussesthe combination of types of relationsonly for the situation involving twoentities, and it is clear that our defini-tion is consistent with his within thislimitation. It is interesting to note

1!A strict test of our interpretation ofJordan's data is not possible since he specifiedfor any given pair of entities only either theL or U relation. We can but guess how thesubjects filled in the missing relationship. Inthe light of our discussion of relations ofdifferent types, in the next section, it appearsthat subjects probably assumed a positiveunit relation when none was specified, sincethere is a marked tendency to experiencenegative liking relations as unpleasant.


that Jordan (11) finds positive likingrelations to be experienced as morepleasant than negative ones. Thisfinding may be interpreted as indicat-ing a tendency toward "positivity"over and above the tendency towardbalance. It is possible, however, thatin the hypothetical situations em-ployed by Jordan the subjects as-sumed positive unit relations betweeneach pair of entities. If this were infact true, then a positive liking rela-tion would form a positive cycle oflength 2 with the positive unit rela-tion, and a negative liking relationwould form a negative cycle of length2 with the positive unit relation.And, according to the theory of bal-ance, the positive cycle should pro-duce more pleasantness than thenegative one. This interpretation canbe tested only through further re-search in which the two relations areindependently varied.

Empirical applicability of concept ofbalance. Heider's discussion of bal-ance refers to a cognitive structure, orthe life space of a single person.Newcomb suggests that a similar con-ception may be applicable to interper-sonal systems objectively described.Clearly, our definition of balance maybe employed whenever the terms"point" and "signed line" can bemeaningfully coordinated to empiricaldata of any sort. Thus, one shouldbe able to characterize a communica-tion network or a power structure asbalanced or not. Perhaps it would befeasible to use the same definition indescribing neural networks. It mustbe noted, however, that it is a matterfor empirical determination whetheror not a tendency to achieve balancewill actually be observed in any par-ticular kind of situation, and what theempirical consequences of not bal-anced configurations are. Before ex-tensive utilization of these notions can

be accomplished, certain further con-ceptual problems regarding balancemust be solved.

One of the principal unsolved prob-lems is the development of a sys-tematic treatment of relations of vary-ing strength. We believe that it ispossible to deal with the strength ofrelations by the concept of a graph ofstrength or, suggested by Harary andNorman (8).

SUMMARY

In this article we have developed ageneralization of Heider's theory ofbalance by use of concepts from themathematical theory of linear graphs.By defining balance in graph-theoreticterms, we have been able to removesome of the ambiguities found inprevious discussions of balance, and tomake the concept applicable to awider range of empirical situationsthan was previously possible. By in-troducing the concept degree of balance,we have made it possible to treatproblems of balance in statistical andprobabilistic terms. It should beeasier, therefore, to make empiricaltests of hypotheses concerning balance.

Although Heider's theory was origi-nally intended to refer only to cog-nitive structures of an individual per-son, we propose that the definition ofbalance may be used generally indescribing configurations of manydifferent sorts, such as communicationnetworks, power systems, sociometricstructures, systems of orientations, orperhaps neural networks. Only fu-ture research can determine whethertheories of balance can be establishedfor all of these configurations. Thedefinitions developed here do, in anycase, give a rigorous method for de-scribing certain structural aspects ofempirical configurations.


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(Received January 11, 1956)

Structural balance: A generalization of Heider's theory

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