VOL. 63, No. 5 SEPTEMBER, 1956 THE PSYCHOLOGICAL REVIEW

STRUCTURAL BALANCE: A GENERALIZATION OF HEIDER'S THEORY'

DORWIN CARTWRIGHT AND FRANK HARARY Research Center for Group Dynamics,

A persistent problem of psychology will arise to change it toward a state has been how to deal conceptually of balance. These pressures may with patterns of interdependent prop- work to change the relation of affec- erties. This problem has been cen- tion between P and 0, the relation of tral, of course, in the theoretical responsibility between 0 and X, or the treatment by Gestalt psychologists of relation of evaluation between P phenomenal or neural configurations andX. or fields (12, 13, 15). It has also The purpose of this paper is to been of concern to social psychologists present and develop the consequences and sociologists who attempt to em- of a formal definition of balance which ploy concepts referring to social sys- is consistent with Heider's conception tems (18). and which may be employed in a more Heider (19), reflecting the general general treatment of empirical con- field-theoretical approach, has con- figurations. The definition is stated sidered certain aspects of cognitive in terms of the mathematical theory fields which contain perceived people of linear graphs (8, 14) and makes use and impersonal objects or events. of a distinction between a given rela- His analysis focuses upon what he calls tion and its opposite relation. Some the P-O-X unit of a cognitive field, of the ramifications of this definition consisting of P (one person), 0 (an- are then examined by means of other person), and X (an impersonal theorems derivable from the definition entity). Each relation among the and from . parts of the unit is conceived as inter- dependent with each other relation. HEIDER'S CONCEPTION OF Thus, for example, if P has a relation BALANCE of affection for 0 and if 0 is seen as responsible for X, then there will be In developing his analysis of bal- a tendency for P to like or approve anced cognitive units, Heider dis- of X. If the nature of X is such that tinguishes between two major types of it would "normally" be evaluated as relations. The first concerns atti- bad, the whole P-O-X unit is placed tudes, or the relation of liking or in a state of imbalance, and pressures evaluating. It is represented sym- 1 bolically as L when positive and as ~L This paper was prepared as part of a when negative. Thus, PLO means P project sponsored in the Research Center for Group Dynamics by the Rockefeller likes, loves, values, or approves 0, and Foundation. JP~LO means P dislikes, negatively 277 278 DORWIN CARTWRIGHT AND FRANK HARARY values, or disapproves 0. The second others as a point of departure for type of relation refers to cognitive further theoretical and empirical work. unit formation, that is, to such specific We shall summarize briefly some of relations as similarity, possession, the major results of this work. causality, proximity, or belonging. Horowitz, Lyons, and Perlmutter It is written as U or ~U. Thus, (10) attempted to demonstrate ten- according to Heider, PUX means that dencies toward balance in an experi- P owns, made, is close to, or is associ- ment employing members of a discus- ated with X, and P~ILY means that sion group as subjects. At the end of P does not own, did not make, or is a discussion period each subject was not associated with X. asked to indicate his evaluation of an A balanced state is then defined in event (PLX or P~LX) which had terms of certain combinations of these occurred during the course of the dis- relations. The definition is stated cussion. The event selected for evalu- separately for two and for three ation was one which would be clearly entities. seen as having been produced by a In the case of two entities, a balanced state single person (OUX). The liking re- exists if the relation between them is positive lation between each P and 0 (PLO (or negative) in all respects, i.e., in regard to or P~LO) had been determined by all meanings of L and U . . . . In the case a sociometric questionnaire admin- of three entities, a balanced state exists if all three relations are positive in all respects, or istered before the meeting. Would if two are negative and one positive (9, p. 110). P's evaluation of the event be such as to produce a balanced P-O-X unit? These are examples of balanced If so, P's evaluation of 0 and X should states: P likes something he made be of the same sign. The experi- (PUX, FIX); P likes what his friend mental data tend to support the likes (PLO, OLX, PLX); P dislikes hypothesis that a P-O-X unit tends what his friend dislikes (PLO, 0~LX, toward a balanced state.2 P~LX"); P likes what his enemy dis- The social situation of a discussion likes (P~LO, 0~LX, PLX); and P's group can be better analyzed, accord- son likes what P likes (PU0, PLX, ing to Horowitz, Lyons, and Perl- OLX). mutter, by considering a somewhat Heider's basic hypothesis asserts more complex cognitive unit. The that there is a tendency for cognitive evaluation of X made by P, they units to achieve a balanced state. argue, will be determined not only by Pressures toward balance may pro- P's evaluation of 0 but also by his duce various effects. perception of the evaluation of X If no balanced state exists, then forces given by others (Qs) in the group. towards this state will arise. Either the dy- The basic unit of such a social situa- namic characters will change, or the unit tion, then, consists of the subject, a relations will be changed through action or through cognitive reorganization. If a change 2 One of the attractive features of this study is not possible, the state of imbalance will is that it was conducted in a natural "field" produce tension (9, pp. 107-109). setting, thus avoiding the dangers of artifici- ality. At the same time the setting placed The theory, stated here in sketchy certain restrictions on the possibility of outline, has been elaborated by Heider manipulation and control of the variables. so as to treat a fuller richness of cog- The data show a clear tendency for P to place nitive experience than would be sug- a higher evaluation on Xs produced by more attractive Os. It is not clearly demonstrated gested by our brief description. It that P likes Xs produced by liked Os and has been used, too, by a number of dislikes Xs produced by disliked Os. STRUCTURAL BALANCE 279

person who is responsible for the Jordan (11) presented subjects with event, and another person who will be 64 different hypothetical situations in seen by the subject as supporting or which the L and U relations between rejecting the event. This is called each pair of elements was systemati- a P-O-Q-X unit. The additional data cally varied. The subject was asked needed to describe these relations were to place himself in each situation by obtained from the sociometric ques- taking the part of P, and to indicate tionnaire which indicated P's evalua- on a scale the degree of pleasantness tion of Q (PLQ or P~L@), and from or unpleasantness he experienced. a question designed to reveal P's per- Unpleasantness was assumed to reflect ception of Q's support or rejection of the postulated tension produced by X, treated by the authors as a unit imbalanced units. Jordan's data tend relation (QUX or Q~UZ).3 to support Heider's hypothesis that Although these authors indicate the imbalanced units produce a state of possibility of treating the P-O-Q-X tension, but he too found that addi- unit in terms of balance, they do not tional factors need to be considered. develop a formal definition of a bal- He discovered, for example, that nega- anced configuration consisting of four tive relations were experienced as un- elements. They seem to imply that pleasant even when contained in the P-O-Q-X unit will be balanced if balanced units. This unpleasantness the P-O-X and the P-Q-X units are was particularly acute when P was a both balanced. They do not consider part of the negative relation. Jordan's the relation between Q and 0, nor the study permits a detailed analysis of logically possible components of which these additional influences, which we it could be a part. Their analysis is shall not consider here. concerned primarily with the two Newcomb (17), in his recent theory triangles (P-O-X and P-Q-X), which of interpersonal communication, has are interdependent, since both contain employed concepts rather similar to the relation of P's liking of X. We those of Heider. He conceives of the noted above that the data tend to simplest communicative act as one in support the hypothesis that the P- which one person A gives information 0-X unit will tend toward balance. to another person B about something The data even more strongly support the hypothesis when applied to the X. The similarity of this A-B-X P-Q-X unit; P's evaluation of X and model to Heider's P-O-X unit, to- his perception of Q's attitude toward gether with its applicability to objec- X tend to agree when P likes Q, and tive interpersonal relations (rather to disagree when P dislikes Q. It than only to the cognitive structure of should be noted, however, that there a single person), may be seen in the was also a clear tendency for P to see following quotations from Newcomb: Q's evaluation of X as agreeing with A-B-X is ... regarded as constituting a his own whether or not he likes Q. system. That is, certain definable relation- In a rather different approach to ships between A and B, between A and X, the question of balanced P-O-X units, and between B and X are all viewed as inter- dependent. . . . For some purposes the sys- 3 Whether this relation should be treated as tem may be regarded as a phenomenal one U or L is subject to debate. For testing within the life space of A or B, for other Heider's theory of balance, however, the purposes as an "objective" system including issue is irrelevant, since he holds that the two all the possible relationships as inferred from relations are interchangeable in defining observations of A's and B's behavior (17, balance. p. 393). 280 DORWIN CARTWRIGHT AND FRANK HARARY

Newcomb then develops the con- and that it may also be important in cept of "strain toward symmetry," interpersonal relations. The concept which appears to be a special instance of balance, however, has been defined of Heider's more general notion of so as to apply to a rather limited range "tendency toward balance." "Strain of situations, and it has contained cer- toward symmetry" is reflected in tain ambiguities. We note five spe- several manifestations of a tendency cific problems. for A and B to have attitudes of the 1. Unsymmetric relations. Should same sign toward a common X. all relations be conceived as sym- Communication is the most common metric? The answer is clearly that and usually the most effective mani- they should not; it is possible for P festation of this tendency. to like 0 while 0 dislikes P. In fact, By use of this conception Newcomb Tagiuri, Blake, and Bruner (21) have reinterprets several studies (1, 4, 5, intensively studied dyadic relations 16, 20) which have investigated the to discover conditions producing sym- interrelations among interpersonal at- metric relations of actual and per- traction, tendencies to communicate, ceived liking. Theoretical discussions pressures to uniformity of opinion of balance have sometimes recognized among members of a group, and this possibility—Heider, for example, tendencies to reject deviates. The states that unsymmetric liking is un- essential hypothesis in this analysis is balanced—but there has been no stated thus: general definition of balance which If A is free either to continue or not to covers unsymmetric relations. The continue his association with B, one or the empirical studies of balance have as- other of two eventual outcomes is likely: sumed that the relations are sym- (a) he achieves an equilibrium characterized metric. by relatively great attraction toward B and 2. Units containing more than three by relatively high perceived symmetry, and the association is continued; or (i) he achieves entities. Nearly all theorizing about an equilibrium characterized by relatively balance has referred to units of three little attraction toward B and by relatively entities. While Horowitz, Lyons, and low perceived symmetry, and the association Perlmutter studied units with four is discontinued (17, p. 402). entities, they did not define balance Newcomb's outcome a is clearly a for such cases. It would seem desir- balanced state as denned by Heider. able to be able to speak of the balance Outcome b cannot be unambiguously of even larger units. translated into Heider's terms. If by 3. Negative relations. Is the nega- "relatively little attraction toward B" tive relation the complement of the is meant a negative L relation between relation or its opposite! All of the A and B, then this outcome would also discussions of balance seem to equate seem to be balanced. Newcomb's these, but they seem to us to be quite "continuation or discontinuation of different, for the complement of a rela- the association between A and B" tion is expressed by adding the word appear to correspond to Heider's U "not" while the opposite is indicated and ~TJ relations. by the prefix "dis" or its equivalent. Thus, the complement of "liking" is STATEMENT OF THE PROBLEM "not liking"; the opposite of "liking" This work indicates that the ten- is "disliking." In general, it appears dency toward balance is a significant that ~L has been taken to mean determinant of cognitive organization, "dislike" (the opposite relation) while STRUCTURAL BALANCE 281

~U has been used to indicate "not THE CONCEPTS OF GRAPH, DIGRAPH, associated with" (the complementary AND relation). Thus, for example, Jordan says: "Specifically, '+L' symbolizes Our approach to this problem has a positive attitude, '—L' symbolizes two primary antecedents: (a) Lewin's a negative attitude, '+TT symbol- treatment (IS) of the concepts of whole, differentiation, and unity, to- izes the existence of unit formation, 1 and '— IT symbolizes the lack of unit gether with Bavelas extension (2) of formation" (11, p. 274). this work to group structure; and (b) 4. Relations of different types. Hei- the mathematical theory of linear der has made a distinction between graphs. two types of relations—one based Many of the graph-theoretic defini- upon liking and one upon unit forma- tions given in this section are con- tion. The various papers following tained in the classical reference on up Heider's work have continued to graph theory, Konig (14), as well as use this distinction. And it seems in Harary and Norman (8). We reasonable to assume that still other shall discuss, however, those concepts types of relations might be designated. which lead up to the theory of balance. A linear graph, or briefly a graph, How can a definition of balance take 4 into account relations of different consists of a finite collection of points types ? Heider has suggested some of A,B, C, • • • together with a prescribed the ways in which liking and unit subset of the set of all unordered pairs relations may be combined, but a of distinct points. Each of these un- general formulation has yet to be ordered pairs, AB, is a line of the developed. graph. (From the viewpoint of the theory of binary relations,6 a graph 5. Cognitive fields and social systems. 6 Heider's intention is to describe bal- corresponds to an irreflexive sym- ance of cognitive units in which the metric relation on points A, B, C, Alternatively a graph may be repre- entities and relations enter as experi- 7 enced by a single individual. New- sented as a matrix. ) comb attempts to treat social systems Figure 1 depicts a graph of four which may be described objectively. points and four lines. The points In principle, it should be possible also might represent people, and the lines to study the balance of sociometric some relationship such as mutual lik- structures, communication networks, ing. With this interpretation, Fig. 1 patterns of power, and other aspects indicates that mutual liking exists be- of social systems. tween those pairs of people A, B, C, We shall attempt to define balance and D joined by lines. Thus D is in so as to overcome these limitations. the relation with all other persons, Specifically, the definition should (a) while C is in the relation only with D. encompass unsymmetric relations, (&) hold for units consisting of any finite 'Points are often called "vertices" by number of entities, (c) preserve the mathematicians and "nodes" by electrical engineers. distinction between the complement 5 This is the approach used by Heider. and the opposite of a relation, (d) "A relation is irreflexive if it contains no apply to relations of different types, ordered pairs of the form (a, a), i.e., if no and (e) serve to characterize cognitive element is in this relation to itself. 7 This treatment is discussed in Festinger units, social systems, or any configura- (3). The logical equivalence of relations, tion where both a relation and its graphs, and matrices is taken up in Harary opposite must be specified. and Norman (8). 282 DORWIN CARTWRIGHT AND FRANK HARARY

distinguished from an unordered pair by designating one of the points as the first point and the other as the second. Thus, for example, the fact that a message can go from A to B is repre- sented by the ordered pair (A, B), or equivalently, by the line AB, as in Fig. 2. Similarly, the fact that A and FIG. 1. A linear graph of four points and D choose each other is represented by four lines. The presence of line AB indicates the existence of a specified symmetric relation- the two directed lines AD and DA. ship between the two entities A and B. A signed graph, or briefly an s-graph, is obtained from a graph when one Figure 1 could be used, of course, to regards some of the lines as positive represent many other kinds of rela- and the remaining lines as negative. tionships between many other kinds Considered as a geometric representa- of entities. tion of binary relations, an s-graph It is apparent from this definition of serves to depict situations or struc- graph that relations are treated in an tures in which both a relation and its all-or-none manner, i.e., either a rela- opposite may occur, e.g., like and tion exists between a given pair of dislike. Figure 3 depicts an s-graph, points or it does not. Obviously, employing the convention that solid however, many relationships of in- lines are positive and dashed lines terest to psychologists (liking, for negative; thus A and B are repre- example) exist in varying degrees. sented as liking each other while A This fact means that our present use and C dislike each other. of graph theory can treat only the Combining the concepts of digraph structural, and not the numerical, and s-graph, we obtain that of an aspects of relations. While our treat- s-digraph. A signed digraph, or an ment is thereby an incomplete repre- s-digraph, is obtained from a digraph sentation of the strength of relations, by taking some of its lines as positive we believe that conceptualization of and the rest as negative. the structural properties of relations A graph of type 2 (8), introduced to is a necessary first step toward a more depict structures in which two differ- adequate treatment of the more com- plex situations. Such an elaboration, however, goes beyond the scope of B this paper. A , or a digraph, con- sists of a finite collection of points together with a prescribed subset of the set of all ordered pairs of distinct points. Each of these ordered pairs AB is called a line of the digraph. Note that the only difference between FIG. 2. A directed graph of four points the definitions of graph and digraph and five directed lines. An AB line indicates the existence of a specified ordered relation- is that the lines of a graph are un- ship involving the two entities A and B. ordered pairs of points while the lines Thusjbr example, if A and B are two people, of a digraph are ordered pairs of the AB line might indicate that a message points, An ordered pair of points is can go from A to B or that A chooses B. STRUCTURAL BALANCE 283 ent relations defined on the same set of elements occur, is obtained from a graph by regarding its lines as being of two different colors (say), and by permitting the same pair of points to be joined by two lines if these lines have different colors. A graph of type r, T = 1, 2, 3, •••, is denned simi- larly. In an s-graph or s-digraph of FIG. 3. A signed graph of four points and type 2, there may occur lines of two five lines. Solid lines have a positive sign different types in which a line of either and dashed lines a negative sign. If the points stand for people and the lines indicate color may be positive or negative. the existence of a liking relationship, this An example of an s-graph of type 2 s-graph shows that A and B have a relation- might be one depicting for the same ship of liking, A and C have one of disliking, P-O-X unit both U and L relations and B and D have a relationship of indifference among the entities, where the sign of (neither liking or disliking). these relations is indicated. now develop a rigorous generalization A path is a collection of lines of a of Heider's concept of balance. graph of the form AB, BC, • • •, DE, It should be evident that Heider's where the points A, B, C, • • • ,D,E, terms, entity, relation, and sign of a are distinct. A cycle consists of the relation may be coordinated to the above path together with the line EA. graphic terms, point, directed line, and The length of a cycle (or path) is the sign of a directed line. Thus, for ex- number of lines in it; an n-cycle is a ample, the assertion that P likes cycle of length n. Analogously to 0 (PLO) may be depicted as a directed graphs, a path of a digraph consists of line of positive sign PO. It should directed lines of the form AB, BC, • • •, also be clear that Heider's two differ- DE, where the points are distinct. A ent kinds of relations (L and U) may cycle consists of this path together be treated as lines of different type. with the line EA. In the later discus- It follows that a graphic representa- sion of balance of an s-digraph we tion of a P-O-X unit having positive shall use the concept of a semicycle. or negative L and U relations will be A semicycle is a collection of lines an s-digraph of type 2. obtained by taking exactly one from For simplicity of discussion we first each pair_AB or BA^BC or CB, •••, consider the situation containing only DE or ED, and EA or AE. We symmetric relations of a single type illustrate semicycles with the digraph (i.e., an s-graph of type 1). Figure 4 of Fig. 2. There are three semicycles shows four such s-graphs. It will be noted that each of these s-graphs con- in_this digraph^ AD± DA; AD, DB, tains one cycle: AB, BC, CA. We BA ; and AD, DB, BA. The last two now need to define the sign of a cycle. of these semicycles are not cycles. The sign of a cycle is the product of Note that every cycle is a semicycle, the signs of its lines. For conveni- and a semicycle of length 2 is neces- ence we denote the sign of a line by sarily a cycle. +1 or — 1 when it is positive or nega- BALANCE tive. With this definition we see that the cycle, AB, BC, CA is positive in With these concepts of graphs, di- s-graph a (-f-l'+l'+l), positive in graphs, and signed graphs we may s-graph b (+1 • — 1 • — 1), negative 284 DORWIN CARTWRIGHT AND FRANK HARARY

Thus we define an s-graph (containing any number of points) as balanced if all of its cycles are positive. Figure 5 illustrates this definition for four s-graphs containing four points. In each of these s-graphs Vc there 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 will be seen that in s-graphs a and b all seven cycles are positive, and these A s-graphs are therefore balanced. In \ s-graphs c and d the cycle, AB, BC, \ CA, is negative (as are several others), \ and these s-graphs are therefore not \ / balanced. It is obvious that this definition of balance is applicable to Vc structures containing any number of entities.8 c d FIG. 4. Four s-graphs of three points and three lines each. Structure a and b are balanced, but c and d are not balanced. in s-graph c ( + 1 • +1 • —1), and nega- tive in s-graph d ( —!• —1 • —1). To generalize, a cycle is positive if it con- tains an even number of negative lines, and it is negative otherwise. Thus, in particular, a cycle containing only positive lines is positive, since the number of negative lines is zero, an even number. In discussing the concept of balance, Heider states (see 9, p. 110) that when there are three entities a balanced state exists if all three relations are positive or if two are negative and one A positive. According to this defini- tion, s-graphs a and b are balanced \Y/ while s-graphs c and d are not (Fig. 4). V We note that in the examples cited D Heider's balanced state is depicted as an s-graph of three points whose cycle is positive. FIG. 5. Four s-graphs containing four In generalizing Heider's concept of points and six lines each. Structures a and 6 balance, we propose to employ this are balanced, but c and d are not balanced. characteristic of balanced states as a 8 If an s-graph contains no cycles, we say general criterion for balance of struc- that it is "vacuously" balanced, since all (in tures with any number of entities. this case, none) of its cycles are positive. STRUCTURAL BALANCE 285

The extension of this definition of balance to s-digraphs containing any number of points is straightforward. Employing the same definition of sign of a semicyde for an s-digraph as for an ordinary s-graph, we similarly de- fine an s-digraph as balanced if all of its semicycles are positive. Consider now Heider's P-O-X unit, containing two persons P and 0 and an impersonal entity X, in which we are concerned only with liking rela- tions. Figure 6 shows three of the possible 3-point s-digraphs which may represent such P-O-X units. A posi- tive PO line means that P likes 0, a negative PO line means that P dis- likes 0. We assume that a person can like or dislike an impersonal entity but that an impersonal entity FIG. 6. Three s-digraphs representing Heider's can neither like nor dislike a person.9 P-O-X units. Only structure 6 is balanced. We also rule out of consideration here "ambivalence," where a person may dislikes him, the s-digraph must be simultaneously like and dislike an- not balanced (s-digraph c is not bal- other person or impersonal entity. anced). These conclusions are con- In each of these s-digraphs there are sistent with Heider's discussion of P-O-X units and with Newcomb's three semicycles: PO, OP; PO, OX, treatment of the A-B-X model. XP; and PO, OX, XP. If we confine The further extension of the notion our discussion to the kind of structures of balance to s-graphs of type 2 re- represented in Fig. 6 (i.e., where there mains to be made. The simplest pro- is no ambivalence and where all cedure would be simply to ignore the possible positive or negative lines are types of lines involved. Then we present), it will be apparent that: would again define an s-graph of type 2 when P and 0 like each other, the to be balanced if all of its cycles are s-digraph is balanced only if both positive. This definition appears to persons either like or dislike X (s- be consistent with Heider's intention, digraph a is not balanced); when P at least as it applies to a situation con- and 0 dislike each other, the s-digraph taining only two entities. For in is balanced only if one person likes X speaking of such situations having and the other person dislikes X (s- both L and U relations, he calls them digraph b is balanced); and when one balanced if both relations between the same pair of entities are of the same person likes the other but the other sign (see 9, p. 110). There remains 9 In terms of digraph theory we define an some question as to whether this defi- object as a point with zero output. Thus a nition will fit empirical findings for completely indifferent person is an object. cycles of greater length. Until further If, psychologically, an impersonal entity is active and likes or dislikes a person or another evidence is available, we advance the impersonal entity, then in terms of digraph above formulation as a tentative defi- theory it is not an object. nition. Obviously the definition of 286 DORWIN CARTWRIGHT AND FRANK HARARY balance can be given for s-graphs of theorem, by considering each pair of general type r in the same way. possible points and verifying that all possible paths joining them have the SOME THEOREMS ON BALANCE same sign. For example, all the By definition, an s-graph is bal- paths between points A and E are anced if and only if each of its cycles negative, all paths joining A and C is positive. In a given situation are positive, etc. represented by an s-graph, however, it The following structure theorem has the advantage that it is useful in de- may be impractical to single out each termining whether or not a given cycle, determine its sign, and then declare that it is balanced only after s-graph is balanced without an ex- the positivity of every cycle has been haustive check of the sign of every checked. Thus the problem arises of cycle, or of the signs of all paths join- deriving a criterion for determining ing every pair of points. whether or not a given graph is Structure theorem. An s-graph is balanced without having to revert to balanced if and only if its points can the definition. This problem is the be separated into two mutually ex- clusive subsets such that each positive subject matter of a separate paper (6), in which two necessary and sufficient line joins two points of the same sub- conditions for an s-graph to be bal- set and each negative line joins points anced are developed. The first of from different subsets. Using the structure theorem, one these is no more useful than the defini- tion in determining by inspection can see at a glance that the s-graph of whether an s-graph is balanced, but it Fig. 7 is balanced, for A, B, C,D, and does give further insight into the E, F, G, H are clearly two disjoint notion of balance. Since the proofs subsets of the set of all points which of these theorems may be found in the satisfy the conditions of the structure other paper, we shall not repeat theorem. them here. It is not always quite so easy to Theorem. An s-graph is balanced determine balance of an s-graph by if and only if all paths joining the same inspection, for it is not always neces- pair of points have the same sign. sarily true that the points of each of Thus, we can ascertain that the the two subsets are connected to each s-graph of Fig. 7 is balanced either by other. Thus the two s-graphs of Fig. listing each cycle separately and veri- 8 are balanced, even though neither of fying that it is positive, or, using this the two disjoint subsets is a connected subgraph. However, the structure theorem still applies to both of the s-graphs of Fig. 8. In the first graph the appropriate subsets of points are A, D, E, H and B, C, F, G; while in the 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 further information about the nature of bal- FIG. 7. An s-graph of eight points and thirteen lines which, by aid of the structure ance. Thus if we regard the s-graph theorem, can be readily seen as balanced. as representing Heider's L-relation in STRUCTURAL BALANCE 287 a group, then the structure theorem tells us that the group is necessarily de- composed into two subgroups (cliques) within which the relationships that occur are positive and between which they are negative. The structure theorem, however, does not preclude r the possibility that one of the two subsets may be empty—as, for ex- D E ample, when a connected graph con- a tains only positive lines. The first theorem also leads to some interesting consequences. Suppose it were true, for example, that when two -»B, people 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 other they can only influence each other ^ negatively (i.e., produce changes oppo- site to those intended). An s-graph depicting the liking relations among a group of people will, then, also depict the potential influence structure of the group. Suppose that Fig. 7 repre- sents such a group. If A attempts to get H to approve of something, H will 48, react by disapproving. If H at- tempts, in turn, to get G to disapprove of the same thing, he will succeed. FIG. 8. Two s-graphs whose balance cannot Thus A's (indirect) influence upon G be determined easily by visual inspection. is negative. The first theorem tells us that A's influence upon G must be noted that in this discussion we give negative, regardless of the path along the term "" a special meaning, which the influence passes, since the as above.) Thus, under the assumed s-graph is balanced. In general, the conditions, any exerted influence re- sign of the influence exerted by any garding opinions will tend to produce point upon any other will be the same, homogeneity within cliques and op- no matter what path is followed, since posing opinions between cliques. the graph is balanced. Although we have illustrated these By use of the structure theorem it theorems by reference to social groups, can be shown that in a balanced group it should be obvious that they hold for any influence from one point to any empirical realizations of s-graphs. another within the same clique must be positive, even if it passes through FURTHER CONCEPTS IN THE individuals outside of the clique, and THEORY OF BALANCE the influence must be negative if it The concepts of balance as de- goes from a person in one clique to a veloped up to this point are clearly person in the other. (It should be oversimplifications of the full com- 288 DORWIN CAETWRIGHT AND FRANK HARARY

plexity of situations with which we want to deal. To handle such com- plex situations more adequately, we need some further concepts. Thus far we have only considered whether a given s-graph is balanced or not balanced. But it is intuitively clear that some unbalanced s-graphs are "more balanced" than others! FIG. 10. An s-graph which is 3-balanced This suggests the introduction of some but not 4-balanced. scale of balance, along which the "amount" of balance possessed by an answer to this question depends on the unbalanced s-graph may be measured. possible values which b(G) may as- Accordingly we define the degree of sume. This in turn depends on the balance of an s-graph as the ratio of structure of the s-graph G. Thus, if the number of positive cycles to the G is the of 3 points total number of cycles. In symbols, and G is not balanced, then the only let G be an s-graph, possible value is b(G) — 0, since there is only one cycle. Similarly, if the c(G) = the number of cycles of G, lines of G are as in Fig. 9, some of c+(G) = the number of positive which may be negative, and if G is not cycles of G, and balanced, then the only possible value b(G) = the degree of balance of G. is b(G) = i and b(G) = 50% does Then not even occur for this structure. c+(G) b(G) = Thus any interpretation given to the c(G) numerical value of b(G) must take into account the distribution curve for Since the number c+(G) can range b(G), which is determined by the from zero to c(G) inclusive, it is clear structure of G. that b(G) lies between 0 and 1. Ob- We now consider the corresponding viously b(G) = 1 if and only if G is concept of the degree of balance for balanced. We can give the number s-diagraphs. Since an s-digraph is b(G) the following probabilistic inter- balanced if all of its semicycles are pretation : the degree of balance of an positive, the degree of balance of an s-graph is the probability that a s-digraph is taken as the ratio of the randomly chosen cycle is positive. number of positive semicycles to the Does b(G) = 50% mean that G total number of semicycles. is exactly one-half balanced? The In a given s-graph which represents the signed structure of some psycho- B logical situation, it may happen that only cycles of length 3 and 4 are im- portant for the purpose of determining balance. Thus in an s-graph repre- senting the relation L in a complex group, it will not matter at all to the group as a whole whether a cycle of length 100, say, is positive. To FIG. 9. A graph of four points which can acquire degrees of balance of only .33 and handle this situation rigorously, we 1.00 regardless of the assignment of positive define an s-graph to be N-balanced if and negative signs to its five lines. all its cycles of length not exceeding STRUCTURAL BALANCE 289

N are positive. Of course the degree theorem on local balance, without of ./V-balance is definable and com- proof. putable for any s-graph. Examples Theorem. If a connected s-graph G can be given of unbalanced s-graphs is balanced at P, and Q is a point on which are, however, jY-balanced for a cycle passing through P, where Q some N. Figure 10 illustrates this is not an articulation point, then G is phenomenon for N = 3, since all of also balanced at Q. its 3-cycles are positive, but it has a Figure 11 serves to illustrate this negative 4-cycle.10 theorem, for the s-graph is balanced For certain problems, one may wish at A, and is also balanced at B but is to concentrate only on one distin- not balanced at D, which is an articu- guished point and determine whether lation point. an s-graph is balanced there. This In actual practice, both local bal- can be accomplished by the notion of ance and ./V-balance may be employed. local balance. We say an s-graph is This can be handled by introducing locally balanced at point P if all cycles the combined concept of local N- through P are positive. Thus the balance. Formally we say that an s-graph of Fig. 11 is balanced at points s-graph is locally N-balanced at P if all A, B, C, and not balanced at D, E, F. cycles of length not exceeding N and If this figure represents a sociometric passing through P are positive. Ob- structure, then the concept of local viously the degree of local ./V-balance balance at A is applicable provided A can be defined analogously to the de- is completely unconcerned about the gree of balance. relations among D, E, F. In summary, the concept of degree Some combinatorial problems sug- of balance removes the limitation of gested by the notions of local balance dealing with only balanced or un- and ./V-balance have been investi- balanced structures, and in addition gated by Harary (7). The principal is susceptible to probabilistic and theorem on local balance, which fol- statistical treatment. The definition lows, uses the term "articulation of local balance enables one to focus point" which we now define. An at any particular point of the struc- articulation pointn of a connected ture. The introduction of TV-balance graph is one whose removal12 results frees us from the necessity of treating all cycles as equally important in de- in a disconnected graph. Thus the termining structural balance. Thus, point D is the only articulation point the extensions of the notion of balance of Fig. 11. We now state the main 10 One way of viewing the definition of N- balance is to regard cycles of length N as having weight 1, and all longer cycles as of weight 0. Of course, it is possible to general- ize this idea by assigning weights to each length, e.g., weight 1/2" to length N. 11A characterization of the articulation points of a graph, or in other words the liaison persons in a group, is given by Ross and Harary (19), using the "structure matrix" of the graph. An exposition of this concept is given in Harary and Norman (8). 12 By the removal of a point of a graph is FIG. 11. An s-graph which is locally meant the deletion of the point and all lines balanced at points A, B, C but not balanced to which it is incident. at points D, E, F. 200 DORWIN CARTWKIGHT AND FRANK HARARY

developed in this section permit a By stating the definition of balance in study of more complicated situations terms of s-digraphs, we are able to than does the original definition of include in one conceptual scheme both Heider. symmetric and unsymmetric relation- ships. And it is interesting to observe ADEQUACY OF THE GENERAL that, according to this definition, THEORY OF BALANCE whenever the lines PO and OP are of In any empirical science the evalua- different signs, the s-digraph contain- tion of a formal model must be con- ing them is not balanced. Thus, to cerned with both its formal properties the extent that tendencies toward and its applicability to empirical data. balance have been effective in the An adequate model should account for settings empirically studied, the as- known findings in a rigorous fashion sumption of symmetry has, in fact, and lead to new research. Although been justified. it is not our purpose in this article to Situations containing any finite num- present new data concerning ten- ber of entities. Heider's discussion of dencies toward balance in empirical balance has been confined to struc- systems, we may attempt to evaluate tures containing no more than three the adequacy of the proposed general entities. The definition of of balance in the light of advanced here contains no such limita- presently available research. tion; it is applicable to structures Our review of Heider's theory of containing any finite number of en- balance and of the research findings tities. Whether or not empirical related to it has revealed certain am- theories of balance will be confirmed biguities and limitations concerning by research dealing with larger struc- (a) the treatment of unsymmetric re- tures can only be determined by em- lations; (&) the generalization to sys- pirical work. It is clear, however, tems containing more than three that our generalization is consistent entities; (c) the distinction between with the more limited definition of the complement and the opposite of a Heider. relation; (d) the simultaneous exis- A relation, its complement, and its tence of relationships of different opposite. Using s-graphs and s-di- types; and (e) the applicability of the graphs to depict relationships between concept of balance to empirical sys- entities allows us to distinguish among tems other than cognitive ones. We three situations: the presence of a now comment briefly upon the way in relation (positive line), the presence which our generalization deals with of the opposite of a relation (negative each of these problems. line), and the absence of both (no Unsymmetric relations. 11 was noted line). The empirical utilization of above that, while theoretical discus- this theory requires the ability to sions of balance have sometimes al- distinguish among these three situa- lowed for the possibility of unsym- tions. In our earlier discussion of the metric relations, no rigorous definition literature on balance, we noted, how- of balance has been developed to en- ever, a tendency to distinguish only compass situations containing unsym- the presence or absence of a relation- metric relationships. Furthermore, ship. It is not always clear, there- empirical studies have tended to as- fore, in attempting to depict previous sume that liking is reciprocated, that research in terms of s-graph theory each liking relation is symmetric. whether a given empirical relationship STRUCTURAL BALANCE 291 should be coordinated to no line or to have no ~U relation and thus remain a negative line. balanced. The mean unpleasantness The experiment of Jordan (11) score for these is 39. The remaining illustrates this problem quite clearly. 18 of his "balanced" situations, hav- He employed three entities and speci- ing at least one <~U relation, become fied certain U and L relations between vacuously balanced since no cycle re- each pair of entities. The empirical mains. The mean unpleasantness of realization of these relations was ob- these vacuously balanced situations is tained in the following way: U was 51. Of Jordan's 32 "imbalanced" made into "has some sort of bond or situations, 19 contain at least one ~U relationship with"; ~U into "has no relation, thus also becoming vacuously sort of bond or relationship with;" balanced, and the mean unpleasant- L was made into "like;" and ~L into ness score for these is 51. The re- "dislike." Viewed in the light of maining 13 situations, by having no s-graph theory, it would appear that ~U relations, remain imbalanced, and Jordan created s-graphs of type 2 their mean score is 66. Thus it is (which may contain positive and clear that the difference in pleasant- negative lines of type U and type L). ness between situations classed by It would also appear, however, that Jordan as "balanced" and "imbal- the ~U relation should be depicted as anced" is greatly increased if the the absence of any U-line but that the vacuously balanced situations are re- ~L relation should be depicted as a moved from both classes (balanced, negative L-line. If this interpreta- 39; vacuously balanced, 51; not bal- tion is correct, Jordan's classification anced, 66). These findings lend sup- of his situations as "balanced" and port to our view that the statement "imbalanced" will have to be revised. "has no sort of bond or relationship Instead of interpreting the ~U rela- with" should be represented as the tion as a negative line, we shall have absence of a line.13 to view it as no U-line, with the result Relations of different types. A basic that all of his situations containing feature of Heider's theory of balance ~U relations are vacuously balanced is the designation of two types of by our definition since there are no relations (L and U). Our generaliza- cycles. tion of the definition of balance per- It is interesting to examine Jordan's mits the inclusion of any number of data in the light of this reinterpreta- types of relations. Heider discusses tion. He presented subjects with 64 the combination of types of relations hypothetical situations, half of which only for the situation involving two were "balanced" and half "imbal- entities, and it is clear that our defini- anced" by his definition. He had tion is consistent with his within this subjects rate the degree of pleasant- limitation. It is interesting to note ness or unpleasantness experienced in 1!A strict test of our interpretation of each situation (a high score indicat- Jordan's data is not possible since he specified ing unpleasantness). For' 'balanced" for any given pair of entities only either the L or U relation. We can but guess how the situations the mean rating was 46 and subjects filled in the missing relationship. In for "imbalanced" ones, 57. the light of our discussion of relations of If, however, we interpret Jordan's different types, in the next section, it appears <~U relation as the absence of a line, that subjects probably assumed a positive unit relation when none was specified, since his situations must be reclassified. there is a marked tendency to experience Of his 32 "balanced" situations, 14 negative liking relations as unpleasant. 292 DORWIN CARTWRIGHT AND FRANK HARARY that Jordan (11) finds positive liking be accomplished, certain further con- relations to be experienced as more ceptual problems regarding balance pleasant than negative ones. This must be solved. finding may be interpreted as indicat- One of the principal unsolved prob- ing a tendency toward "positivity" lems is the development of a sys- over and above the tendency toward tematic treatment of relations of vary- balance. It is possible, however, that ing strength. We believe that it is in the hypothetical situations em- possible to deal with the strength of ployed by Jordan the subjects as- relations by the concept of a graph of sumed positive unit relations between strength or, suggested by Harary and each pair of entities. If this were in Norman (8). fact true, then a positive liking rela- tion would form a positive cycle of SUMMARY length 2 with the positive unit rela- tion, and a negative liking relation In this article we have developed a would form a negative cycle of length generalization of Heider's theory of 2 with the positive unit relation. balance by use of concepts from the And, according to the theory of bal- mathematical theory of linear graphs. ance, the positive cycle should pro- By defining balance in graph-theoretic duce more pleasantness than the terms, we have been able to remove negative one. This interpretation can some of the ambiguities found in be tested only through further re- previous discussions of balance, and to search in which the two relations are make the concept applicable to a independently varied. wider range of empirical situations Empirical applicability of concept of than was previously possible. By in- balance. Heider's discussion of bal- troducing the concept degree of balance, ance refers to a cognitive structure, or we have made it possible to treat the life space of a single person. problems of balance in statistical and Newcomb suggests that a similar con- probabilistic terms. It should be ception may be applicable to interper- sonal systems objectively described. easier, therefore, to make empirical Clearly, our definition of balance may tests of hypotheses concerning balance. be employed whenever the terms Although Heider's theory was origi- "point" and "signed line" can be nally intended to refer only to cog- meaningfully coordinated to empirical nitive structures of an individual per- data of any sort. Thus, one should son, we propose that the definition of be able to characterize a communica- balance may be used generally in tion network or a power structure as describing configurations of many balanced or not. Perhaps it would be different sorts, such as communication feasible to use the same definition in networks, power systems, sociometric describing neural networks. It must structures, systems of orientations, or be noted, however, that it is a matter perhaps neural networks. Only fu- for empirical determination whether ture research can determine whether or not a tendency to achieve balance theories of balance can be established will actually be observed in any par- for all of these configurations. The ticular kind of situation, and what the definitions developed here do, in any empirical consequences of not bal- case, give a rigorous method for de- anced configurations are. Before ex- scribing certain structural aspects of tensive utilization of these notions can empirical configurations. STRUCTURAL BALANCE 293

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