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In the psychology of motivation, balance theory is a theory of attitude change, proposed by Fritz Heider.^[1]^[2] It conceptualizes the cognitive consistency motive as a drive toward psychological balance. The consistency motive is the urge to maintain one's values and beliefs over time. Heider proposed that "sentiment" or liking relationships are balanced if the affect valence in a system multiplies out to a positive result.

Structural balance theory in social network analysis is the extension proposed by Dorwin Cartwright and Frank Harary.^[3] It was the framework for the discussion at a Dartmouth College symposium in September 1975.^[4]

P-O-X model

For example: a Person ( $P$ ) who likes ( $+$ ) an Other ( $O$ ) person will be balanced by the same valence attitude on behalf of the other. Symbolically, $P(+)>O$ and $P<(+)O$ results in psychological balance.

This can be extended to things or objects ( $X$ ) as well, thus introducing triadic relationships. If a person $P$ likes object $X$ but dislikes other person $O$ , what does $P$ feel upon learning that person $O$ created the object $X$ ? This is symbolized as such:

$P(+)>X$
$P(-)>O$
$O(+)>X$

Cognitive balance is achieved when there are three positive links or two negatives with one positive. Two positive links and one negative like the example above creates imbalance or cognitive dissonance.

Multiplying the signs shows that the person will perceive imbalance (a negative multiplicative product) in this relationship, and will be motivated to correct the imbalance somehow. The Person can either:

Decide that $O$ isn't so bad after all,
Decide that $X$ isn't as great as originally thought, or
Conclude that $O$ couldn't really have made $X$ .

Any of these will result in psychological balance, thus resolving the dilemma and satisfying the drive. (Person $P$ could also avoid object $X$ and other person $O$ entirely, lessening the stress created by psychological imbalance.)

To predict the outcome of a situation using Heider's balance theory, one must weigh the effects of all the potential results, and the one requiring the least amount of effort will be the likely outcome.

Determining if the triad is balanced is simple math:

$+++=+$ ; Balanced.

$-+-=+$ ; Balanced.

$-++=-$ ; Unbalanced.

Examples

Balance theory is useful in examining how celebrity endorsement affects consumers' attitudes toward products.^[5] If a person likes a celebrity and perceives (due to the endorsement) that said celebrity likes a product, said person will tend to like the product more, in order to achieve psychological balance.

However, if the person already had a dislike for the product being endorsed by the celebrity, they may begin disliking the celebrity, again to achieve psychological balance.

Heider's balance theory can explain why holding the same negative attitudes of others promotes closeness.^[6]^: 171 See The enemy of my enemy is my friend.

Signed graphs and social networks

Dorwin Cartwright and Frank Harary looked at Heider's triads as 3-cycles in a signed graph. The sign of a path in a graph is the product of the signs of its edges. They considered cycles in a signed graph representing a social network.

A balanced signed graph has only cycles of positive sign.

Harary proved that a balanced graph is polarized, that is, it decomposes into two entirely positive subgraphs that are joined by negative edges.^[7]

In the interest of realism, a weaker property was suggested by Davis:^[8]

No cycle has exactly one negative edge.

Graphs with this property may decompose into more than two entirely positive subgraphs, called clusters.^[6]^: 179 The property has been called the clusterability axiom.^[9] Then balanced graphs are recovered by assuming the

Parsimony axiom: The subgraph of positive edges has at most two components.

The significance of balance theory for social dynamics was expressed by Anatol Rapoport:

The hypothesis implies roughly that attitudes of the group members will tend to change in such a way that one's friends' friends will tend to become one's friends and one's enemies' enemies also one's friends, and one's enemies' friends and one's friends' enemies will tend to become one's enemies, and moreover, that these changes tend to operate even across several removes (one's friends' friends' enemies' enemies tend to become friends by an iterative process).^[10]

Note that a triangle of three mutual enemies makes a clusterable graph but not a balanced one. Therefore, in a clusterable network one cannot conclude that "the enemy of my enemy is my friend," although this aphorism is a fact in a balanced network.

Criticism

Claude Flament^[11] expressed a limit to balance theory imposed by reconciling weak ties with relationships of stronger force such as family bonds:

One might think that a valued algebraic graph is necessary to represent psycho-social reality, if it is to take into account the degree of intensity of interpersonal relationships. But in fact it then seems hardly possible to define the balance of a graph, not for mathematical but for psychological reasons. If the relationship AB is +3, the relationship BC is –4, what should the AC relationship be in order that the triangle be balanced? The psychological hypotheses are wanting, or rather they are numerous and little justified.

At the 1975 Dartmouth College colloquium on balance theory, Bo Anderson struck at the heart of the notion:^[12]

In graph theory there exists a formal balance theory that contains theorems that are analytically true. The statement that Heider's psychological balance can be represented, in its essential aspects, by a suitable interpretation of that formal balance theory should, however, be regarded as problematical. We cannot routinely identify the positive and negative lines in the formal theory with the positive and negative "sentiment relations", and identify the formal balance notion with the psychological idea of balance or structural tension. .. It is puzzling that the fine structure of the relationships between formal and psychological balance has been given scant attention by balance theorists.

References

^ Heider, Fritz (1946). "Attitudes and Cognitive Organization". The Journal of Psychology. 21: 107–112. doi:10.1080/00223980.1946.9917275. PMID 21010780.
^ Heider, Fritz (1958). The Psychology of Interpersonal Relations. John Wiley & Sons.
^ Cartwright, D.; Harary, Frank (1956). "Structural balance: a generalization of Heider's theory" (PDF). Psychological Review. 63 (5): 277–293. doi:10.1037/h0046049. PMID 13359597. Archived (PDF) from the original on 2020-12-01. Retrieved 2020-12-04.
^ Paul W. Holland & Samuel Leinhardt (editors) (1979) Perspectives on Social Network Research, Academic Press ISBN 9780123525505
^ John C. Mowen and Stephen W. Brown (1981), "On Explaining and Predicting the Effectiveness of Celebrity Endorsers", in Advances in Consumer Research Volume 08, eds. Kent B. Monroe, Advances in Consumer Research Volume 08: Association for Consumer Research, Pages: 437-441.
^ ^a ^b Gary Chartrand (1977) Graphs as Mathematical Models, chapter 8: Graphs and Social Psychology, Prindle, Webber & Schmidt, ISBN 0-87150-236-4
^ Frank Harary (1953) On the Notion of Balance of a Signed Graph Archived 2018-06-02 at the Wayback Machine, Michigan Mathematical Journal 2(2): 153–6 via Project Euclid MR0067468
^ James A. Davis (May 1967) "Clustering and structural balance in graphs", Human Relations 20:181–7
^ Claude Flament (1979) "Independent generalizations of balance", in Perspectives on Social Network Research
^ Anatol Rapoport (1963) "Mathematical models of social interaction", in Handbook of Mathematical Psychology, v. 2, pp 493 to 580, especially 541, editors: R.A. Galanter, R.R. Lace, E. Bush, John Wiley & Sons
^ Claude Flament (1963) Application of Graph Theory to Group Structure, translators Maurice Pinard, Raymond Breton, Fernand Fontaine, chapter 3: Balancing Processes, page 92, Prentice-Hall
^ Bo Anderson (1979) "Cognitive Balance Theory and Social Network Analysis: Remarks on some fundamental theoretical matters", pages 453 to 69 in Perspectives on Social Network Research, see page 462.