Editing Balance theory (section)

==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 (graph theory)|path]] in a [[graph (discrete mathematics)|graph]] is the product of the signs of its edges. They considered [[cycle (graph theory)|cycle]]s 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.<ref>[[Frank Harary]] (1953) [https://projecteuclid.org/euclid.mmj/1028989917 On the Notion of Balance of a Signed Graph] {{Webarchive|url=https://web.archive.org/web/20180602010351/https://projecteuclid.org/euclid.mmj/1028989917 |date=2018-06-02 }}, [[Michigan Mathematical Journal]] 2(2): 153–6 via [[Project Euclid]] {{mr|id=0067468}}</ref>

In the interest of realism, a weaker property was suggested by Davis:<ref>[[James A. Davis]] (May 1967) "Clustering and structural balance in graphs", [[Human Relations]] 20:181–7</ref>
:No cycle has exactly one negative edge.
Graphs with this property may decompose into more than two entirely positive subgraphs, called '''clusters'''.<ref name=GC/>{{rp|179}} The property has been called the ''clusterability axiom''.<ref>Claude Flament (1979) "Independent generalizations of balance", in ''Perspectives on Social Network Research''</ref> Then balanced graphs are recovered by assuming the 
: Parsimony axiom: The subgraph of positive edges has at most two [[component (graph theory)|component]]s.

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 [[friend of a friend|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).<ref>[[Anatol Rapoport]] (1963) "Mathematical models of social interaction", in [https://archive.org/details/handbookofmathem017893mbp ''Handbook of Mathematical Psychology'', v. 2], pp 493 to 580, especially 541, editors: R.A. Galanter, R.R. Lace, E. Bush, [[John Wiley & Sons]]</ref>
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<ref>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]]</ref> expressed a limit to balance theory imposed by reconciling [[weak ties]] with relationships of stronger force such as [[human bonding|family bonds]]:
:One might think that a [[graph (abstract data type)|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:<ref>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.</ref>
: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.