Structural balance

Structural balance is a theory from social psychology, formalized by Fritz Heider in 1946, which states that relationships in a social network tend toward configurations that minimize cognitive dissonance. In a triad of three individuals, a balanced state is one in which the product of the three relationship signs is positive — informally, \'the friend of my friend is my friend\' and \'the enemy of my enemy is my friend.\' An unbalanced triad, such as three mutual enemies or two friends who are enemies with a third, creates psychological tension that drives the network toward reconfiguration.

The theory was given a graph-theoretic foundation by Harary through the concept of signed graphs. A signed graph is balanced if and only if its vertices can be partitioned into two groups with all positive edges within groups and all negative edges between groups. This characterization transforms a psychological hypothesis into a combinatorial theorem and connects structural balance to graph bipartition and community detection.