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. See The enemy of my enemy is my friend. Signed graphs and social networks Frank Harary and Dorwin Cartwright 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. In the interest of realism, a weaker property was suggested by Davis: No cycle has exactly one negative edge. Graphs with this property may decompose into more than two entirely positive subgraphs, called clusters. The property has been called the clusterability axiom. 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). 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 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