Evaluating balance on social networks from their simple cycles

Signed networks have long been used to represent social relations of amity (+) and enmity (-) between individuals. Group of individuals who are cyclically connected are said to be balanced if the number of negative edges in the cycle is even and unbalanced otherwise. In its most natural formulation, the balance of a social network is thus defined from its simple cycles, cycles which do not visit any vertex more than once. Because of the inherent difficulty associated with finding such cycles on very large networks, social balance has always been studied via other, less-direct means. In this article we present the balance as measured from the simple cycles and primitive orbits of social networks. We use a Monte Carlo implementation of a novel exact formula for counting the simple cycles on any weighted directed graph. We show that social networks exhibit strong inter-edge correlations favoring balanced situations and we determine the corresponding correlation length ξ. For longer simple cycles, the percentage of unbalanced simple cycles undergoes a rapid transition to values expected from an uncorrelated model. Our method is more generally applicable to evaluate arbitrary functions over the simple cycles and simple paths of any weighted directed graph and can also answer vertex-specific questions.