Assessing frustration in real-world signed networks: A statistical theory of balance

According to the so-called strong version of structural balance theory, actors in signed social networks avoid establishing triads with an odd number of negative links. Generalizing, the weak version of balance theory allows for nodes to be partitioned into any number of blocks with positive internal links, mutually connected by negative links. If this prescription is interpreted rigidly, i.e., without allowing for statistical noise in the observed link signs, then most real graphs will appear to require a larger number of blocks than the actual one, or even to violate both versions of the theory. This might lead to conclusions invoking even more relaxed notions of balance. Here, after rephrasing structural balance theory in statistically testable terms, we propose an inference scheme to unambiguously assess whether a real-world signed graph is balanced. We find that the proposed statistical balance theory leads to interpretations that are quite different from those derived from the current deterministic versions of the theory.