S1 Appendix -

<div><p>Social network extensions of Heider’s balance theory have led to a plethora of adaptations, often inconsistent with Heider and each other. We present a general model that permits the description and testing of specific balance theoretic predictions as Heider had originally proposed them. We formulate balance statements as a comparison of two conditional probabilities of a tie: , conditioned on 2-path relations vs. their counterfactual negation, . Here <i>q</i>, <i>r</i> and <i>s</i>, are indices that identify elements in a set of mutually exclusive and exhaustive relations in a multigraph (their sum produces a complete graph). This relaxes the dichotomous assumption of a signed graph. We identify neutral ties distinctive from negative and positive ties and convert the underlying signed graph into a restricted multigraph. The point-biserial correlations of relation <i>q</i> with the count of 2-paths (through relations <i>r</i> and <i>s</i>) describe the difference in conditional probabilities, or the prevalence for any stipulated balance configuration. In total 18 such unique balance correlations are identified for any trichotomous multigraph, although 27 theoretical statements exist. Two major advantages of balance correlations are: direct comparison between any two correlations over time, even if network sizes and densities differ; and the evaluation of specific balanced and unbalanced behaviors and predictions. We apply the model to a large data set consisting of friendly vs hostile relations between countries from 1816 to 2007. We find strong evidence for one of the balance statements, and virtually no evidence for unbalanced predictions. However, there is ample stable evidence that “neutral” ties are important in balancing the relations among nations. Results suggest that prevalence of balance-driven behavior varies over time. Indeed, other behaviors may prevail in certain eras.</p></div>