Balance Correlations, Agentic Zeros, and Networks: The Structure of 192 Years of War and Peace
Social network extensions of Heider's balance theory have not always been consistent. Structural balance theory primarily focuses on graph partitioning, thereby assuming, homogeneity in balance-driven behavior of nodes. We present a general model and formal notation that permit testing such behavioral assumptions. Specifically, we formulate statements as a comparison of two conditional probabilities of a tie, $Ego\stackrel{q}{\text{-}}Alter$, first conditional on 2-paths $Ego\, \stackrel{r}{\text{-}}\,X\,\stackrel{s}{\text{-}}\,Alter$, and second conditional on all others, $\neg (Ego\,\stackrel{r}{\text{-}}\,X\,\stackrel{s}{\text{-}}\,Alter)$. The key here is that $q$, $r$ and $s$ represent indices of relations in a set of mutually exclusive and exhaustive relations (their sum produces a complete graph). This relaxes the assumption of a signed graph dichotomy. Here we identify neutral as distinct from negative and positive ties. Descriptive statistics measuring the difference in conditional probabilities, or the prevalence for any stipulated balance configuration, are given by the point bi-serial correlations of relation $q$ with the count of $2$-paths (through relations $r$ and $s$). Two major advantages are: direct comparison, even if network sizes and densities differ, and evaluation of specific (un)balance behaviors. We apply this approach on a data set with friendly vs hostile relations between countries from 1816 to 2007. We find strong evidence for one of the four classic Heiderian balance theory predictions, and virtually no evidence in support of the unbalanced predictions. However, we do find stable and surprising evidence that the neutral ties are important in balancing the relations among nations. Results further suggest that prevalence of balance driven behavior varies over time, and that other triadic motivated behaviors prevail among countries in certain eras.