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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., Comment: 29 pages, presented at Networks 2021, Bloomington, USA, Sunbelt 2022, Cairns, Sunbelt 2023, Portland, Sunbelt 2023, Portland, USA, ION IX, Lexington, USA, and EUSN 2023 Ljubljana, Slovenia

The rise of socially targeted marketing suggests that decisions made by consumers can be predicted not only from their personal tastes and characteristics, but also from the decisions of people who are close to them in their networks. One obstacle to consider is that there may be several different measures for "closeness" that are appropriate, either through different types of friendships, or different functions of distance on one kind of friendship, where only a subset of these networks may actually be relevant. Another is that these decisions are often binary and more difficult to model with conventional approaches, both conceptually and computationally. To address these issues, we present a hierarchical model for individual binary outcomes that uses and extends the machinery of the auto-probit method for binary data. We demonstrate the behavior of the parameters estimated by the multiple network-regime auto-probit model (m-NAP) under various sensitivity conditions, such as the impact of the prior distribution and the nature of the structure of the network, and demonstrate on several examples of correlated binary data in networks of interest to Information Systems, including the adoption of Caller Ring-Back Tones, whose use is governed by direct connection but explained by additional network topologies.