Triadic relationships are accepted to play a key role in the dynamics of social and political networks. Building on insights gleaned from balance theory in social network studies and from Boltzmann-Gibbs statistical physics, we propose a model to quantitatively capture the dynamics of the four types of triadic relationships in a network. Central to our model are the triads’ incidence rates and the idea that those can be modeled by assigning a specific triadic energy to each type of triadic relation. We emphasize the role of the degeneracy of the different triads and how it impacts the degree of frustration in the political network. In order to account for a persistent form of disorder in the formation of the triadic relationships, we introduce the systemic variable temperature. In order to learn about the dynamics and motives, we propose a generic Hamiltonian with three terms to model the triadic energies. One term is connected with a three-body interaction that captures balance theory. The other terms take into account the impact of heterogeneity and of negative edges in the triads. The validity of our model is tested on four datasets including the time series of triadic relationships for the standings between two classes of alliances in a massively multiplayer online game (MMOG). We also analyze real-world data for the relationships between the “agents” involved in the Syrian civil war, and in the relations between countries during the Cold War era. We find emerging properties in the triadic relationships in a political network, for example reflecting itself in a persistent hierarchy between the four triadic energies, and in the consistency of the extracted parameters from comparing the model Hamiltonian to the data.

Structural balance in social network theory starts from signed networks with active relationships (friendly or hostile) to establish a hierarchy between four different types of triadic relationships. The lack of an active link also provides information about the network. To exploit the information that remains uncovered by structural balance, we introduce the inactive relationship that accounts for both neutral and nonexistent ties between two agents. This addition results in ten types of triads, with the advantage that the network analysis can be done with complete networks. To each type of triadic relationship, we assign an energy that is a measure for its average occupation probability. Finite temperatures account for a persistent form of disorder in the formation of the triadic relationships. We propose a Hamiltonian with three interaction terms and a chemical potential (capturing the cost of edge activation) as an underlying model for the triadic energy levels. Our model is suitable for empirical analysis of political networks and allows to uncover generative mechanisms. It is tested on an extended data set for the standings between two classes of alliances in a massively multi-player on-line game (MMOG) and on real-world data for the relationships between countries during the Cold War era. We find emergent properties in the triadic relationships between the nodes in a political network. For example, we observe a persistent hierarchy between the ten triadic energy levels across time and networks. In addition, the analysis reveals consistency in the extracted model parameters and a universal data collapse of a derived combination of global properties of the networks. We illustrate that the model has predictive power for the transition probabilities between the different triadic states. arXiv:1807.09042v3 [physics.soc-ph] 17 Dec 2018Heider [1] and its extension to graphs by Cartwright [2] is based on active relationships that can be friendly ("+") or unfriendly ("−"). The four types of emerging triadic relationships are categorized in two stable (= balanced) and two unstable (= unbalanced) ones, whereby one anticipates an overall tendency to create more balanced triads. Balanced triads [+ + +] and [+ − −] have an even number of "−" edges. Unbalanced [+ + −] and [− − −] triads, however, are a key ingredient in real-life political networks. Balance theory has found applications in many branches of sciences including psychology [1], studies of international networks [3,4,5,6,7], sociology [8,9,10,11,12] and ecology [13].As balance theory introduces correlations between the edge attributes in triads, in a physics framework [14,15,16,17] it maps onto a system with predominant three-edge interactions. Associating the existence of unbalanced triads to the occurrence of a non-vanishing excitation energy, the principles of social balance can be mapped onto a model with variations in an energy landscape. Marvel et al. [14] investigated the energy landscape of a social-balance inspired system and stressed the occurrence ...

Games involving virtual worlds are popular in several segments of the population and societies. The online environment facilitates that players from different countries interact in a common virtual world. Virtual worlds involving social and economic interactions are particularly useful to test social and economic theories. Using data from EVE Online, a massive online multi-player game simulating a fantasy galaxy, we analyse the relation between the real-world context in which players live and their in-game behaviour at the country level. We find that in-game aggressiveness to non-player characters is positively related to real-world levels of aggressiveness as measured by the Global Peace Index and the Global Terrorist Index at the country level. The opposite is true for in-game aggressiveness towards other players, which seems to work as a safety valve for real-world player aggressiveness. The ability to make in-game friends is also positively related to real-world levels of aggressiveness in much the same way. In-game trading behaviour is dependent on the macroeconomic environment where players live. The unemployment rate and exchange rate make players trade more efficiently and cautiously in-game. Overall, we find evidence that the real-world environment affects in-game behaviour, suggesting that virtual worlds can be used to experiment and test social and economic theories, and to infer real-world behaviour at the country level.