The remarkable robustness of many social systems has been associated with a peculiar triangular structure in the underlying social networks. Triples of people that have three positive relations (e.g., friendship) between each other are strongly overrepresented. Triples with two negative relations (e.g., enmity) and one positive relation are also overrepresented, and triples with one or three negative relations are drastically suppressed.

For almost a century, the mechanism behind these very specific (“balanced”) triad statistics remained elusive. Here, we propose a simple realistic adaptive network model, where agents tend to minimize social tension that arises from dyadic interactions. Both opinions of agents and their signed links (positive or negative relations) are updated in the dynamics. The novelty of the model arises from the fact that agents only need information about their local neighbors in the network and do not require (often unrealistic) higher order network information for their relation and opinion updates.

We demonstrate the quality of the model on detailed temporal relation data of a society of thousands of players of a massive multiplayer online gamewhere we can observe triangle formation directly. It not only successfully predicts the distribution of triangle types but also, explains empirical group size distributions, which are essential for social cohesion. We discuss the details of the phase diagrams behind the model and their parameter dependence, andwe comment on to what extent the resultsmight apply universally in societies.

Tuan Minh Pham, Rudolf Hanel, Jan Korbel, Stefan Thurner, Empirical social triad statistics can be explained with dyadic homophylic interactions, PNAS 119 (6) (2022) e2121103119