# Signed Relations and Structural Balance in Complex Systems: From Data to Models

According to the theory of structural balance, interacting systems balance the (positive or negative) relations between different system elements such that local conflicts are minimized. Hence, structural imbalances induce a dynamics to resolve such conflicts. This dynamics plays a vital role in evolutionary processes because a multitude of possible solutions exists. At the same time, if these solutions cannot be reached, this can hamper the functionality of systems. This general problem also occurs in social systems, where instead of a more balanced state, for instance, the polarization of opinions emerges. Are we able to address this problem from a formal perspective? Do we have data available to study it in real systems? Can we develop models that help us to understand when structural balance fails, and how it can be mitigated?

Data about interactions between system elements (agents) is ubiquitous. This applies particularly to interactions between individuals, thanks to new communication technologies, sensor recordings, and online interactions. To analyze, to model and to interpret these data, however, is one of the biggest challenges for data science. Instead of applying existing methodologies to just another data set, we need to implement system specific concepts. Such problems do not just concern data analysis, they pertain even more to the modeling of such systems.

This proposal makes an important step forward, by addressing these methodological challenges. To understand structural balance, the differences between many-particle systems and socio-economic multi-agent systems need to be addressed. While in physical systems with given spin-spin interactions, spins try to align such that the local frustration is minimized, in social systems agents have the opportunity to also change the sign of their relations, to obtain a better balanced state. But how do we know about their signed relations, for example their friend-or-foe relationship? The data in almost all cases records only the observed interactions, but to model and to understand the problem of structural balance, we need their relations. Therefore, in this project, we first solve the methodological problem of inferring signed relations from interactions by developing a novel statistical approach to analyze such data.