Mathematical analysis of signed networks: structure and dynamics

Signed networks, characterized by the presence of both positive and negative edges, offer a powerful framework for modeling complex systems with antagonistic interactions. These systems are ubiquitous across various scientific domains, including social sciences, ecology, molecular biology, economics, neuroscience, and international relations. Despite this prevalence, most approaches have neglected the information conveyed by negative edges, thus failing to capture the full complexity of these systems. Recently, however, this has begun to change, with researchers paying increasing attention to the development of mathematical tools to analyze signed networks.

This thesis contributes to this collective effort by analyzing the structure and dynamics of signed networks from a walk-based perspective. We show that walk enumeration can be leveraged to capture structural properties of signed networks, such as local balance, effective alliances or enmities, signed Euclidean distances, similarity metrics, and correlation coefficients. This thesis also investigates how the structure of signed networks impacts dynamical processes taking place on them, focusing on standard and anomalous diffusion models, where structural balance is shown to influence the convergence towards a stationary state. Finally, we prove the practical utility of our theoretical results by applying them to the analysis of various real-world datasets. In particular, we show how our metrics can be used to detect factions of antagonizing agents, uncover hierarchies of alliances, and identify polarized nodes. A particular emphasis is placed on the analysis of international relations throughout history. Our metrics are able to effectively detect significant historical events and periods of conflict, offering a quantitative approach that complements traditional narrative-driven historical research. Overall, this thesis provides a rigorous mathematical foundation for the mathematical analysis of signed networks, as well as a set of versatile tools for analyzing empirical systems that exhibit a mix of cooperation and conflict.