Complex systems with time-varying interactions

[eng] The nonlinear interactions among components of complex systems give rise to intricate dynamics. Understanding such behaviors requires not only modeling the internal dynamics of the components, but also capturing how their interaction structure evolves over time. This evolution can be random, chaotic, and driven by external environmental factors. Developing tools to characterize such evolving patterns is essential for uncovering the mechanisms that shape complex system behavior. Several approaches have been proposed to study the temporal evolution of complex systems, including methods from time series analysis, dynamical systems theory, and temporal network models. These tools have advanced our understanding of systems with time-varying interactions. However, many aspects of the dynamics of networks remain poorly understood, and the development of general tools for their analysis is still lacking. Moreover, it is possible that other types of dynamical behaviors are present but remain hidden, as current methods are not yet capable of detecting or characterizing them. This thesis addresses these challenges by first examining how environmental changes affect the dynamics of complex systems. The analysis identifies the conditions under which temporal variability in the environment influences system behavior, and to what extent these effects manifest in the dynamics. In the second part, a new approach to network dynamics is introduced, based on the assumption that a temporal network can be viewed as a trajectory in graph space. Time series analysis metrics—such as empirical autocorrelation and Lyapunov exponent estimation—are applied to characterize this evolution for both labelled and unlabelled temporal networks. Finally, the proposed framework is applied to empirical brain network data, providing insights into how the predictability of neural dynamics varies across different pathological conditions. These contributions provide a unified framework for analyzing the dynamics of systems with time-varying interactions, offering methodological tools that can be applied across domains where evolving structures play a central role.