Quantifying the robustness of interconnected systems from their collective dynamics

Complex networks are powerful models to investigate real-world physical, biophysical and socio-technical systems. However, in some cases, the underlying structure cannot be observed directly and connections have to be statistically inferred by observing, on nodes, the time course of relevant physical quantities. This is a well-known inverse problem which to date has not been satisfactorily solved, yet, although it is a very common scenario when modelling interconnected systems. Solving this problem would allow to characterize the macroscopic features of a networked system, from mesoscale organization to critical behaviour. To this aim, a thesis work is here proposed to use the observation of collective dynamics, in terms of multivariate time series, to gain insights about the structural and functional robustness of the underlying complex network to external perturbations. Specifically, we will explore the limits of the fuzzy network approach, recently introduced, where uncertainty about the existence of edges results in an ensemble network reconstruction. Besides being a new approach for network inference, it has not been used to date to investigate the critical behaviour of a complex network, both from a theoretical and computational viewpoint. Our theoretical analysis aims at extending the well-known generating function formalism, widely used in statistical mechanics, to this framework. In parallel, the computational approach considers synthetic networks and Kuramoto dynamics on top of them to characterize robustness through the time course of oscillator phases far from equilibrium. By assuming no knowledge about the network behind the Kuramoto dynamics, the analysis focuses on proving, under the right conditions, whether the network ensemble allows one to recover the robustness properties of the synthetic network underneath.