Supplementary materials and Figures from Finding the direction of lowest resilience in multivariate complex systems

The dynamics of complex systems, such as ecosystems, financial markets and the human brain emerge from the interactions of numerous components. We often lack the knowledge to build reliable models for the behaviour of such network systems. This makes it difficult to predict potential instabilities. We show that one could use the natural fluctuations in multivariate time series to reveal network regions with particularly slow dynamics. The multidimensional slowness points to the direction of minimal resilience, in the sense that simultaneous perturbations on this set of nodes will take longest to recover. We compare an autocorrelation-based method to a variance-based method and compare them for different time-series lengths, data resolution and different noise regimes and show that the autocorrelation-based method is less robust for short time series or time series with a low resolution but more robust for varying noise levels. This novel approach may help to identify unstable regions of multivariate systems, or to distinguish safe from unsafe perturbations.