Linear-stability measures for assessing the state of a dynamical system are inherently local, and thus insufficient to quantify stability against substantial perturbations. The volume of a state’s basin of attraction offers a powerful alternative—and points towards a plausible explanation for regularity in real-world networks. The human brain1,2, power grids3, arrays of coupled lasers4 and the Amazon rainforest5,6 are all characterized by multistability7. The likelihood that these systems will remain in the most desirable of their many stable states depends on their stability against significant perturbations, particularly in a state space populated by undesirable states. Here we claim that the traditional linearization-based approach to stability is too local to adequately assess how stable a state is. Instead, we quantify it in terms of basin stability, a new measure related to the volume of the basin of attraction. Basin stability is non-local, nonlinear and easily applicable, even to high-dimensional systems. It provides a long-sought-after explanation for the surprisingly regular topologies8,9,10 of neural networks and power grids, which have eluded theoretical description based solely on linear stability11,12,13. We anticipate that basin stability will provide a powerful tool for complex systems studies, including the assessment of multistable climatic tipping elements14.

Multivariate transfer entropy (TE) is a model-free approach to detect causalities in multivariate time series. It is able to distinguish direct from indirect causality and common drivers without assuming any underlying model. But despite these advantages it has mostly been applied in a bivariate setting as it is hard to estimate reliably in high dimensions since its definition involves infinite vectors. To overcome this limitation, we propose to embed TE into the framework of graphical models and present a formula that decomposes TE into a sum of finite-dimensional contributions that we call decomposed transfer entropy. Graphical models further provide a richer picture because they also yield the causal coupling delays. To estimate the graphical model we suggest an iterative algorithm, a modified version of the PC-algorithm with a very low estimation dimension. We present an appropriate significance test and demonstrate the method's performance using examples of nonlinear stochastic delay-differential equations and observational climate data (sea level pressure).

Geoscientific measurements often provide time series with irregular time sampling, requiring either data reconstruction (interpolation) or sophisticated methods to handle irregular sampling. We compare the linear interpolation technique and different approaches for analyzing the correlation functions and persistence of irregularly sampled time series, as Lomb-Scargle Fourier transformation and kernel-based methods. In a thorough benchmark test we investigate the performance of these techniques. All methods have comparable root mean square errors (RMSEs) for low skewness of the inter-observation time distribution. For high skewness, very irregular data, interpolation bias and RMSE increase strongly. We find a 40 % lower RMSE for the lag-1 autocorrelation function (ACF) for the Gaussian kernel method vs. the linear interpolation scheme,in the analysis of highly irregular time series. For the cross correlation function (CCF) the RMSE is then lower by 60 %. The application of the Lomb-Scargle technique gave results comparable to the kernel methods for the univariate, but poorer results in the bivariate case. Especially the high-frequency components of the signal, where classical methods show a strong bias in ACF and CCF magnitude, are preserved when using the kernel methods. We illustrate the performances of interpolation vs. Gaussian kernel method by applying both to paleo-data from four locations, reflecting late Holocene Asian monsoon variability as derived from speleothem δ¹⁸O measurements. Cross correlation results are similar for both methods, which we attribute to the long time scales of the common variability. The persistence time (memory) is strongly overestimated when using the standard, interpolation-based, approach. Hence, the Gaussian kernel is a reliable and more robust estimator with significant advantages compared to other techniques and suitable for large scale application to paleo-data.

While it is an important problem to identify the existence of causal associations between two components of a multivariate time series, a topic addressed in Runge, Heitzig, Petoukhov, and Kurths [Phys. Rev. Lett. 108, 258701 (2012)], it is even more important to assess the strength of their association in a meaningful way. In the present article we focus on the problem of defining a meaningful coupling strength using information-theoretic measures and demonstrate the shortcomings of the well-known mutual information and transfer entropy. Instead, we propose a certain time-delayed conditional mutual information, the momentary information transfer (MIT), as a lag-specific measure of association that is general, causal, reflects a well interpretable notion of coupling strength, and is practically computable. Rooted in information theory, MIT is general in that it does not assume a certain model class underlying the process that generates the time series. As discussed in a previous paper [Runge, Heitzig, Petoukhov, and Kurths, Phys. Rev. Lett. 108, 258701 (2012)], the general framework of graphical models makes MIT causal in that it gives a nonzero value only to lagged components that are not independent conditional on the remaining process. Further, graphical models admit a low-dimensional formulation of conditions, which is important for a reliable estimation of conditional mutual information and, thus, makes MIT practically computable. MIT is based on the fundamental concept of source entropy, which we utilize to yield a notion of coupling strength that is, compared to mutual information and transfer entropy, well interpretable in that, for many cases, it solely depends on the interaction of the two components at a certain lag. In particular, MIT is, thus, in many cases able to exclude the misleading influence of autodependency within a process in an information-theoretic way. We formalize and prove this idea analytically and numerically for a general class of nonlinear stochastic processes and illustrate the potential of MIT on climatological data.