The heart of the scientific enterprise is a rational effort to understand the causes behind the phenomena we observe. In large-scale complex dynamical systems such as the Earth system, real experiments are rarely feasible. However, a rapidly increasing amount of observational and simulated data opens up the use of novel data-driven causal methods beyond the commonly adopted correlation techniques. Here, we give an overview of causal inference frameworks and identify promising generic application cases common in Earth system sciences and beyond. We discuss challenges and initiate the benchmark platform causeme.net to close the gap between method users and developers.

A common goal in the control of a large network is to minimize the number of driver nodes or control inputs. Yet, the physical determination of control signals and the properties of the resulting control trajectories remain widely under-explored. Here we show that: (i) numerical control fails in practice even for linear systems if the controllability Gramian is ill-conditioned, which occurs frequently even when existing controllability criteria are satisfied unambiguously; (ii) the control trajectories are generally nonlocal in the phase space, and their lengths are strongly anti-correlated with the numerical success rate and number of control inputs; (iii) numerical success rate increases abruptly from zero to nearly one as the number of control inputs is increased, a transformation we term numerical controllability transition. This reveals a trade-off between nonlocality of the control trajectory in the phase space and nonlocality of the control inputs in the network itself. The failure of numerical control cannot be overcome in general by merely increasing numerical precision-successful control requires instead increasing the number of control inputs beyond the numerical controllability transition.

We study optimal synchronization in networks of heterogeneous phase oscillators. Our main result is the derivation of a synchrony alignment function that encodes the interplay between network structure and oscillators' frequencies and can be readily optimized. We highlight its utility in two general problems: constrained frequency allocation and network design. In general, we find that synchronization is promoted by strong alignments between frequencies and the dominant Laplacian eigenvectors, as well as a matching between the heterogeneity of frequencies and network structure. PACS numbers: 05.45.Xt, 89.75.Hc A central goal of complexity theory is to understand the emergence of collective behavior in large ensembles of interacting dynamical systems. Synchronization of networkcoupled oscillators has served as a paradigm for understanding emergence [1][2][3][4], where examples arise in nature (e.g., flashing of fireflies [5] and cardiac pacemaker cells [6]), engineering (e.g., power grid [7] and bridge oscillations [8]), and at their intersection (e.g., synthetic cell engineering [9]). We consider the dynamics of N network-coupled phase oscillators θ i for i = 1, . . . , N , whose evolution is governed bẏHere ω i is the natural frequency of oscillator i, K > 0 is the coupling strength, [A ij ] is a symmetric network adjacency matrix, and H is a 2π-periodic coupling function [10]. We treat H(θ) with full generality so long as H ′ (0) > 0. The choices H(θ) = sin(θ) and H(θ) = sin(θ − α) with the phase-lag parameter α ∈ (−π/2, π/2) yield the classical Kuramoto [10] and Sakaguchi-Kuramoto models [11].Considerable research has shown that the underlying structure of a network plays a crucial role in determining synchronization [12][13][14][15][16][17][18][19][20][21], yet the precise relationship between the dynamical and structural properties of a network and its synchronization remains not fully understood. One unanswered question is, given an objective measure of synchronization, how can synchronization be optimized? One application lies in synchronizing the power grid [22], where sources and loads can be modeled as oscillators with different frequencies. To this end, we ask: what structural and/or dynamical properties should be present to optimize synchronization?We measure the degree of synchronization of an ensemble of oscillators using the Kuramoto order parameterHere re iψ denotes the phases' centroid on the complex unit circle, with the magnitude r ranging from 0 (incoherence) to 1 (perfect synchronization) [10]. In general, the question of optimization (maximizing r) is challenging due to the fact that the macroscopic dynamics depend on both the natural frequencies and the network structure. To quantify the interplay between node dynamics and network structure, we derive directly from Eqs. (1) and (2) a synchrony alignment function which is an objective measure of synchronization and can be used to systematically optimize a network's synchronization. We highlight this result by addressing two classes of op...

Abstract. The broad abundance of time series data, which is in sharp contrast to limited knowledge of the underlying network dynamic processes that produce such observations, calls for a rigorous and efficient method of causal network inference. Here we develop mathematical theory of causation entropy, an information-theoretic statistic designed for model-free causality inference. For stationary Markov processes, we prove that for a given node in the network, its causal parents forms the minimal set of nodes that maximizes causation entropy, a result we refer to as the optimal causation entropy principle. Furthermore, this principle guides us to develop computational and data efficient algorithms for causal network inference based on a two-step discovery and removal algorithm for time series data for a network-couple dynamical system. Validation in terms of analytical and numerical results for Gaussian processes on large random networks highlight that inference by our algorithm outperforms previous leading methods including conditioned Granger causality and transfer entropy. Interestingly, our numerical results suggest that the number of samples required for accurate inference depends strongly on network characteristics such as the density of links and information diffusion rate and not necessarily on the number of nodes.

We derive a master stability function (MSF) for synchronization in networks of coupled dynamical systems with small but arbitrary parametric variations. Analogous to the MSF for identical systems, our generalized MSF simultaneously solves the linear stability problem for near-synchronous states (NSS) for all possible connectivity structures. We also derive a general sufficient condition for stable near-synchronization and show that the synchronization error scales linearly with the magnitude of parameter variations. Our analysis underlines significant roles played by the Laplacian eigenvectors in the study of network synchronization of near-identical systems.