Inferring causal networks of dynamical systems through transient dynamics and perturbation

General optimization framework for accurate and efficient reconstruction of symmetric complex networks from dynamical data

The challenging problem of network reconstruction from dynamical data can in general be formulated as an optimization task of solving multiple linear equations. Existing approaches are of the two types: Point-by-point (PBP) and global methods. The local PBP method is computationally efficient, but the accuracies of its solutions are somehow low, while a global method has the opposite traits: High accuracy and high computational cost. Taking advantage of the network symmetry, we develop a novel framework integrating the advantages of both the PBP and global methods while avoiding their shortcomings: i.e., high reconstruction accuracy is guaranteed, but the computational cost is orders of magnitude lower than that of the global methods in the literature. The mathematical principle underlying our framework is block coordinate descent (BCD) for solving optimization problems, where the various blocks are determined by the network symmetry. The reconstruction framework is validated by numerical examples with a variety of network structures (i.e., sparse and dense networks) and dynamical processes. Our success is a demonstration that the general principle of exploiting symmetry can be extended to tackling the challenging inverse problem or reverse engineering of complex networks. Since solving a large number of linear equations is key to a plethora of problems in science and engineering, our BCD-based network reconstruction framework will find broader applications.

Usefulness and limitations of convergent cross sorting and continuity scaling methods for their application in simulated and real-world time series

Causality detection methods are valuable tools for detecting causal links in complex systems. The efficiency of continuity scaling (CS) and the convergent cross sorting (CSS) methods to detect causality was analysed. Usefulness and limitations of both methods in their application to simulated and real-world time series was explored under different scenarios. We find that CS is more robust and efficient than the CSS method for all simulated systems, even when increasing noise levels were considered. Both methods were not able to infer causality when time series with a marked difference in their main frequencies were analysed. Minimum time-series length required for the detection of a causal link depends on intrinsic system dynamics and on the method selected to detect it. Using simulated time series, only the CS method was capable to detect bidirectional causality. Causality detection, using the CS method, should at least include: (i) causality strength convergence analysis, (ii) statistical tests of significance, (iii) time-series standardization, and (iv) causality strength ratios as a strength indicator of relative causality between systems. Causality cannot be detected by either method in simulated time series that exhibit generalized synchronization.

Extreme transient dynamics

We study the extreme transient dynamics of four self-excited pendula coupled via the movable beam. A slight difference in the pendula lengths induces the appearance of traveling phase behavior, within which the oscillators synchronize, but the phases between the nodes change in time. We discuss various scenarios of traveling states (involving different pendula) and their properties, comparing them with classical synchronization patterns of phase-locking. The research investigates the problem of transient dynamics preceding the stabilization of the network on a final synchronous attractor, showing that the width of transient windows can become extremely long. The relation between the behavior of the system within the transient regime and its initial conditions is examined and described. Our results include both identical and non-identical pendula masses, showing that the distribution of the latter ones is related to the transients. The research performed in this paper underlines possible transient problems occurring during the analysis of the systems when the slow evolution of the dynamics can be misinterpreted as the final behavior.

Embedding entropy: a nonlinear measure of dynamical causality

Research on concepts and computational methods of causality has a long history, and there are various concepts of causality as well as corresponding computing algorithms based on measured data. Here, by considering causes and effects from a dynamical perspective, we present a unified mathematical framework for the so-called dynamical causality (DC), which can detect causal interactions over time; notably, this framework covers Granger causality, transfer entropy, embedding causality and their conditional versions. Based on this framework, we further propose a causality criterion called embedding entropy (EE) to measure the DC between two variables. Moreover, its conditional version, conditional embedding entropy (cEE), is also derived for detecting conditional/direct causality. The significant advantages of EE and cEE are that they can be employed for solving not only nonlinear causal inference but also the non-separability problem, and they can reduce the scale bias in numerical calculation. The performance and robustness of EE and cEE were demonstrated through numerical simulations, and causal inference on various real-world datasets validated their effectiveness.