Causality structures in nonlinear dynamical systems

Causal relationships in a complex system determine the system’s dynamical processes and the subsequent output time series. Detecting causalities from the time series is a basic task in the data-based identification of the system. However, a causal relationship may be present only in some specific time durations. Even in a case where a causal relationship in the underlying dynamical mechanism is persistent, its effect on the time series generally changes, e.g., it is significant in some specific time durations and becomes weak or even disappears in other time durations. In the currently used causality detection methods, the causal effects are measured with some scalar quantities obtained from the time series with a statistical average procedure, i.e., the details of causal effects are lost. In this work, a concept called the structure of causality is proposed to preserve the details of causal effect. Technically, a phase space reconstruction method is employed to embed the time series for a pair of elements into a high-dimensional phase space, the phase points in which are subsequently mapped into graph-lets using the visibility graph algorithm to retain their structural information. The co-occurrences for all the pairs of unique graph-lets form a co-occurrent matrix. One finds the pairs that can reflect the causality relationship with high significance, forming a set of co-occurrent unique graph-let pairs, called the structure of causality. Causalities in several dynamical models and several real-world complex systems turn out to have their rich specific causality structures, respectively. Possible applications of the causality structure include, e.g., the improvement of performances for causality detection methods and the designation of intervention and/or control strategies.