Effective precursors for self-organization of complex systems into a critical state based on dynamic series data

Abstract

Many different precursors are known, but not all of which are effective, i.e., giving enough time to take preventive measures and with a minimum number of false early warning signals. The study aims to select and study effective early warning measures from a set of measures directly related to critical slowing down as well as to the change in the structure of the reconstructed phase space in the neighborhood of the critical transition point of sand cellular automata. We obtained a dynamical series of the number of unstable nodes in automata with stochastic and deterministic vertex collapse rules, with different topological graph structure and probabilistic distribution law for pumping of automata. For these dynamical series we computed windowed early warning measures. We formulated the notion of an effective measure as the measure that has the smallest number of false signals and the longest early warning time among the set of early warning measures. We found that regardless of the rules, topological structure of graphs, and probabilistic distribution law for pumping of automata, the effective early warning measures are the embedding dimension, correlation dimension, and approximation entropy estimated using the false nearest neighbors algorithm. The variance has the smallest early warning time, and the largest Lyapunov exponent has the greatest number of false early warning signals. Autocorrelation at lag-1 and Welch’s estimate for the scaling exponent of power spectral density cannot be used as early warning measures for critical transitions in the automata. The efficiency definition we introduced can be used to search for and investigate new early warning measures. Embedding dimension, correlation dimension and approximation entropy can be used as effective real-time early warning measures for critical transitions in real-world systems isomorphic to sand cellular automata such as microblogging social network and stock exchange.