Complex systems, emergence, and multiscale analysis: A tutorial and brief survey

Abstract

Mankind has long been fascinated by emergence in complex systems. With the rapidly accumulating big data in almost every branch of science, engineering, and society, a golden age for the study of complex systems and emergence has arisen. Among the many values of big data are to detect changes in system dynamics and to help science to extend its reach, and most desirably, to possibly uncover new fundamental laws. Unfortunately, these goals are hard to achieve using black-box machine-learning based approaches for big data analysis. Especially, when systems are not functioning properly, their dynamics must be highly nonlinear, and as long as abnormal behaviors occur rarely, relevant data for abnormal behaviors cannot be expected to be abundant enough to be adequately tackled by machine-learning based approaches. To better cope with these situations, we advocate to synergistically use mainstream machine learning based approaches and multiscale approaches from complexity science. The latter are very useful for finding key parameters characterizing the evolution of a dynamical system, including malfunctioning of the system. One of the many uses of such parameters is to design simpler but more accurate unsupervised machine learning schemes. To illustrate the ideas, we will first provide a tutorial introduction to complex systems and emergence, then we present two multiscale approaches. One is based on adaptive filtering, which is excellent at trend analysis, noise reduction, and (multi)fractal analysis. The other originates from chaos theory and can unify the major complexity measures that have been developed in recent decades. To make the ideas and methods better accessed by a wider audience, the paper is designed as a tutorial survey, emphasizing the connections among the different concepts from complexity science. Many original discussions, arguments, and results pertinent to real-world applications are also presented so that readers can be best stimulated to apply and further develop the ideas and methods covered in the article to solve their own problems. This article is purported both as a tutorial and a survey. It can be used as course material, including summer extensive training courses. When the material is used for teaching purposes, it will be beneficial to motivate students to have hands-on experiences with the many methods discussed in the paper. Instructors as well as readers interested in the computer analysis programs are welcome to contact the corresponding author.