Multiscale structural complexity of natural patterns

Significance Structural complexity of patterns, systems, and processes is a very basic and intuitively clear concept in human’s perception of reality that is very difficult to describe quantitatively. A demand in a mathematical notion that properly reflects complexity of hierarchical nonrandom structures exists in many areas of science, from geology to social sciences. Here, we propose an easy to compute, robust, and universal definition of complexity based on interscale dissimilarity of patterns. Using classical magnetic patterns as an example, we demonstrate that our approach leads to maximization of complexity for the most visually nontrivial patterns and can be used to detect phase transitions with high accuracy, making it a promising tool for studying pattern formation in a variety of systems. Complexity of patterns is key information for human brain to differ objects of about the same size and shape. Like other innate human senses, the complexity perception cannot be easily quantified. We propose a transparent and universal machine method for estimating structural (effective) complexity of two-dimensional and three-dimensional patterns that can be straightforwardly generalized onto other classes of objects. It is based on multistep renormalization of the pattern of interest and computing the overlap between neighboring renormalized layers. This way, we can define a single number characterizing the structural complexity of an object. We apply this definition to quantify complexity of various magnetic patterns and demonstrate that not only does it reflect the intuitive feeling of what is “complex” and what is “simple” but also, can be used to accurately detect different phase transitions and gain information about dynamics of nonequilibrium systems. When employed for that, the proposed scheme is much simpler and numerically cheaper than the standard methods based on computing correlation functions or using machine learning techniques.