Towards a Universal Measure of Complexity

Abstract

Recently, it has been argued that entropy can be a direct measure of complexity, where the smaller value of entropy indicates lower system complexity, while its larger value indicates higher system complexity. We dispute this view and propose a universal measure of complexity that is based on Gell-Mann’s view of complexity. Our universal measure of complexity is based on a non-linear transformation of time-dependent entropy, where the system state with the highest complexity is the most distant from all the states of the system of lesser or no complexity. We have shown that the most complex is the optimally mixed state consisting of pure states, i.e., of the most regular and most disordered which the space of states of a given system allows. A parsimonious paradigmatic example of the simplest system with a small and a large number of degrees of freedom is shown to support this methodology. Several important features of this universal measure are pointed out, especially its flexibility (i.e., its openness to extensions), suitability to the analysis of system critical behaviour, and suitability to study the dynamic complexity.

Numerical studies for an ab initio investigation into the Boltzmann prescription in statistical mechanics of large systems

“…These studies typically assume a non-logarithmic entropy, leading to power law distributions (e.g., Zipf-Mandelbrot distributions [6,7,8]) instead of exponential distributions of energy. Tsallis statistical mechanics falls into this category and it has found many applications throughout the years [11,12,13,14,15,16,17,18,19,20]. On the other hand, not everybody is fully convinced about the validity of such approaches to statistical mechanics [21,22].…”

Three Decades in Econophysics—From Microscopic Modelling to Macroscopic Complexity and Back

“…We wish to highlight the close relationship between the micro and the macro, specifically showing how and where models from physics applied to economic and financial entities bring about the emergent properties that make the field of econophysics so challenging and interesting. From the early insights and models [ 13 ], their extensions [ 14 , 15 ] through what has become known as stylized facts [ 16 ] of the financial markets, including behaviour of prices and their volatility, and to the inherent connectivity driving global risk [ 17 ], tools and methods from physics and complexity sciences [ 18 , 19 ], such as phase transitions [ 2 , 3 , 20 ], fractal and multifractal analysis [ 21 ] and network science [ 22 , 23 , 24 , 25 ], all help to understand the intricacies of our economic and financial lives. The following sections will move back and forth between the micro and macro perspective highlighting some recent research driving econophysics forward.…”

Multiscale Entropy Analysis of Short Signals: The Robustness of Fuzzy Entropy-Based Variants Compared to Full-Length Long Signals

“…Quantifying the complexity level from realizations of the system’s dynamics, i.e., time series, is still a challenge. While different interpretations of complexity may be assumed, entropy certainly plays a role in the estimation of time series complexity [ 2 , 3 ].…”