March 24, 2015
This work introduces a complexity measure which addresses some conflicting issues between existing ones by using a new principle - measuring the average amount of symmetry broken by an object. It attributes low (although different) complexity to either deterministic or random homogeneous densities and higher complexity to the intermediate cases. This new measure is easily computable, breaks the coarse graining paradigm and can be straightforwardly generalised, including to continuous cases and general networks. By applying this measure to a series of objects, it is shown that it can be consistently used for both small scale structures with exact symmetry breaking and large scale patterns, for which, differently from similar measures, it consistently discriminates between repetitive patterns, random configurations and self-similar structures.
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November 1, 2017
The concept of complexity appears in virtually all areas of knowledge. Its intuitive meaning shares similarities across fields, but disagreements between its details hinders a general definition, leading to a plethora of proposed measurements. While each might be appropriated to the problems it addresses, the lack of an underlying fundamental principle prevents the development of a unified theory. Here it is shown that the statistics of the amount of symmetry broken by system...
September 29, 2019
Complexity is a multi-faceted phenomenon, involving a variety of features including disorder, nonlinearity, and self-organisation. We use a recently developed rigorous framework for complexity to understand measures of complexity. We illustrate, by example, how features of complexity can be quantified, and we analyse a selection of purported measures of complexity that have found wide application and explain whether and how they measure complexity. We also discuss some of the...
May 15, 2002
A measure of complexity based on a probabilistic description of physical systems is proposed. This measure incorporates the main features of the intuitive notion of such a magnitude. It can be applied to many physical situations and to different descriptions of a given system. Moreover, the calculation of its value does not require a considerable computational effort in many cases of physical interest.
June 20, 2010
In this review we establish various connections between complex networks and symmetry. While special types of symmetries (e.g., automorphisms) are studied in detail within discrete mathematics for particular classes of deterministic graphs, the analysis of more general symmetries in real complex networks is far less developed. We argue that real networks, as any entity characterized by imperfections or errors, necessarily require a stochastic notion of invariance. We therefor...
February 2, 2022
Nanostructured surfaces usually exhibit complicated morphologies that cannot be described in terms of Euclidean geometry. Simultaneously, they do not constitute fully random noise fields to be characterized by simple stochastics and probability theory. In most cases, nanomorphologies consist of complicated mixtures of order and randomness, which should be described quantitatively if one aims to control their fabrication and properties. In this work, inspired by recent develop...
May 14, 2002
I present my viewpoint on complexity, stressing general arguments and using a rather simple language.
March 10, 2020
Complexity of patterns is a 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- and three-dimensional patterns that can be straightforwardly generalized onto other classes of objects. It is based on multi-step renormalization of the pattern of intere...
October 22, 2022
The best way to model, understand, and quantify the information contained in complex systems is an open question in physics, mathematics, and computer science. The uncertain relationship between entropy and complexity further complicates this question. With ideas drawn from the object-relations theory of psychology, this paper develops an object-relations model of complex systems which generalizes to systems of all types, including mathematical operations, machines, biologica...
February 12, 2009
We propose a new type of entropic descriptor that is able to quantify the statistical complexity (a measure of complex behaviour) by taking simultaneously into account the average departures of a system's entropy S from both its maximum possible value Smax and its minimum possible value Smin. When these two departures are similar to each other, the statistical complexity is maximal. We apply the new concept to the variability, over a range of length scales, of spatial or grey...
May 9, 2012
Concepts used in the scientific study of complex systems have become so widespread that their use and abuse has led to ambiguity and confusion in their meaning. In this paper we use information theory to provide abstract and concise measures of complexity, emergence, self-organization, and homeostasis. The purpose is to clarify the meaning of these concepts with the aid of the proposed formal measures. In a simplified version of the measures (focusing on the information produ...