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 classic information-theoretic measures from the 1980s and 1990s. This work gives the reader a tool kit for quantifying features of complexity across the sciences.
Similar papers 1
January 28, 2002
This article summarises a Web-book on "Complexity" that was developed to introduce undergraduate students to interesting complex systems in the biological, physical and social sciences, and the common tools, principles and concepts used for their study.
June 30, 2010
If a concept is not well defined, there are grounds for its abuse. This is particularly true of complexity, an inherently interdisciplinary concept that has penetrated very different fields of intellectual activity from physics to linguistics, but with no underlying, unified theory. Complexity has become a popular buzzword used in the hope of gaining attention or funding -- institutes and research networks associated with complex systems grow like mushrooms. Why and how did i...
May 14, 2002
I present my viewpoint on complexity, stressing general arguments and using a rather simple language.
January 2, 2001
Numerous definitions for complexity have been proposed over the last half century, with little consensus achieved on how to use the term. A definition of complexity is supplied here that is closely related to the Kolmogorov Complexity and Shannon Entropy measures widely used as complexity measures, yet addresses a number of concerns raised against these measures. However, the price of doing this is to introduce context dependence into the definition of complexity. It is argue...
May 6, 2008
The term {\em complexity} is used informally both as a quality and as a quantity. As a quality, complexity has something to do with our ability to understand a system or object -- we understand simple systems, but not complex ones. On another level, {\em complexity} is used as a quantity, when we talk about something being more complicated than another. In this chapter, we explore the formalisation of both meanings of complexity, which happened during the latter half of the...
November 16, 2011
At present, there is a great deal of confusion regarding complexity and its measures (reviews on complexity measures are found in, e.g. Lloyd, 2001 and Shalizi, 2006 and more references therein). Moreover, there is also confusion regarding the nature of life. In this situation, it seems the task of determining the fundamental complexity measures of life is especially difficult. Yet this task is just part of a greater task: obtaining substantial insights into the nature of bio...
August 16, 2012
We review possible measures of complexity which might in particular be applicable to situations where the complexity seems to arise spontaneously. We point out that not all of them correspond to the intuitive (or "naive") notion, and that one should not expect a unique observable of complexity. One of the main problems is to distinguish complex from disordered systems. This and the fact that complexity is closely related to information requires that we also give a review of i...
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...
August 31, 2011
The term complexity derives etymologically from the Latin plexus, which means interwoven. Intuitively, this implies that something complex is composed by elements that are difficult to separate. This difficulty arises from the relevant interactions that take place between components. This lack of separability is at odds with the classical scientific method - which has been used since the times of Galileo, Newton, Descartes, and Laplace - and has also influenced philosophy and...
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.