Complex systems research is becoming ever more important in both the natural and social sciences. It is commonly implied that there is such a thing as a complex system, different examples of which are studied across many disciplines. However, there is no concise definition of a complex system, let alone a definition on which all scientists agree. We review various attempts to characterize a complex system, and consider a core set of features that are widely associated with complex systems in (...) the literature and by those in the field. We argue that some of these features are neither necessary nor sufficient for complexity, and that some of them are too vague or confused to be of any analytical use. In order to bring mathematical rigour to the issue we then review some standard measures of complexity from the scientific literature, and offer a taxonomy for them, before arguing that the one that best captures the qualitative notion of the order produced by complex systems is that of the Statistical Complexity. Finally, we offer our own list of necessary conditions as a characterization of complexity. These conditions are qualitative and may not be jointly sufficient for complexity. We close with some suggestions for future work. (shrink)

Murray Gell-Mann has proposed the concept of effective complexity as a measure of information content. The effective complexity of a string of digits is defined as the algorithmic complexity of the regular component of the string. This paper argues that the effective complexity of a given string is not uniquely determined. The effective complexity of a string admitting a physical interpretation, such as an empirical data set, depends on the cognitive and practical interests of investigators. The effective complexity of a (...) string as a purely formal construct, lacking a physical interpretation, is either close to zero, or equal to the string's algorithmic complexity, or arbitrary, depending on the auxiliary criterion chosen to pick out the regular component of the string. Because of this flaw, the concept of effective complexity is unsuitable as a measure of information content. (shrink)