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)

Complexity is heterogenous, involving nonlinearity, self-organisation, diversity, adaptive behaviour, among other things. It is therefore obviously worth asking whether purported measures of complexity measure aggregate phenomena, or individual aspects of complexity and if so which. This paper uses a recently developed rigorous framework for understanding complexity to answer this question about measurement. The approach is two-fold: find measures of individual aspects of complexity on the one hand, and explain measures of complexity on the other. We illustrate the conceptual framework of (...) complexity science and how it links the foundations to the practised science with examples from different scientific fields and of various aspects of complexity. Furthermore, we analyse a selection of purported measures of complexity that have found wide application and explain why and how they measure aspects of complexity. This work gives the reader a tool to take any existing measure of complexity and analyse it, and to take any feature of complexity and find the right measure for it. (shrink)