DOI
https://doi.org/10.1007/s13194-012-0056-8

Summary
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.

Relevance for Complex Systems Knowledge
In the literature, there are many definitions of how to measure complexity. Such measures of complexity may be applied to three targets: the methods used to study certain systems; the data that are obtained from certain systems; the system itself. Some say that the complexity sciences are simply those that use certain characteristic methods. This approach is a pragmatic account of complexity. Those measures that are applied directly to systems are physical accounts of complexity. Examples are the theories of Logical Depth, Thermodynamic Depth, and Effective Complexity. Finally, examples of measures of complexity applied to the data are the Statistical Complexity and the Effective Measure Complexity.

Point of Strength
Three questions are usually posed when attempting to quantify the complexity of the system under study. They are: (1) How hard is it to describe the system? (2) How hard is it to create that system? (3) What is the degree of organization? Question (1) represents the difficulty of description; question (2) represents the difficulty of creation; question (3) represents the degree of organization. The difficulty of description corresponds to the complexity of the data, whereas the difficulty of creation corresponds to the complexity of the system.