# What is a complex system?

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## Abstract

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.

## Keywords

Complexity Statistical complexity Information Complex system## Notes

### Acknowledgements

We are extremely grateful to several anonymous referees for this journal and to the editor for very helpful comments and criticisms, and also to the students of the Bristol Centre for the Complexity Sciences doctoral programme over several years for their comments on our ideas. James Ladyman acknowledges the support of the AHRC Foundations of Structuralism project. Karoline Wiesner acknowledges funding through EPSRC grant EP/E501214/1.

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