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.
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Notes
The relationship between macrostates and microstates is key to the complex sciences because very often what is interesting about the system is the way that a stable causal structure arises that can be described at a higher level than that of the properties of the parts (see Section 2.5 on emergence below).
One anonymous referee claimed that it is not possible to define chaos, but on the contrary unlike complexity chaos can readily be defined as systems that exhibit so-called strong mixing. Moreover, recently, Charlotte Werndl has shown that there is a kind of unpredictability unique to chaos (2008). Note that chaos as in chaos theory is always deterministic chaos.
Note that we are not here talking about whether the system that produces the data is deterministic or not. Of course, the Shannon entropy of a probability distribution is insensitive to whether that probability distribution was produced by deterministic or an indeterministic system. Our point is just that a good measure of complexity will not be maximal for random data strings.
For a proof consider the following. For a given number of causal states the Statistical Complexity (Eq. 4) has a unique maximum at uniform probability distribution over the states. This is achieved by a perfectly periodic sequence with period equal to the number of states. As soon as deviations occur the probability distribution will likely not be uniform anymore and the Shannon entropy and with it the Statistical Complexity will decrease. Hence, the Statistical Complexity scores highest for perfectly ordered strings.
We are grateful to an anonymous reviewer whose criticisms based on the example of the climate forced us to clarify these points.
References
Anderson, P. W. (1972). More is different: Broken symmerty and the nature of the hierarchical structure of science. Science, 177, 393–396.
Brian Arthur, W. (1999). Complexity and the economy. Science, 284, 107–109.
Badii, R., & Politi, A. (1999). Complexity: Hierarchical structures and scaling in physics. Cambridge University Press.
Bennett, C. H. (1988). Logical depth and physical complexity. In R. Herken, (Ed.), The universal Turing machine, a half-century survey (pp. 227–257). Oxford: Oxford University Press.
Cooper, J. M. (Ed.) (1997). Phaedrus in Plato complete works. Hackett.
Cover, T. M., & Thomas, J. A. (2006). Elements of information theory (2nd edn.). Wiley-Blackwell, September.
Crutchfield, J. P. (1994). The calculi of emergence: Computation, dynamics and induction. Physica D: Nonlinear Phenomena, 75(1–3), 11–54.
Crutchfield, J. P., & Shalizi, C. R. (1999). Thermodynamic depth of causal states: Objective complexity via minimal representations. Physical Review E, 59(1), 275–283.
Crutchfield, J. P., & Young, K. (1989). Inferring statistical complexity. Physical Review Letters, 63, 105.
Crutchfield, J. P., & Young, K. (1990). Computation at the onset of chaos. Entropy, Complexity and the Physics of Information, SFI Studies in the Sciences of Complexity, VIII, 223–269.
Dennett, D. (1991). Real patterns. The Journal of Philosophy, 88(1), 27–51.
Editorial. (2009). No man is an island. Nature Physics, 5, 1.
Feynman, R. (2000). Feynman lectures on computation. Westview Press.
Foote, R. (2007). Mathematics and complex systems. Science, 318, 410–412.
Gell-Mann, M. (1995). What is complexity. Complexity, 1, 1.
Gell-Mann, M., & Lloyd, S. (1996). Information measures, effective complexity, and total information. Complexity, 2(1), 44–52.
Gell-Mann, M., & Lloyd, S. (2004). Effective complexity. In M. Gell-Mann, & C. Tsallis, (Eds.), Nonextensive entropy – interdisciplinary applications. The Santa Fe Institute, OUP USA.
Goldenfeld, N., & Kadanoff, L. P. (1999). Simple lessons from complexity. Science, 284, 87–89.
Grassberger, P. (1986). Toward a quantitative theory of self-generated complexity. International Journal of Theoretical Physics, 25(9), 907–938.
Grassberger, P. (1989). Problems in quantifying self-generated complexity. Helvetica Physica Acta, 62, 489–511.
Holland, J. H. (1992). Adaptation in natural and artificial systems: An introductory analysis with applications to biology, control and artificial intelligence (new edn.). MIT Press, July.
Holland, J. H. (1992). Complex adaptive systems. Daedalus, 121(1), 17–30.
Jaynes, E. T. (1957a). Information theory and statistical mechanics. The Physical Review, 106(4), 620–630.
Jaynes, E. T. (1957b). Information theory and statistical mechanics, ii. The Physical Review, 108(2), 171–190.
Kolmogorov, A. N. (1965). Three approaches to the quantitive definition of information. Problems of Information Transmission, 1, 1–17.
Kolmogorov, A. N. (1983). Combinatorial foundations of information theory and the calculus of probabilities. Russian Mathematical Surveys, 38(4), 29–40.
Ladyman, J., Ross, D., Spurrett, D., & Collier, J. (2007). Everything must go: Metaphysics naturalized. Oxford University Press.
Li, C.-B., Yang, H., & Komatsuzaki, T. (2008). Multiscale complex network of protein conformational fluctuations in single-molecule time series. Proceedings of the National Academy of Sciences, 105(2), 536–541.
Li, M., & Vitnyi, P. M. B. (2009). An introduction to Kolmogorov complexity and its applications (3rd ed.). Springer, March.
Lloyd, S. (2001). Measures of complexity: A nonexhaustive list. Control Systems Magazine, IEEE, 21, 7–8.
Lloyd, S., & Pagels, H. (1988). Complexity as thermodynamic depth. Annals of Physics, 188, 186–213.
MacKay, R. S. (2008). Nonlinearity in complexity science. Nonlinearity, 21, T273.
Mainzer, K. (1994). Thinking in complexity: The complex dynamics of matter, mind and mankind. Springer.
Merricks, T. (2001). Objects and persons. Oxford University Press.
Mitchell, S. (2009). Unsimple truths: Science, complexity, and policy. University of Chicago Press.
Morin, E., & Belanger, J. L. R. (1992). Method: Towards a study of humankind : The nature of nature: 001. Peter Lang Pub Inc, November.
Paley, W. (2006). Natural theology. Oxford University Press.
Palmer, A. J., Fairall, C. W., & Brewer, W. A. (2000). Complexity in the atmosphere. IEEE Transactions on Geoscience and Remote Sensing, 38(4), 2056–2063.
Parrish, J. K., & Edelstein-Keshet, L. (1999). Complexity, pattern, and evolutionary trade-offs in animal aggregation. Science, 284, 99–101.
Rind, D. (1999). Complexity and climate. Science, 284, 105–107.
Ross, D. (2000). Rainforrest realism: A Dennettian theory of existence. In D. Ross (Ed.), Dennett’s philosophy: A comprehensive assessment (Chapter 8, pp. 147–168). MIT Press.
Shalizi, C. R., & Crutchfield, J. P. (2001). Computational mechanics: Pattern and prediction, structure and simplicity. Journal of Statistical Physics, 104(3), 817–879.
Shalizi, C. R., & Moore, C. (2003). What is a macrostate? Subjective observations and objective dynamics. cond-mat/0303625.
Shalizi, C. R., Shalizi, K. L., & Haslinger, R. (2004). Quantifying self-organization with optimal predictors. Physical Review Letters, 93(11), 118701.
Shannon, C. E. (1948). A mathematical theory of communication. Bell System Technical Journal, 27, 379–423; 623–656.
Simon, H. A. (1962). The architecture of complexity. Proceedings of the American Philosophical Society, 106(6), 467–482.
Timpson, C. G. (2006). The grammar of teleportation. British Journal of Philosophy of Science, 57(3), 587–621.
Wackerbauer, R., Witt, A., Atmanspacher, H., Kurths, J., & Scheingraber, H. (1994). A comparative classification of complexity measures. Chaos, Solitons & Fractals, 4(1), 133–173.
Wallace, D. (2003). Everett and structure. Studies In History and Philosophy of Modern Physics, 34(1), 87–105.
Weng, G., Bhalla, U. S., & Iyengar, R. (1999). Complexity in biological signaling systems. Science, 284, 92–96.
Werner, B. T. (1999). Complexity in natural landform patterns. Science, 284, 102–104.
Whitesides, G. M., & Ismagilov, R. F. (1999). Complexity in chemistry. Science, 284, 89–92.
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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Ladyman, J., Lambert, J. & Wiesner, K. What is a complex system?. Euro Jnl Phil Sci 3, 33–67 (2013). https://doi.org/10.1007/s13194-012-0056-8
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DOI: https://doi.org/10.1007/s13194-012-0056-8