IMA: Complex Systems
Robert MacKay Presidential Address: Complex System in Science and Society Mathematics Today Vol. 49. No. 6 Dec 2013
This nicely provides an underpinning for various familiar models of complexity in terms of stochastic dynamical systems.
- A system is complex when it consists of many interdependent components. Complexity is a Goldilocks phenomena, arising from intermediate levels of interdependence.
- Emergence is where there are one or more probability distributions for the ‘state’ of the system compatible with the underlying ‘laws’.
- For dynamical systems the phases are probability distributions for trajectories compatible with having run from the infinite past.
- Non-uniqueness of phase for an infinite system may be reflected in very long-lived metastable phases for large finite versions.
- A system exhibits strong emergence if it has more than one phase.
- Changes in characteristic patterns can lead to ‘tipping points’ that qualitatively resemble cusp catastrophes.
[For] many applications, especially social ones, I think the framework of multi-component stochastic processes is much more relevant.
[The] results of relying on inadequate understanding can be disastrous, e.g. the 2008 financial crisis, the current national debt crises, the riots in England during August 2011, and the various recent animal and infectious disease epidemics. It is essential to base management of socio-economic systems on sound foundations … .
I interpret ‘management as control of phases’. [It] is reasonable to expect governments to control probability distributions.
Critics of [the] use of Bayes’ theorem ask where the prior probability comes from, but I take the view that we have been observing the system for a while and any effects of the initial choice of prior probability have been washed out, by dynamics or observations.
[The] big data revolution … is likely to create a huge demand for mathematically skilled data scientists.
Points to note
- We are told that climate, ecosystems, cell biology, electricity distribution, cities, the internet and government are all complex ‘systems’, but there is no discussion of what is meant by ‘a system’ other than to say that they have ‘states’ subject to ‘laws’.
- Emergence seems rather limited: there can be no emergence of laws. The whole of politics seems to take place within a very circumscribed arena, with only ‘internal’ innovation within unchanging ‘laws’. This is not how it seems.
- Examples of tipping points are said to include the end of apartheid, the run on Northern Rock, the Arab Spring and the 2011 UK riots.
- There is no discussion of how the approach of the paper could have been applied to any of these.
- There is no description or critique of ‘big data’. There is no hint of a need for mathematics to help critique its use.
The article can be read as promoting the use of ‘sophisticated’ mathematics to support decision-making in the face of complex systems. It refers to the ‘2008 financial crisis’, which some blamed on the use of such mathematics.
In terms of the paper, many social phenomena (such as bank runs, social disorder, epidemics) can be thought of as having multiple (potential) phases with changes in parameters (such as the ease of communication) creating catastrophy-theory like situations and hence tipping points. But this was already part of general understanding in the 1970s (as is clear from the paper’s references).
The financial crisis is more interesting. It was recognized, e.g. by business cycle theory, that different phases had been possible, but supposed that things had changed. Some had it that the causes of the undesirable phases had been removed, others that the system was being maintained in a ‘safe’ state, so that although the phases existed in principle, the were not a danger in practice. Both view proved wrong: we tipped.
Some argue that the ‘sophisticated mathematics’ of time drove us towards the catastrophe quicker, and perhaps even helped create or develop it. How might the new insights be applied differently? In the first place, one might look out for other phases and not dismiss them. But some were already using mathematics to highlight and illustrate different phases. The problem was not the lack of mathematics, but the pragmatic certainty of the clients that only one phase mattered.
There is a more general problem here. So-called ‘rational’ decision-theory is ‘pragmatic’ in so far as it commends using the current model, with its definite probability distribution, unless and until it should prove broken. Thus one only worries about the current phase until one goes past a tipping point, in which case one worries about the new phase. The paper’s approach is much too ‘metaphysical’. The article has an example of how changes in oceanic circulation could become a tipping point. This might be developed further.
Finally, the financial crisis reminds us of Keynes’ contribution in this area. He questions whether the things of interest really are state-full systems in the sense that this paper supposes. Is there really one over-arching set of laws, in place since the big-bang, which govern all these interactions, or might not there be more radical emergence? I have always found this issue to divide people, and the following fudge helpful.
Consider an (idealised) infinite-dimensional system together with a computable (and hence finite-dimensional) Bayesian model. As the current phase endures the model will tend to a model of the current phase, and so be falsified by a catastrophe. More generally, suppose that for a period the phases happen to be variations on each other, and that each phase is identifiable. Then we can have a common model for all these phases, where each phase corresponds to a different set of parameters, as in the paper’s examples. But even this model can be falsified by the infinite-dimensional system. It is as if we only distinguish between a small set of effective states, but from time to time need to be more discerning. Thus even if the things of interest are systems in the sense that is claimed, they are sufficiently complex to warn us against the dangers of treating them as ‘known’ or even ‘knowable’ systems.
‘Big data’ would seem to have striking analogues to finance. There are opportunities for mathematically skilled data scientists, but there seems also to be a need to question the underlying assumptions, which we may ignore at our peril. It would be good to see a worked example of how the mathematics being advocated would apply.