Les complexités de la physique

I will start by proposing a definition of complexity precise enough so that it goes beyond the commonly vague use of the term and general enough to encompass most of the reasonable attempts to put it at work. Concentrating on physical science, it will then be shown by relying on some specific examples that where a description of the physical situation naturally evokes complexity (in the sense defined), its specific analysis in fact avoids it. One could probably go so far as to claim that physical theory is based on the determined effort to eschew dealing with complexity as such. The possibility of such an escape is intimately linked to the constitutive mathematisation of physics. But we will argue that the spectre of complexity, driven out of the objects of physics, comes back to haunt its practice,which consists indeed, of complex modes of operation resulting from processes that are less simple and linear than is often believed.

Les complexités de la physique - Marc Barthelemy

I will try to present the point of view of a "practitioner" of complexity. Being a theoretical physicist interested in cities, I am confronted to the difficulties - not of quantifying the degree of complexity of urban systems - but mainly how to model such a system. In this respect, I advocate for a pragmatic approach, mostly inspired by statistical physics.

The fundamental starting point for such an approach is the availability of data and the existence of a reproducible empirical fact. In order to illustrate this type of approach, we can consider a recent study [1] about Zipf's law. If we rank cities in a given country according to their population, Zipf [2] proposed almost 100 years ago that the population of a city is inversely proportional to its rank. This law has been tested on many periods and for many countries and seems to be (roughly) correct.

The second step - and the most difficult one - is then to find a theoretical explanation for this observation and to write an equation for the temporal evolution of urban populations. The modelling of such a complex system relies on the fundamental idea that there is a hierarchy of mechanisms: some processes are dominant and should be able to explain Zipf's law. Other mechanisms are "details" and merely represent negligible second-order corrections. The important guide is then the comparison of predictions of the model with the observation and which allows for possible corrections to the theory. There are however many traps at this modelling stage. The first one is to propose a mathematical theory with predictions in qualitative agreement only with empirical observations. This problem appears often in economics and is for example the case of Krugman's model [3] for the agglomeration of industrial activity in cities: Krugman proposed a nonlinear equation displaying concentration effects but without any quantitative empirical validation. The second trap is maybe trickier (but also very common): it is in general not (too) difficult to construct a mathematical model whose predictions are in agreement with the observations but that relies on unrealistic assumptions. This is what happened for Zipf's law: the economist Gabaix proposed a stochastic equation that predicts this law [4] (and this approach is considered so far to be the correct explanation) but it relies on the incorrect assumption that cities cannot disappear. In order to avoid such a problem, the model should be based on realistic assumptions and also gives more than one prediction. In general, it is the tension between the smallest number of assumptions (and parameters) and the largest number of predictions empirically verified that leads to a good modelling. This is what we applied in the case of Zipf's law [1]: starting from first principles, we were able to derive an equation for urban population. Interestingly enough this derivation is based on the generalized central limit theorem, a simple mathematical illustration of how details are irrelevant at large scales (we could surmise that the presence of the sum of many terms is an important condition for the possibility of constructing a mathematical description of such complex systems). This equation explains why Zipf's law is in general an approximation and in fact, most of the time not valid. It also explains dynamical effects such as the rise and fall of cities. In addition, this mathematical modelling allows us to identify the main ingredient for understanding the temporal evolution of urban population and which is the existence of very large and rare inter-urban migratory shocks. This shows in particular the importance of political decisions and planning and more generally, how a theoretical approach to complex systems can be helpful for practical considerations.

 [1] Zipf, G. K. Human Behavior and the Principle of Least Effort (Addison-Wesley, 1949).

[2] V. Verbavatz & M. Barthelemy, "The growth equation of cities", Nature 587, pages 397–401 (2020).

[3] P. Krugman. The self-organizing economy, Blackwell, 1996.

[4] X. Gabaix. "Zipf's law for cities: an explanation." The Quarterly journal of economics 114.3 (1999): 739-767.

Article de Marc Barthelemy : here