PHYS 522: Statistical Physics

Power Law in Chess

Zipf’s Law in the Popularity Distribution of Chess Openings We perform a quantitative analysis of extensive chess databases and show that the frequencies of opening moves are distributed according to a power law with an exponent that increases linearly with the game depth, whereas the pooled distribution of all opening weights follows Zipf’s law with universal exponent.We propose a simple stochastic process that is able to capture the observed playing statistics and show that the Zipf law arises from the self-similar nature of the game tree of chess. Thus, in the case of hierarchical fragmentation the scaling is truly universal and independent of a particular generating mechanism. Our findings are of relevance in general processes with composite decisions.

Confronting the Mystery of Urban Hierarchy

The size distribution of cities in the United States is startlingly well described by a simpler power law: the number of cities whose population exceedsSis proportional to 1/S. This simple regularity is puzzling; even more puzzling is the fact that it has apparently remained true for at least the past century. Standard models of urban systems offer no explanation of the power law. A random growth model proposed by Herbert Simon 40 years ago is the best try to date—but while it can explain a power law, it cannot reproduce one with the right exponent. At this point we are in the frustrating postion of having a striking empirical regularity with no good theory to account for it.J. Japan. Int. Econ., December 1996,10(4), pp. 399–418.

What is the most competitive sport?

We present an extensive statistical analysis of the results of all sports competitions in five major sports leagues in England and the United States. We characterize the parity among teams by the variance in the winning fraction from season-end standings data and quantify the predictability of games by the frequency of upsets from game results data. We introduce a mathematical model in which the underdog team wins with a fixed upset probability. This model quantitatively relates the parity among teams with the predictability of the games, and it can be used to estimate the upset frequency from standings data. We propose the likelihood of upsets as a measure of competitiveness.

The scaling laws of human travel

The website WheresGeorge.com invites its users to enter the serial numbers of their US dollar bills and track them across America and beyond. Why? "For fun and because it had not been done yet", they say. But the dataset accumulated since December 1998 has provided the ideal raw material to test the mathematical laws underlying human travel, and that has important implications for the epidemiology of infectious diseases. Analysis of the trajectories of over half a million dollar bills shows that human dispersal is described by a 'two-parameter continuous-time random walk' model: our travel habits conform to a type of random proliferation known as 'superdiffusion'. And with that much established, it should soon be possible to develop a new class of models to account for the spread of human disease.