Pondering the power law

Hamish Johnston

It’s a long-running joke in the physicsworld.com newsroom that physicists see power laws everywhere. Indeed, a quick scan of the arXiv preprint server reveals physics papers that apply power-law analysis to a wide range of topics from cosmology to geography.

But how many of these studies actually produce useful results? Not many, argue two applied mathematicians in the UK.

A power-law description of nature says that a physical quantity or probability distribution is proportional to an exponential power of another quantity. A simple example is the inverse-square law that describes the gravitational attraction between two masses. A more statistical formulation is the Gutenberg–Richter law, which describes the number of earthquakes experienced in a location as a function of earthquake magnitude.

But what does power-law analysis actually tell us about the physical properties of a system? Its proponents argue that if different things – say earthquake frequency and measles outbreaks – share the same power law, then there must be something similar about the fundamental dynamics that drives both systems. This train of thought has already proved very useful in the study of thermodynamic phase transitions, for example, where seemingly unrelated systems change phase in exactly the same manner.

But should physicists expect the same success when power laws are applied to other systems? Writing in today’s issue of Science, Michael Stumpf of Imperial College London and Mason Porter of the University of Oxford argue that, so far, the track record is not very promising.

They argue that many power-law studies have poor statistical underpinnings and don’t shed much light on the underlying mechanisms of the systems of interest. Indeed, they write that “even the most statistically successful calculations of power laws offer little more than anecdotal value”. That is fighting talk, so expect a robust response from the power-law community in the letters pages of Science.

Hmm, I wonder if the time gaps between letters will follow a power law? You can read all about that particular effect in this Physics World article by Albert-László Barabási, who is one of the discipline’s leading exponents.

You can read Stumpf and Porter’s article here, but it may require a subscription.