Do mysterious laws link tennis rankings and earthquakes?

Similar mathematical rules may describe both the growth of cities and the movements of the people in them

Recent studies suggest maths laws may govern everything from the timing of earthquakes to tennis player rankings. But are these rules just a mirage?

In the 1930s, linguist George Kingsley Zipf spotted an unusual trend in languages.

In a given work of literature, the second-most commonly used word occurs half as often as the most-used. The third-most used word occurs one-third as often as the first, and so on.

He had discovered what has become the most widely known "power law".

It was a striking finding - that something as fluid and untamed as language can appear to be so incredibly ordered.

Power laws show up frequently in physics, where mathematical order is no surprise.

They describe how far the Earth moves between solstices, the precise colour of a hot electric stove, and how much a rubbed balloon attracts hair.

But other, seemingly more random systems seem to follow power laws, such as the distribution of times and sizes of earthquakes, or how far people move around in daily life.

Recent years have seen an explosion of publications in the scientific literature purporting to show instances of power laws in all sorts of places.

These include the characteristic patterns of motion of people with clinical depression, the numbers of webpages within websites, even the rankings of players of different sports or the diversity of different countries' exports are just a few examples.

The idea has arisen in most of these cases that some single, unknown mechanism is at work, that nature itself is imposing order in a way we simply do not yet understand. Why would the distributions of wealth among tennis players and fencing champions be identical if nature didn't decree it?

Fashionable topic

In the current issue of Science, Michael Stumpf of Imperial College London and Mason Porter from the University of Oxford have suggested that in a lot of cases, such thinking is misguided.

The power laws behind the timing and size of earthquakes are less contentious

The result, Dr Porter told BBC News, was a proliferation of high-profile publications on power laws that are unconnected - or simply not there - lending more mystery to their perceived ubiquity.

"Science goes through fashions just like anything else does, and certainly finding power laws has been fashionable," he said.

Dr Porter and Prof Stumpf say that many cases of reported power laws may simply be a trick of the numbers.

They may instead result from the same mathematics that brings about the "bell curve".

Give a large class an exam and plot out how many people got which score, and out comes a bell shape on a graph: many people have middling score (the central curve of the bell), with very few perfect scores or perfect failures (making the bell shape taper out on either side of the curve).

Dr Porter said that a number of unrelated effects jumbled into one equation could result in a power law.

Selective evidence

Albert-Laszlo Barabasi of Northwestern University - who has studied power laws in people's movement and the structure of the internet - said that the authors were covering well-trodden ground, but were themselves misguided.

"Every three to five years, someone wakes up in the morning, and says, 'I will write today a piece to point out again that we need to be careful with the fitting process, and that some power laws do not have mechanistic support'," he told BBC News.

"These papers differ only in the level of evidence they are willing to muster to support their argument - and this evidence is always selective."

Prof Barabasi argues that power laws in networks such as the internet have been explained, and that the authors' analysis of whether a power law is really there or not is too restrictive.

Fractals show the "self-similarity" that may turn out to be universal

This is, in many ways, the standard rumble among scientists disagreeing about how to explain the patterns they see.

Geoffrey West of the Southwest Research Institute, whose recent work includes laying out the power laws that appear to underlie growth, wealth and crime in urban environments, agreed that some of the recent spate of science papers on power laws may be on shaky ground.

He told BBC News that the situation reminded him of the early days of what are called fractals.

Another mathematical construction, fractals gave rise to many a psychedelic poster on a student wall because of their "self-similarity".

This means that if you zoom in on the poster's pattern, it is repeated on a smaller level. Zoom in some more, into infinity in fact, and that same pattern is repeated again and again.

"If you go back 15 or 20 years to the beginning of fractals, everybody was finding fractals in anything - and in fact, that was a very important observation," he said.

But he said some of this work was now being questioned.

The Science paper's authors concede that Prof West's work on the scaling of animals' size with their metabolism seems solid.

From shrew to elephant and beyond, and down to the cells that make them up, the amount of energy they consume scales up in the same way - a power law that works across a factor of one thousand million billion in size.

Like fractals, zoom in on the curve that describes it, and it always looks the same.

"My hunch is that for the vast majority of systems... [underlying them] is some version of self-similarity, and from that, some generalised power law," Prof West said.

While the literature abounds with new sightings of power laws in the wild, and scientists argue whether they are real - or whether they mean anything - there may be something behind it all. With time, we will know if it is instead just mathematical coincidence.