In crowds, mathematics governs walking

Path-seeking behaviour unconsciously honours three famous mathematicians, researchers reveal. Andrew Masterson reports.

Individuals in crowds reflect power laws in their walking patterns.

Salvatore D'alia / EyeEm/Getty Images

Large numbers of pedestrians moving in opposite directions down crowded streets and passageways, or across busy zebra crossings, spontaneously adopt a walking mode named by one famous mathematician in honour of another.

In a paper published in the Journal of the Royal Society Interface, researchers led by Hisashi Murakami from the University of Tokyo in Japan report the results of experiments conducted in “mock corridors”. In the tests, the researchers counted step patterns and direction changes among groups of people moving bidirectionally in enclosed spaces.

The study sought to build on previous work that analysed the spontaneous organisation that emerges whenever crowds of people – or insects, or fish, for that matter – have to reach various destination points.

This work found that such path-seeking behaviours involve individuals deviating from their most direct routes in order to pass more efficiently through oncoming pedestrian traffic – a trade-off between direction and speed.

Individuals decide their routes by anticipating the motions of neighbours. They do not seek to become involved in collective action – but such action emerges as a consequence of individual decisions.

“However,” Murakami and colleagues note, “the strategies that individuals adopt for the behaviour and how the deviation of individual movements impact the emergent organisation are poorly understood.”

To better understand the factors involved, the researchers monitored closely the way individuals in a bidirectional crowd moved, and concluded that they unconsciously adopted an approach known as the “Lévy walk process”.

“The Lévy walk is a specific class of random walk in which the walker takes rare long steps and many smaller steps, leading to a power-law distribution of step lengths,” the researchers explain.

More often known as Lévy flight, the concept was named after the French mathematician Paul Lévy and describes a random walk in which step-lengths have a “heavy-tailed” probability.

Lévy flight was named by another famous mathematician, Benoit Mandelbrot – he of fractal fame. And, in a perhaps appropriately recursive measure, it could also be described as a Markov process, a way of assigning conditional probability distribution named after the Russian mathematician Andrey Markov.

For pedestrians on a busy crosswalk trying to negotiate human obstacles moving in the opposite direction, however, the history of mathematics is probably the last thing on their minds. The Lévy walk, Murakami and colleagues suggest, emerges as a result of other, very ancient imperatives.

It is, they note, “considered optimal when searching unpredictably distributed resources” and may therefore “also facilitate path-seeking by pedestrians in a crowd”.

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