Math to match pedestrian behavior is all about timing

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The behavior of pedestrians can seem unpredictable — especially in a big crowd. Yet a group of scientists has developed an accurate mathematical representation of real-life pedestrian movement. By factoring in people’s ability to anticipate impending collisions, the researchers created simulations in which pedestrians naturally form lanes to walk in opposing directions and slow as an area gets more congested, just as real pedestrians do. “It’s the most humanlike model that exists right now,” says Stephen Guy, a computer scientist at the University of Minnesota, Twin Cities, who led the work.

The research, reported in the Dec. 5 Physical Review Letters, could help planners control crowds and design more pedestrian-friendly public spaces. It also may inspire other physicists to try quantifying human interactions that at first glance seem too convoluted for mathematical simplification.

Previous simulations of crowd movement treated people like electrons, which avoid collisions with each other because of a repulsive force that depends on the distance between particles. But those simulations never quite reproduced the movement of pedestrians.

So Guy and his colleagues looked for patterns in a dataset of 1,500 pedestrian trajectories from various environments, including sparsely populated public squares and bottlenecked doorways. The researchers quickly realized that people behave differently from particles, says study coauthor Brian Skinner, a physicist now at Argonne National Laboratory in Illinois who usually studies electron transport. The data showed no correlation between pedestrian separation and “interaction energy,” a measure analogous to an electron’s repulsive force (or in human terms, the level of discomfort). That makes sense, Skinner says, since the distance between people walking side by side shouldn’t matter as much as the distance between people headed straight for each other.

Instead the researchers identified a key variable that was not identified in previous studies: the amount of time until an impending collision, which depends on people’s ability to anticipate. Whether pedestrians were strolling through a wide-open campus quad or squeezing through a narrow doorway, their interaction energy would increase fourfold when their expected time until impact with another person was cut in half.

Such a relationship is called a power law because the change in one variable is determined by some exponent of a change in the other variable. (In this case, the exponent is two: Decreasing the collision time by a factor of two increases the interaction energy by a factor of two squared.) Scientists love power-law relationships because they reveal underlying order in complex systems. Power laws help describe distributions of wealth, sizes of cities and magnitudes of earthquakes.

Armed with the newly discovered power law, the researchers created simulations of various crowd scenarios that match actual pedestrian behavior. The sim people move at different speeds, just like real people do, and they are steered by two forces: one based on the calculated time to collision and the other directing them toward a destination.

“It’s brilliant,” says Uri Alon, a biological physicist at the Weizmann Institute of Science in Rehovot, Israel, who was not involved in the study. “It gives a way to understand how people in crowds orient their motion.”

Guy hopes the research will enable architects to create safer buildings and urban planners to design public spaces with more efficient crowd flow. Skinner is looking at the bigger picture. If analyzing pedestrian data can lead to an effective crowd simulation, he says, then perhaps other datasets could enable an understanding of, say, how people drive on the highway or how workers perform on deadline. “We found a method for taking data and using it to quantify social behavior,” he says. “That’s exciting to me.”

MEETING UP  The best-ever simulation of pedestrians moving through a crowd relies on a new formula that encapsulates people’s ability to anticipate collisions. Credit: Karamouzas et al./PRL 2014.