Why do some crowds move in an orderly fashion while others degenerate into chaotic messes? New research led by an MIT mathematician may finally solve the thorny crowd problem.
Navigating through a busy crowd can often be awkward, but sometimes it feels a lot easier than other times. In a crowded hallway, people seem to spontaneously organize themselves into lines, while in an open city square, people are moving in all directions, running from one side to the other.
Karol Bacsik, a mathematician at MIT, and his colleagues have developed a mathematical theory that accurately predicts pedestrian flow and the point at which it transitions from organized lanes to a confused crowd. The work, which they reported in the journal PNAS on March 24, 2025, could help architects and urban planners design safer and more efficient public spaces that encourage orderly crowd movement.
The team began by creating a mathematical model of moving crowds in different spaces, using fluid dynamics equations to analyze pedestrian movement in different scenarios.
“If you think about the movement of the entire crowd, rather than individual people, you can use flow-like descriptions,” Bacsik said in a statement. “If you’re only interested in global characteristics, like whether there are lanes or not, then you can make predictions without detailed knowledge of everyone in the crowd.”
Crowd, stock photos
Both the width of the space and the angles at which people moved through it had a significant impact on the overall order of the crowd. Bacsik’s team identified “angular distribution” — the number of people moving in different directions — as a key factor in whether people organized into lanes.
Where the spread of people moving in different directions is relatively small—for example, in a narrow corridor or on a sidewalk—pedestrians tend to form lanes and meet oncoming traffic head-on. However, a wider range of individual directions of movement—for example, in an open square or an airport concourse—makes the likelihood of disorder much greater as pedestrians weave and skirt each other to get to their individual destinations.
According to this theoretical analysis, the tipping point was at a scatter angle of about 13 degrees, meaning that orderly lanes could turn into a chaotic flow once pedestrians started moving at more extreme angles.
“It’s all very common sense,” Bacik said. “But now we have a way to quantify when to expect streaks — that spontaneous, organized, safe flow — versus a disordered, less efficient, potentially more dangerous flow.”
But the researchers were interested in finding out whether the reality of human crowds supported this theory, so they designed an experiment that simulated a busy intersection. Volunteers, each wearing a paper hat with a unique barcode, were assigned different starting and ending positions and asked to walk between opposite sides of a gym without colliding with other participants. An overhead camera recorded each scenario, tracking both the movement of individual pedestrians and the overall movement of the crowd.
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A subsequent analysis of 45 trials confirmed the importance of angular dispersion, showing the transition from ordered lanes to disordered movement at angles close to the theoretically predicted 13 degrees. Additionally, as disorder increased, pedestrians were forced to move more slowly to avoid collisions, with a speed reduction of about 30% for random crowds compared to ordered lanes, the team found.
Bacik’s team is now trying to test these predictions in real-world conditions and hopes their work will ultimately help improve conditions in crowded places.
“We would like to analyse the footage and compare it with our theory,” he said. “We can imagine that for anyone designing a public space, if they want to have a safe and efficient pedestrian flow, our work could provide a simpler guide or some practical rules.”
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