The next occasion you traverse a bustling plaza, crosswalk, or airport terminal, pay attention to the flow of pedestrians. Are individuals progressing in organized lines, one after another, towards their respective destinations? Or is it a chaotic mix of personal routes, as people dodge and weave through the throng?

Karol Bacik, an instructor at MIT, along with his team, investigated the movement of human crowds and created a pioneering method to forecast when pedestrian pathways will shift from organized to chaotic. Their insights could aid in the design of public areas that cultivate safe and efficient pathways.

In a study published this week in the Proceedings of the National Academy of Sciences, the researchers examine a frequent situation where pedestrians navigate a busy crosswalk. The team scrutinized this scenario through mathematical models and simulations, contemplating the various angles at which people may traverse and the dodging actions they may execute as they endeavor to reach their destinations while steering clear of colliding with other walkers along the way.

The researchers also conducted controlled crowd experiments and observed how actual participants navigated through a crowd to reach specific locations. Through their mathematical and experimental research, the team identified a critical metric that determines if pedestrian traffic remains ordered, allowing clear lanes to form, or disordered, devoid of any recognizable paths through the mass of people. This measure, referred to as “angular spread,” illustrates the number of individuals moving in diverse directions.

A crowd exhibiting a relatively minor angular spread indicates that most pedestrians are walking in opposing directions and confront oncoming traffic directly, such as in a crosswalk. In this situation, more organized, lane-like movement is anticipated. Conversely, if a crowd displays a larger angular spread, like in a terminal, it suggests there are many more pathways that individuals can take, increasing the likelihood of disorder.

Indeed, the researchers determined the threshold at which a mobile crowd transitions from order to disorder. They found that this threshold occurs at an angular spread of approximately 13 degrees, indicating that if pedestrians fail to walk directly across, but instead an average individual deviates at an angle greater than 13 degrees, it could propel a crowd into disordered flow.

“This concept is fundamentally straightforward,” remarks Bacik, an applied mathematics instructor at MIT. “The essential question is whether we can address it with precision and mathematically, and identify where the transition occurs. We now possess a method to quantify when to anticipate lanes—this spontaneous, structured, safe flow—versus disordered, inefficient, and potentially more hazardous movement.”

The study’s co-authors include Grzegorz Sobota and Bogdan Bacik from the Academy of Physical Education in Katowice, Poland, and Tim Rogers from the University of Bath in the United Kingdom.

Right, left, center

Bacik, who specializes in fluid dynamics and granular flow, began researching pedestrian flow in 2021, when he and his collaborators examined the effects of social distancing and how people might navigate around each other while maintaining safe distances. This research motivated them to explore the broader dynamics of crowd flow.

In 2023, he and his team investigated “lane formation,” a phenomenon where particles, grains, and, yes, people have been observed to instinctively form lanes, moving in single-file when compelled to cross an area from two opposing directions. In that research, the group identified the mechanism through which such lanes develop, which Bacik summarizes as “an imbalance of turning left versus right.” Essentially, they discovered that once something in a crowd resembles a lane, individuals nearby that emerging lane either join or are compelled to the sides, walking parallel to the original lane, which others can then follow. Thus, a crowd can organically organize into regular, structured lanes.

“Now we are questioning how robust this mechanism is,” Bacik states. “Does it function exclusively in this very idealized scenario, or can lane formation accommodate some imperfections, such as individuals not moving perfectly straight, as might occur in a crowd?”

Lane change

For their latest study, the team aimed to pinpoint a crucial transition in crowd flow: When do pedestrians shift from orderly, lane-like movement to less organized, chaotic flow? The researchers initially tackled this question mathematically, employing an equation typically used to describe fluid dynamics, concerning the average motion of numerous individual particles.

“If you consider the entire crowd flowing, rather than individuals, you can apply fluid-like descriptions,” Bacik clarifies. “It’s the art of averaging, where, even if some individuals may cross more confidently than others, these effects are likely to neutralize in a sufficiently large crowd. If your focus is on the overall characteristics, like whether lanes exist or not, you can make predictions without needing detailed knowledge of every individual in the crowd.”

Bacik and his team utilized fluid flow equations, applying them to the context of pedestrians moving across a crosswalk. They adjusted specific parameters within the equation, such as the width of the fluid channel (in this scenario, the crosswalk), and the angle at which molecules (or people) crossed, along with various routes that individuals can take to “dodge” around one another to prevent collisions.

Based on these computations, the researchers discovered that pedestrians in a crosswalk are more inclined to form lanes when they walk relatively straight across from opposing directions. This order largely persists until individuals begin to veer across at more extreme angles. At this point, the equation predicts that pedestrian movement is likely to become disordered, with little to no lane formation occurring.

The researchers were eager to ascertain whether the theoretical findings align with real-world scenarios. To that end, they conducted experiments in a gymnasium, where they captured the movements of pedestrians through an overhead camera. Each participant donned a paper hat featuring a unique barcode that the camera could track.

During the experiments, the team assigned participants various starting and ending points along opposite sides of a simulated crosswalk, instructing them to walk across simultaneously to their designated location without colliding with anyone. They repeated the experiment countless times, changing participants’ starting and ending positions each round. Ultimately, the researchers compiled visual data on multiple crowd flows, with individuals adopting various crossing angles.

Upon analyzing the data to note when lanes formed spontaneously and when they did not, the team discovered that, akin to the equation’s predictions, the angular spread was significant. Their experiments confirmed that the shift from ordered to disordered flow occurred approximately at the theorized 13 degrees. In other words, if an average individual deviated more than 13 degrees from straight ahead, the pedestrian flow could descend into disorder, with minimal lane formation. Additionally, they found that the greater the chaos in the crowd, the less effectively it moved.

The team intends to examine their predictions in real-world environments and pedestrian pathways.

“We aim to analyze footage and compare it with our theoretical framework,” Bacik remarks. “And we envision that, for anyone crafting a public space, if they seek a safe and efficient pedestrian flow, our research could provide simpler guidelines or practical rules of thumb.”

This research is partly funded by the Engineering and Physical Sciences Research Council of UK Research and Innovation.