SIAM News Blog

Disorder and Disasters in High-Density Crowds

By Arianna Bottinelli and Jesse L. Silverberg

While mass gatherings are usually safe, sometimes things can go catastrophically wrong.

For the most part, concerts, protests, rallies, inaugurations, pilgrimages, festivals, sporting events, and Black Friday sales come and go with few or no injuries. From time to time, however, an event takes a turn for the worse, and we hear stories of medical emergencies and tragic outcomes. For example, an oversold crowd at a 2012 Halloween dance party in Madrid resultehd in five fatalities, all of whom were women between the ages of 17 and 20. A few years earlier in 2010, the infamous Love Parade disaster in Germany led to 21 deaths and roughly 500 injuries.

In both cases, asphyxiation was broadly found to be a lethal factor behind the tragedies. But how do these types of disasters occur? Given that most people are generally good-natured, why would anyone want to hurt someone else?

The truth is that disasters at mass gatherings usually aren’t driven by malicious intent. Instead, a phenomenon called “crowd turbulence” was the principle culprit from the Halloween and Love Parade tragedies. In the same way that waves travel through water, this large-scale emergent collective motion is essentially a turbulent wave that moves through a crowd when people are packed at least five or six per square meter. Unfortunately, such compression waves can drive sudden and unexpected increases in pressure that rapidly lead to asphyxiation. 

In other mass gatherings, most notably Black Friday sales, people have tripped and been injured by trampling. Upon falling, they often cannot get back on their feet because the crowd is simply too dense. As with crowd turbulence, no one person is out to cause harm; instead, the underlying problem stems from large-scale collective motion.

Time and time again, we find that emerging patterns in crowds’ collective motion are the real source of disaster. 

Dense crowds with shoulder-to-shoulder contact collectively act like a soft material, with potentially disastrous outcomes. Photograph by Ulrike Biets, used with permission.

In our recent work, published in Physical Review Letters, we set out to understand the emergent collective properties of densely-packed crowds. With an eye towards crowd safety, we focused our efforts on comprehending the basic physical mechanisms underlying the most dangerous types of collective motion. As our studies of simulated crowds unfolded, we identified a recurring theme: emergent collective motion propagates through a disordered network of body-to-body contacts. Stepping back from this initial observation, we began to notice close similarities to research in materials science. In fact, we would soon discover that this analogy to other well-studied physical systems would open a whole new avenue to advance our work.

We realized that the movement of dense groups of people is similar to the way grains settle in a silo, pills pour down assembly lines, and coffee beans pack in a bag. In these examples of granular systems, each “grain”—people, wheat, pills, or coffee beans—experiences contact forces with its neighbors as it finds its way into a static equilibrium configuration. As a result, densely-packed grains create an invisible network of forces, with some members experiencing higher forces and others experiencing less.

Remarkably, this contact network can amplify small perturbations into large-scale collective motion. Moreover, the random and disordered nature of any given contact network ultimately selects preferential directions for the collective motion to propagate. Consequently, it is possible to anticipate—with a remarkable degree of accuracy—the collective behavior of a system of densely packed grains. So the following question naturally arose: can the physics of granular media have explanatory power for crowd disasters?

We began by looking for established mathematical tools to analyze force contact networks. Mode analysis is a particularly useful method that applies to both real and simulated granular materials data. This analytical framework has the benefit of quantifying large-scale dynamical properties from the small-scale motion of each individual grain. At the heart of mode analysis is a correlation matrix that converts seemingly-random displacements of grains about their equilibrium position into a prediction for large-scale collective motion. The correlation matrix’s eigenvalues and eigenvectors are the equations for making these predictions. These quantities contain useful information about coherent large-scale motion, response to perturbations, and local structural stability. In particular, the eigenvectors corresponding to the largest eigenvalues—also called soft modes—are vector fields indicating preferential directions of motion for the system with a given contact network. Even better, soft modes are calculated on a case-by-case basis from the specific structure of each randomly self-organized network, allowing for example-specific predictions of collective motion.

The adaption of mode analysis to analyze our high-density crowd simulations allowed us to reconstruct crowds’ soft modes and extract broader trends. Along these lines, we discovered a variety of surprising results. 

For example, the first mode—which is the easiest to excite—is actually a large-scale density wave that bears a striking resemblance to crowd turbulence. Using mode analysis, we were able track the origin of this collective motion to the general principle of continuous symmetry breaking. Essentially, a mathematical theorem connects any broken continuous symmetry to a low-energy, long-range excitation. These excitations are called Goldstone modes; in the case of broken rotational symmetry, the Goldstone mode is a density wave. Technically speaking, this theorem suggests any crowd packed at high density breaks continuous symmetry and thus is susceptible to a coherent wave that spans the entire crowd. The Goldstone mode is qualitatively extremely similar to crowd turbulence, and if the interpretation is correct, an asphyxia-inducing density wave is an inherent possibility within any dense crowd. This result is particularly concerning for anyone standing next to a wall or barrier, since they are most susceptible to harsh compression pressures when the Goldstone mode is excited. 

While thinking about incidents where people trip and are trampled, we asked whether it’s possible to predict who’s most likely to fall and identify high-risk areas in the crowd. In studies of granular matter, grains—people, wheat, pills, beans, etc.—placed in low-force regions of contact networks have a bit more area to move around. When the system jostles, these so-called “soft spots” are where things move the most. In the specific context of people, our analogy means a sudden disruption to the crowd would forcefully displace people in soft spots the furthest. Interestingly, we often found a soft spot right in the middle of our simulated crowds, which we suspect indicates that people standing there are the ones most at risk of tripping and being subsequently trampled.

Arianna Bottinelli and Jesse Silverberg's study of the mathematics and emergent collective properties of densely-packed crowds could impact safety technologies and the way in which researchers study mass gatherings.

Spurred by our findings, we dug deeper into the dense-human-crowd-as-jammed-granular-media analogy and investigated what happens as crowds become agitated. As one would suspect, the risks get worse. With ideas borrowed from the study of stochastic resonance, we were able to understand why. The extra jostling motion of a pushier crowd suppresses the jamming effects of the shoulder-to-shoulder contact network. Consequently, it’s easier for additional long-range waves, similar to the Goldstone mode, to travel through the crowd. Each of these new waves can propagate in its own direction, creating additional pockets within the crowd at risk for asphyxia-inducing pressures. This means that agitation makes it easier for a crowd crush to initiate, and harder to predict which direction collective waves will travel.

Armed with a newfound understanding of the mechanisms giving risk to crowd disasters, we turn our eyes forward.

In the future, our theoretical framework may have a profound impact on safety technologies and the way in which we look at mass gatherings. By adapting our analysis for real-time video data, we aim to predict where and when dangerous collective motions are most likely to initiate, and in which directions they will travel. The same technology would also help us localize soft spots in large crowds, which in turn could identify trampling risk for stampedes. Ultimately, the goal is to suggest real-world and real-time actions that safety experts can undertake to minimize the risk of disaster at mass gatherings.

But we’re not there yet. 

In the meantime, our big takeaway from this study is a better understanding of how to stay safe. To minimize risk, be aware of your surroundings and how tightly-packed the crowd becomes. Even with the best intentions, people packed should-to-shoulder don’t have full control of their movement. Random, unexpected collisions between neighbors yield serious collective effects. It’s important to participate in the events about which you feel most passionate, but if you find yourself in a crowd that’s noticeably dense, you can protect yourself and others by spreading out and moving to an area with more physical space.

The paper appeared in Physical Review Letters, 117, 228301 (2016). It is also available on the arXiv.

Further Reading:
Detailed list of human stampedes:
An intro to stampedes, crushes, and crashes:
Prior work on the Physics of Mosh Pits by co-author Silverberg (YouTube):
Prior work on the Physics of Mosh Pits by co-author Silverberg et. al. (research paper):

Arianna Bottinelli earned her Ph.D. in applied mathematics at Uppsala University in 2016. Her interests include complex systems and collective behavior in living systems, with a focus on human behavior. She was recently awarded a postdoctoral fellowship at the Nordic Institute of Theoretical Physics in Stockholm, where she will bring forward theoretical and experimental research on the physics of high-density crowds.  
Jesse Silverberg earned his Ph.D. at Cornell University in experimental soft matter physics and is currently engaged in postdoctoral studies at the Wyss Institute for Biologically Inspired Engineering at Harvard University. His work on emergent properties of mechanical networks touches on a wide variety of systems ranging from origami metamaterials and active matter to tissue-scale biomechanics. 
blog comments powered by Disqus