By Lina Sorg
Global patterns, such as animal flocks, schools of fish, traffic jams, and human crowds, pervade our daily life. Researchers in the fields of biology and simulation generally believe that local interactions between community members at the individual level generate these patterns. Although numerous models of collective motion do exist, experimental evidence is still lacking. During a minisymposium at the 2017 SIAM Conference on Applications of Dynamical Systems, William Warren of Brown University presented a behavioral dynamics approach that uses local interactions to explain larger global patterns and the self-organization of human crowds. His local-to-global approach regulates crowd behavior and predicts the resulting collective motion.
Warren began his talk by acknowledging Hermann Haken’s research on pattern formation. “An individual is plugged into local interactions, and somehow global patterns emerge out of all these chaotic local interactions,” he said. Taking a bottom-up approach to a 1977 Haken model, Warren devised a pedestrian model on the individual level using agent-based simulation. He collected data from Brown’s Virtual Environment Navigation Lab (VENLab) for this purpose. The lab’s experiments tracked the navigation of a leader and a follower through a virtual landscape in a 12 x 12-meter display area.
Warren relied on his more recent experiments, conducted with Kevin Rio and Christopher Rhea in 2014, in which the virtual leader changes his speed. In this case, the human follower tracked the leader’s change fairly closely. “We tested half a dozen models on this, and the simplest model that accounted for the data is simply that the follower accelerates or decelerates depending on the difference in speed,” Warren said. “This means that the follower matches the leader’s speed. It’s a fairly simple system.”
He conducted the same experiment again, but this time evaluated the follower’s response to a change in heading (turning and changing direction) rather than speed. The follower still tracked the leader’s movements, but the delay was longer due to the directional shift. “You accelerate or decelerate in order to match the leader’s direction of travel,” Warren said. “If we put a bunch of these things together, we can simulate and eventually predict the collective behavior of a group of people in a crowd.”
Warren then turned his attention to the behavioral dynamics within a neighborhood, rather than a one-on-one interaction. “In a crowd you’re not just influenced by one individual, you’re influenced by a group of individuals,” he said. This added complication begs a few crucial questions. How are pedestrians influenced by multiple neighbors? And how do they determine their velocity based on the velocities of their neighbors? Warren proposed two different classes of models, but neither seemed wholly sufficient. The classical attraction/repulsion models are one such type. He showed an animation of the Boids model, an artificial program that simulates the flocking behavior of birds. “It looks like plausible motion, but there’s no real alignment for how this works,” Warren said, thus ruling out attraction/repulsion models. And Tamas Vicsek’s self-propelled particles model includes many models but unfortunately very little data. “Trying to measure the global motion and infer the local interactions is kind of a doomed enterprise,” he said.
Given the aforementioned model limitations, Warren chose another path. “We’re trying to put a human participant into a virtual crowd and then manipulate that virtual crowd and measure the response of the human under the very general instructions, ‘follow the crowd,’” he said. The participant “walks” with the virtual crowd for 12 meters, during which Warren measures the individual’s lateral change in position or speed. “The response of the subject is proportional to the number of neighbors, so it’s statistically significant,” he said.
Having proved that coupling strength decreases with distance (and zeroes out at four meters), Warren combined his crowd-based model with his earlier experiments of a solo leader and follower. The resulting neighborhood model’s weight decays exponentially with distance, meaning that it has a ‘soft’ radius rather than the ‘hard’ radius commonly found in models of this type. “The neighborhood model is a weighted average of neighbors with an exponential decay with distance and unidirectional coupling,” he said.
Warren collected swarm data with real individuals, who were instructed to walk and arbitrarily turn while staying as a group; the motion-captured data looks like a flock of birds in slow motion, adhering to the principles of self-organization. He measured the individual trajectories using a tractable simulation, but focused on obtaining information about the crowd’s global coherence.
To tie everything together, Warren presented a video of a shoal of foraging fish, moving seemingly randomly but in such a way that they remain connected. When a predator appeared offscreen, the fish immediately formed a coherent school. “The individual fish are being recruited into this collective motion based on the actions of their neighbors,” Warren said. “A visual neighborhood creates a positive feedback that recruits more participants into the collective motion. This confirms that collective behavior emerges from experimentally-specified local interactions.”
In summary, Warren created a novel neighborhood structure—comprised of multiple models on an individual and crowd-based level—to analyze patterns of collective motion. His pedestrian model facilitates general understanding of the emergence of adaptive behavior from dynamic interactions between an individual and its surrounding environment. Pedestrian interactions, devoid of internal models or plans, lead to crowd behavior in a way that is consistent with the principles of self-organization. This realization can find applications in computer animation, architectural design, urban planning, and mobile robotics.