In social psychology, emotional contagion occurs when one individual bases his/her emotional response on the emotions and reactions of surrounding individuals. This phenomenon frequently arises in high-stress crowds — complex, dynamic systems comprised of many single particles acting both independently and collectively. These individuals self-organize and behave in nonlinear ways to produce patterns of emergent behavior.
Kinetic models, which predict a system’s overall performance based on its specific components, are valuable tools for understanding the mechanisms of reactions and forecasting collective behavior in crowds. During a minisymposium presentation at the 2018 SIAM Conference on the Life Sciences, which took place in Minneapolis, Minn., this August, Daniel Balagué Guardia of Case Western Reserve University presented a new agent-based model for the study of kinetic contagion in fearful crowds.
“In kinetic modeling, you can’t avoid talking about particle systems in agent-based models,” Guardia said. “Agent-based models are particle systems at the microscopic level.” Other levels include the hydrodynamic limit and the kinetic level. Rather than evaluating individuals themselves, the kinetic level examines the densities of individuals in terms of position and velocity. “All three scales give you a really nice pattern and information,” Guardia said.
Guardia focuses his research on high-stress crowds, such as those resulting from potentially-dangerous disturbances or emergency situations. He showed the audience two videos of real crowd movement in such scenarios. One captured the immediate collective motion of police in response to a detonated bomb in Minnesota, while the other featured civilians’ unprovoked evacuation during a bomb scare in Italy. “These situations are stressful and scary, and this is what we try to capture with our model,” Guardia said.
Researchers frequently use the Cucker-Smale model—an agent-based, kinetic, deterministic method that describes self-organization of individuals in a population—to examine collective crowd motion. “It captures the basic dynamics of a group of organisms deciding to move together,” Guardia said. To demonstrate, he presented a simulation of arrows representing 1,000 individuals and their velocities. When constrained at a certain angle, all of the arrows moved to the right. A second simulation depicted them spreading out from the origin and moving around independently of one another. “Depending on your initial conditions, your agents can behave in very different ways,” Guardia said.
Simulations of particle movement with the Cucker-Smale model — an agent-based, kinetic, deterministic model that depicts collective crowd motion.
He then referred to a two-dimensional model of collective motion created by Milind Tambe (University of Southern California) and his colleagues. Their model’s variables account for individual particles, a set of agents within a given distance, level of emotion, receptiveness to influence, and strength of both emotion transmission and interactions between particles within a given distance (which one can tweak as necessary). Upon comparing Tambe’s model with the popular Cucker-Smale, Guardia found that Tambe’s resulting equation is equivalent to the velocity evolution described by the Cucker-Smale model.
After introducing these existing kinetic models, Guardia acknowledged that particles in a crowd move away from the origin in a real direction. In response to this movement, he created an agent-based flocking model with a contagious emotional component that allows individuals to change direction over the course of simulation. To demonstrate his model, Guardia exhibited another simulation with two sets of individuals pointing towards one another. Although the individuals initially overlapped, they then began aligning and moving together. While only one velocity is possible when taking the hydrodynamic limit, the kinetic level can account for multiple velocities; this is a distinct benefit of kinetic models.
Because the solution to the particle system is the solution to the kinetic equation, Guardia analyzed the kinetic equation to satisfy certain hypotheses and obtain a notion of the solution. He defined the notion of the solution based on the hypotheses and the model’s relation to the flow map. “Eventually all of the velocities have the same magnitude and move in the same direction,” he said upon simulating the kinetic model. “The emotional component of individuals will be exactly the same.”
To further authenticate his model, Guardia conducted numerical validation in two dimensions and graphed the convergence of the two alphas. A log plot reaffirmed proof of convergence. “We have a numerical angle that even if you’re not constrained in the initial constraint, you can still converge,” he said. “If arrows are pointing in all directions, then the angles are pretty far apart. But in some situations they will also converge.” Convergence simply occurs at a different speed.
Ultimately, Guardia hopes his model will lend clarity to the complex dynamics of crowd behavior in high-stress situations and better predict collective motion.
||Lina Sorg is the associate editor of SIAM News.