Complex Systems
How are large swarms of fireflies able to synchronize their flashes? How does traffic on a busy thoroughfare break down leading to a disruptive traffic jam? Why do some diseases spread quickly and become pandemics while others never spread beyond the community where the outbreak begins? How do videos on the internet suddenly explode in popularity to become viral sensations? All of these questions deal with large groups of elements that are interacting with each other. These are referred to as complex systems due to the fact that their dynamics cannot be explained by looking at individual components, only by looking at the complex interactions between components can one begin to understand the collective behavior that emerges.
Complex systems can be studied from a mathematical perspective using tools from the fields of nonlinear dynamics and network science. There are two fundamental aspects that go into modeling the types of systems:
1. Topology: What is the structure?
The structure of a system can be modeled as a network made up of nodes and links. Typically, each node represents an individual element in the system (i.e. an oscillator like a firefly, a car or intersection, an individual or a community, etc.), and each link represents an interaction between elements. This network structure is quantified using a matrix indicating the existence and strength of connections between nodes. These connections can either be static or dynamic.
2. Dynamics: How does the system evolve over time?
Often the intrinsic properties of nodes in a network are changing (i.e. the phases of oscillators, the number of cars waiting at a traffic light, the number of susceptible or infected people in the population, etc.). These dynamics can be described using systems of (often nonlinear) differential (or difference) equations where the network structure is built into the governing equations.
One of the fundamental questions in the study of complex systems research is: How does the topology affect the dynamics? In graduate school, my primary focus was on studying pattern formation in networks of coupled oscillators with nonlocal coupling (where the interaction strength was related to some metric for distance).
More recently, I have turned my attention to three (somewhat interrelated) phenomena: synchronization, urban traffic flow, epidemics. More specifically I am interested in understanding how to optimize the underlying network structure to enhance and/or diminish a system’s ability to synchronize or to transport people, disease or information. This work is relevant for a variety of applications including preventing blackouts on the power grid, improving traffic signal timing in cities, and minimizing the severity of an epidemic through targeted interventions (vaccination, quarantining, etc.).