Mathematics and Statistics

Speaker: Dr. Trent Marbach, Acadia University

📅 Date: Fri, March 27 ⏰ Time: 1:30pm 📍 Location: HSH 206

Title: From Raindrops to Robbers: The Dynamic Geometry of Network Propagation

Abstract: When a raindrop meets a still puddle, concentric waves spread outward. A spark in dry grass grows into a moving front of flame. Heat migrates invisibly through solid metal. In the physical world, such spreading is shaped by geometry and governed by differential equations.

However, many modern systems do not live in continuous space. Electrical circuits, communication networks, and fire-detection infrastructures are built on networks: collections of nodes joined by edges. In these settings, propagation no longer moves smoothly through space; it travels along discrete pathways.

In discrete systems, geometry does not disappear. It takes a new form. This new form is revealed in how influence travels through a population, how signals propagate through a circuit, or how failures cascade through a sensor network.

In this talk, we study propagation-based processes and discrete pursuit–evasion games on networks, where influence or “fire” spreads from selected nodes along edges. We show how ideas from isoperimetry, the study of how boundary size relates to volume, lead to sharp bounds and structural insights into the speed and shape of this spread. By importing geometric and analytic thinking into discrete settings, we obtain new results and deepen our understanding of key network parameters that measure expansion and constraint.

These insights advance foundational mathematical questions while also illuminating how network structure shapes resilience, detection, and control in real systems. We conclude by reflecting on the mathematical tools we have and the ones still needed to fully understand propagation in complex networks.