Signal Propagation on a Network

Abstract

This thesis investigates the behaviour of signals in networked systems by applying diffusion and reaction-diffusion equations on a variety of network topologies, which include path graphs, tree graphs, Y-shaped graphs, and square grid graphs. We employ mathematical models, including the diffusion equation, the Fisher equation, and the FitzHugh-Nagumo equations, to describe concentration and excitation across networks. Using methods such as eigenvalue analysis, finite- difference methods, and the Method of Lines (MOL), numerical simulations were performed to solve these equations and analyse the impact of network topology on signal propagation. Key findings include the adaptation of continuous diffusion models to discrete network structures, the successful application of the Crank-Nicholson method for solving diffusion equations on networks, and the analysis of pulse dynamics and stability in reaction-diffusion models. The FitzHugh-Nagumo model was particularly useful for exploring excitable systems and the propagation of pulses across networks, showing how topology influences wave formation and stability.