Dynamical processes on structured networks

Complexity science has provided valuable insight and understanding in countless systems across many domains. This insight arises from understanding the inter-play of the interactions between interdependent discrete dynamical systems. In this thesis we consider the role that the structure of the interconnections has on the overall resultant dynamics.

First we consider pattern formation models in complex networks. These are non-linear reaction-diffusion systems, where a transition from an unstable steady state to a new steady state occurs, after some form of symmetry is broken. We will explore, using network science, how the structure of the network affects the final shape of the resultant pattern. By further breaking the symmetry of the network we will investigate how the network topology itself creates oscillatory patterns.  

In the second half of the thesis we are concerned with contagion spreading. Since it is, in practice, impossible to exactly solve disease spread models on net-works, we must use approximation methods. We have derived an exact solutions on tree graphs, and use it to correct the mean field approximation. Finally, we return to reaction-diffusion systems, but this time in the context of disease spread. By modifying the diffusion operator, we model social distancing, and study its effects on slowing down an epidemic.