I discuss “simple” dynamical systems on networks and examine how network structure affects dynamics of processes running on top of networks. I consider results based on “locally tree-like” and/or mean-field and pair approximations and examine when they seem to work well, what can cause them to fail, and when they seem to produce accurate results even though their hypotheses are violated fantastically. I’ll also present a new model for multi-stage complex contagions — in which fanatics produce greater influence than mere followers — and examine dynamics on networks with heterogeneous correlations.

This talk discusses joint work with James Gleeson, Sergey Melnik, Peter Mucha, and
Jonathan Ward.