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In this talk I will describe three pieces of work that attempt to build understanding of collective behavior in dynamics on networks.

First I will describe a system of coupled oscillators subject to common forcing. Each feature - coupling and forcing - is known to be able to bring about coherent motion in a population of dissimilar oscillators, though their mechanisms differ. Specifically, coupling brings about order in a bottom-up fashion, though many small-scale pairwise interactions, while forcing imposes order in a centralized, top-down fashion. We examine a special case whose simplicity lets us precisely quantify the trade-off between these two forces.

Next I will talk again about coupled oscillators, this time on a network with a modular structure. It is known that such a system admits a low-dimensional ODE description in the limit that the number of oscillators goes to infinity; we consider the question of whether the same equations can also be used to precisely reduce the dimension of the dynamics when the number of oscillators is small. We find that it is possible in certain cases, with the caveat that the dimension reduction depends jointly on the structure of the coupling network and the natural frequencies.

Finally I will discuss spreading processes on networks. Specifically we consider models where a node can be in one of two states (active or inactive); a node activates when some fraction of its neighbors are active; and a node never de-activates once it is active. Much is known about how to describe the macroscopic effect of activating some small number of nodes at random; we extend these results to include the possibility of preferentially activating high-degree nodes. We find that degree heterogeneity hinders global cascades when seed nodes are selected uniformly at random, while facilitating global cascades when seed nodes are selected according to degree.

There will be Chickpeas