This dissertation addresses three problems concerning collective behavior in dynamics on networks: coupling in a heterogeneous population of oscillators under global forcing, low-dimensional effective laws of motion for oscillator networks, and degree-targeting the spread of cascades on modular, degree-heterogeneous networks. We refer to these problems as coupled entrainment, coarse graining, and degree-targeting cascades, respectively.

From coupled entrainment, we learn that there is not just one way to measure order in a population of dynamical units, just as there is no unique way to achieve order in a population of dynamical units. Different mechanisms an give rise to different types of order. And, these types can be both competitive and synergistic with one another.

From coarse graining, we find that under certain special assumptions, it is possible to leverage the presence of collective behavior to simplify a dynamical model of a system. From degree-targeting cascades, we learn that it can be possible to average over a highly heterogeneous population and arrive at an accurate representation with very few degrees of freedom. Given that reduced model, we can assess in detail the interplay of network structure and a seeding policy in determining the spread of an activation cascade.

In all cases, analytical insight comes from the existence of a low-dimensional object occupying a privileged position in a high-dimensional state space, and the existence of that low-dimensional object is crucially related to symmetries (broadly construed) of the underlying high-dimensional system.

In coupled entrainment, the key calculation is a linear stability analysis of a particular fixed point of an infinite-dimensional dynamical system. The fixed point in question is already part of a special family of states in which an oscillator with a given natural

frequency is pinned to a given phase. The linearization in question has an eigenvalue −1/π with infinite degeneracy and a single eigenmode with eigenvalue −1/π + K/2 that passes through zero as the parameter K is increased. Both stages of simplification relied on the delicate symmetry built into the natural frequency distribution and the forcing function. Specifically, the choice of natural frequencies, together with the forcing function, brought about a fixed point in which the phases of all oscillators were spread around the unit circle in a perfectly symmetrical way.

In coarse-graining, existence of a low-dimensional attractor in a large (but not infinite) oscillator system is the central theme. Dimension reduction analyses that are valid in the thermodynamic limit guide our search in finite dimensions, and we find reasonable and numerically computable conditions under which such dimension reduction is possible. Here we find that identifying which degrees of freedom admit a reasonable model relies not only on knowing which phases form cohesive groups, but also which oscillators are equivalent to which others in terms of their couplings to other oscillators. The result is joint insight into mesoscale synchronization in the Kuramoto model, on the one hand, and general constraints on discovering low-dimensional dynamical models, on the other.

Finally, in studying cascades, the underlying dynamics are discrete, which differentiates them from the oscillators considered previously. Nonetheless, we find that passing to probabilities of activation leads us to a continuous model that can be drastically compressed due to statistical regularities (or symmetries) of the network that hosts the dynamics. In this case, the symmetries are that nodes in the same module are statistically equivalent in terms of their probability to be connected to an active node.