  We investigate parametric resonance in oscillator networks subjected to
periodically time-varying oscillations in the edge strengths. Such models are
inspired by the well-known parametric resonance phenomena for single
oscillators, as well as the potential rich phenomenology when such parametric
excitations are present in a variety of applications like deep brain
stimulation, AC power transmission networks, as well as vehicular flocking
formations. We consider cases where a single edge, a subgraph, or the entire
network is subjected to forcing, and in each case, we characterize an
interesting interplay between the parametric resonance modes and the
eigenvalues/vectors of the graph Laplacian. Our analysis is based on a novel
treatment of multiple-scale perturbation analysis that we develop for the
underlying high-dimensional dynamic equations.
