  Oscillatory activity is ubiquitous in natural and engineered network systems.
The interaction scheme underlying interdependent oscillatory components governs
the emergence of network-wide patterns of synchrony that regulate and enable
complex functions. Yet, understanding, and ultimately harnessing, the
structure-function relationship in oscillator networks remains an outstanding
challenge of modern science. Here, we address this challenge by presenting a
principled method to prescribe exact and robust functional configurations from
local network interactions through optimal tuning of the oscillators'
parameters. To quantify the behavioral synchrony between coupled oscillators,
we introduce the notion of functional pattern, which encodes the pairwise
relationships between the oscillators' phases. Our procedure is computationally
efficient and provably correct, accounts for constrained interaction types, and
allows to concurrently assign multiple desired functional patterns. Further, we
derive algebraic and graph-theoretic conditions to guarantee the feasibility
and stability of target functional patterns. These conditions provide an
interpretable mapping between the structural constraints and their functional
implications in oscillator networks. As a proof of concept, we apply the
proposed method to replicate empirically recorded functional relationships from
cortical oscillations in a human brain, and to redistribute the active power
flow in different models of electrical grids.
