  Oscillatory activity is ubiquitous in natural and engineered network systems.
Yet, understanding the structure-function relationship in oscillator networks
remains an unanswered fundamental question of modern science. In this work, we
present a method to prescribe exact and robust patterns of functional relations
from local network interactions of the oscillators. To quantify the behavioral
synchrony between agents we introduce the notion of functional patterns, which
encode the pairwise relationships between the oscillators' phases -- akin to
the Pearson correlation coefficient between two time series. The main
contribution of this work is the development of a method to enforce a desired
functional pattern by optimally tailoring the oscillators' parameters.
Importantly, our method is agnostic to the scale at which a network system is
studied, computationally efficient, theoretically sound, and presents an
interpretable mapping between the structural principles and the functional
implications of 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 an electrical grid. Our theory finds applications in the design,
analysis, and control of oscillator network systems.
