  In most systems, its division into interacting constituent elements gives
rise to a natural network structure. Analyzing the dynamics of these elements
and the topology of these natural graphs gave rise to the fields of (nonlinear)
dynamics and network science, respectively. However, just as an object in a
potential well can be described as both a particle (real space representation)
and a wave (reciprocal or Fourier space representation), the `natural' network
structure of these interacting constituent elements is not unique. In
particular, in this work we develop a formalism for Fourier Transforming these
networks to create a new class of interacting constituent elements $-$ the
coupled wave network $-$ and discuss the nontrivial experimental realizations
of these structures. This perspective unifies many previously distinct
structures, most prominently the set of local nonlinear lattice models, and
reveals new forms of order in nonlinear media. Notably, by analyzing the
topological characteristics of nonlinear scattering processes, we can control
the system's dynamics and isolate the different dynamical regimes that arise
from this reciprocal network structure, including the bounding scattering
topologies.
