  The goal of this work is to identify steady-state solutions to dynamical
systems defined on large, random families of networks. We do so by passing to a
continuum limit where the adjacency matrix is replaced by a non-local operator
with kernel called a graphon. This graphon equation is often more amenable to
analysis and provides a single equation to study instead of the infinitely many
variations of networks that lead to the limit. Our work establishes a rigorous
connection between steady-states of the continuum and network systems.
Precisely, we show that if the graphon equation has a steady-state solution
whose linearization is invertible, there exists related steady-state solutions
to the finite-dimensional networked dynamical system over all sufficiently
large graphs converging to the graphon. The proof involves setting up a
Newton--Kantorovich type iteration scheme which is shown to be a contraction on
a suitable metric space. Interestingly, we show that the first iterate of our
defined operator in general fails to be a contraction mapping, but the second
iterate is proven to contract on the space. We extend our results to show that
linear stability properties further carry over from the graphon system to the
graph dynamical system. Our results are applied to twisted states in a Kuramoto
model of coupled oscillators, steady-states in a model of neuronal network
activity, and a Lotka--Volterra model of ecological interaction.
