  We study asymptotic dynamics in networks of coupled quadratic nodes. While
single map complex quadratic iterations have been studied over the past
century, considering ensembles of such functions, organized as coupled nodes in
a network, generate new questions with potentially interesting applications to
the life sciences.
  We investigate how traditional Fatou-Julia results may generalize in the case
of networks. We discuss extensions of concepts like escape radius, Julia and
Mandelbrot sets (as parameter loci in $\mathbb{C}^n$, where $n$ is the size of
the network). We study topological properties of these asymptotic sets and of
their two-dimensional slices in $\mathbb{C}$ (defined in previous work). We
find that, while network Mandelbrot sets no longer have a hyperbolic bulb
structure, some of their geometric landmarks are preserved (e.g., the cusp
always survives), and other properties (such as connectedness) depend on the
network structure. We investigate possible extensions of the relationship
between the Mandelbrot set and the Julia set connectedness loci in the case of
network dynamics.
  We discuss possible classifications of asymptotic behavior in networks based
on their underlying graph structure, using the geometry of Julia or Mandelbrot
sets as a classifier. Finally, we propose a method for book-keeping asymptotic
dynamics simultaneously over many networks with a common graph-theoretical
property. \emph{Core} Julia and Mandelbrot sets describe statistically average
asymptotic behavior of orbits over an entire collection of configurations.
