  Many natural systems are organized as networks, in which the nodes interact
in a time-dependent fashion. The object of our study is to relate connectivity
to the temporal behavior of a network in which the nodes are (real or complex)
logistic maps, coupled according to a connectivity scheme that obeys certain
constrains, but also incorporates random aspects. We investigate in particular
the relationship between the system architecture and possible dynamics. In the
current paper we focus on establishing the framework, terminology and pertinent
questions for low-dimensional networks. A subsequent paper will further address
the relationship between hardwiring and dynamics in high-dimensional networks.
  For networks of both complex and real node-maps, we define extensions of the
Julia and Mandelbrot sets traditionally defined in the context of single map
iterations. For three different model networks, we use a combination of
analytical and numerical tools to illustrate how the system behavior (measured
via topological properties of the Julia sets) changes when perturbing the
underlying adjacency graph. We differentiate between the effects on dynamics of
different perturbations that directly modulate network connectivity:
increasing/decreasing edge weights, and altering edge configuration by adding,
deleting or moving edges. We discuss the implications of considering a rigorous
extension of Fatou-Julia theory known to apply for iterations of single maps,
to iterations of ensembles of maps coupled as nodes in a network.
