  Many natural systems are organized as networks, in which the nodes (be they
cells, individuals or populations) interact in a time-dependent fashion. The
dynamic behavior of these networks depends on how these nodes are connected,
which can be understood in terms of an adjacency matrix, and connection
strengths. The object of our study is to relate connectivity to temporal
behavior in networks of coupled nonlinear oscillators. We investigate the
relationship between classes of system architectures and classes of their
possible dynamics, when the nodes are coupled according to a connectivity
scheme that obeys certain constrains, but also incorporates random aspects.
  We illustrate how the phase space dynamics and bifurcations of the system
change when perturbing the underlying adjacency graph. We differentiate between
the effects on dynamics of the following operations that directly modulate
network connectivity: (1) increasing/decreasing edge weights, (2)
increasing/decreasing edge density, (3) altering edge configuration by adding,
deleting or moving edges.
  We discuss the significance of our results in the context of real life
networks. Some interpretations lead us to draw conclusions that may apply to
brain networks, synaptic restructuring and neural dynamics.
