  Attractors of dynamical systems may be networks in phase space that can be
heteroclinic (where there are dynamical connections between simple invariant
sets) or excitable (where a perturbation threshold needs to be crossed to a
dynamical connection between "nodes"). Such network attractors can display a
high degree of sensitivity to noise both in terms of the regions of phase space
visited and in terms of the sequence of transitions around the network. The two
types of network are intimately related---one can directly bifurcate to the
other.
  In this paper we attempt to quantify the effect of additive noise on such
network attractors. Noise increases the average rate at which the networks are
explored, and can result in "macroscopic" random motion around the network. We
perform an asymptotic analysis of local behaviour of an escape model near
heteroclinic/excitable nodes in the limit of noise $\eta\rightarrow 0^+$ as a
model for the mean residence time $T$ near equilibria.
  We also explore transition probabilities between nodes of the network in the
presence of anisotropic noise. For low levels of noise, numerical results
suggest that a (heteroclinic or excitable) network can approximately realise
any set of transition probabilities and any sufficiently large mean residence
times at the given nodes. We show that this can be well modelled in our example
network by multiple independent escape processes, where the direction of first
escape determines the transition. This suggests that it is feasible to design
noisy network attractors with arbitrary Markov transition probabilities and
residence times.
