  Dynamic processes of interacting units on a network are out of equilibrium in
general. In the case of a directed tree, the dynamic cavity method provides an
efficient tool that characterises the dynamic trajectory of the process for the
linear threshold model. However, because of the computational complexity of the
method, the analysis has been limited to systems where the largest number of
neighbours is small. We devise an efficient implementation of the dynamic
cavity method which substantially reduces the computational complexity of the
method for systems with discrete couplings. Our approach opens up the
possibility to investigate the dynamic properties of networks with fat-tailed
degree distribution. We exploit this new implementation to study properties of
the non-equilibrium steady-state. We extend the dynamical cavity approach to
calculate the pairwise correlations induced by different motifs in the network.
Our results suggest that just two basic motifs of the network are able to
accurately describe the entire statistics of observed correlations. Finally, we
investigate models defined on networks containing bi-directional interactions.
We observe that the stationary state associated with networks with symmetric or
anti-symmetric interactions is biased towards the active or inactive state
respectively, even if independent interaction entries are drawn from a
symmetric distribution. This phenomenon, which can be regarded as a form of
spontaneous symmetry-breaking, is peculiar to systems formulated in terms of
Boolean variables, as opposed to Ising spins.
