  Many real-world networks are directed, sparse and hierarchical, with a
mixture of feed-forward and feedback connections with respect to the hierarchy.
Moreover, a small number of 'master' nodes are often able to drive the whole
system. We study the dynamics of pattern presentation and recovery on sparse,
directed, Hopfield-like neural networks using Trophic Analysis to characterise
their hierarchical structure. This is a recent method which quantifies the
local position of each node in a hierarchy (trophic level) as well as the
global directionality of the network (trophic coherence). We show that even in
a recurrent network, the state of the system can be controlled by a small
subset of neurons which can be identified by their low trophic levels. We also
find that performance at the pattern recovery task can be significantly
improved by tuning the trophic coherence and other topological properties of
the network. This may explain the relatively sparse and coherent structures
observed in the animal brain, and provide insights for improving the
architectures of artificial neural networks. Moreover, we expect that the
principles we demonstrate, through numerical analysis, here will be relevant
for a broad class of system whose underlying network structure is directed and
sparse, such as biological, social or financial networks.
