  In many real, directed networks, the strongly connected component of nodes
which are mutually reachable is very small. This does not fit with current
theory, based on random graphs, according to which strong connectivity depends
on mean degree and degree-degree correlations. And it has important
implications for other properties of real networks and the dynamical behaviour
of many complex systems. We find that strong connectivity depends crucially on
the extent to which the network has an overall direction or hierarchical
ordering -- a property measured by trophic coherence. Using percolation theory,
we find the critical point separating weakly and strongly connected regimes,
and confirm our results on many real-world networks, including ecological,
neural, trade and social networks. We show that the connectivity structure can
be disrupted with minimal effort by a targeted attack on edges which run
counter to the overall direction. And we illustrate with example dynamics --
the SIS model, majority vote, Kuramoto oscillators and the voter model -- how a
small number of edge deletions can utterly change dynamical processes in a wide
range of systems.
