  The structure of flow networks determines their function under normal
conditions as well as their response to perturbative damage. Brain vasculature
often experiences transient or permanent occlusions in the finest vessels, but
it is not clear how these micro-clots affect the large scale blood flow or to
what extent they decrease functionality. Motivated by this, we investigate how
flow is rerouted after the occlusion of a single edge in networks with a
hierarchy in edge conductivities. We find that in 2D networks, vessels formed
by highly conductive edges serve as barriers to contain the displacement of
flow due to a localized perturbation. In this way, the vein provides shielding
from damage to surrounding edges. We show that once the conductivity of the
vein surpasses an initial minimal value, further increasing the conductivity
can no longer extend the shielding provided by the vein. Rather, the length
scale of the shielding is set by the network topology. Upon understanding the
effects of a single vein, we investigate the global resilience of networks with
complex hierarchical order. We find that a system of veins arranged in a grid
is able to modestly increase the overall network resilience, outperforming a
parallel vein pattern.
