  We discuss transport on load bearing branching hierarchical networks which
can model diverse systems which can serve as models of river networks, computer
networks, respiratory networks and granular media. We study avalanche
transmissions and directed percolation on these networks, and on the V lattice,
i.e., the strongest realization of the lattice. We find that typical
realizations of the lattice show multimodal distributions for the avalanche
transmissions, and a second order transition for directed percolation. On the
other hand, the V lattice shows power - law behavior for avalanche
transmissions, and a first order (explosive) transition to percolation. The V
lattice is thus the critical case of hierarchical networks. We note that small
perturbations to the V lattice destroy the power-law behavior of the
distributions, and the first order nature of the percolation. We discuss the
implications of our results.
