  The structure of hierarchical networks in biological and physical systems has
long been characterized using the Horton-Strahler ordering scheme. The scheme
assigns an integer order to each edge in the network based on the topology of
branching such that the order increases from distal parts of the network (e.g.,
mountain streams or capillaries) to the "root" of the network (e.g., the river
outlet or the aorta). However, Horton-Strahler ordering cannot be applied to
networks with loops because they they create a contradiction in the edge
ordering in terms of which edge precedes another in the hierarchy. Here, we
present a generalization of the Horton-Strahler order to weighted planar
reticular networks, where weights are assumed to correlate with the importance
of network edges, e.g., weights estimated from edge widths may correlate to
flow capacity. Our method assigns hierarchical levels not only to edges of the
network, but also to its loops, and classifies the edges into reticular edges,
which are responsible for loop formation, and tree edges. In addition, we
perform a detailed and rigorous theoretical analysis of the sensitivity of the
hierarchical levels to weight perturbations. We discuss applications of this
generalized Horton-Strahler ordering to the study of leaf venation and other
biological networks.
