  River networks serve as a paradigmatic example of all branching networks.
Essential to understanding the overall structure of river networks is a
knowledge of their detailed architecture. Here we show that sub-branches are
distributed exponentially in size and that they are randomly distributed in
space, thereby completely characterizing the most basic level of river network
description. Specifically, an averaged view of network architecture is first
provided by a proposed self-similarity statement about the scaling of drainage
density, a local measure of stream concentration. This scaling of drainage
density is shown to imply Tokunaga's law, a description of the scaling of side
branch abundance along a given stream, as well as a scaling law for stream
lengths. This establishes the scaling of the length scale associated with
drainage density as the basic signature of self-similarity in river networks.
We then consider fluctuations in drainage density and consequently the numbers
of side branches. Data is analyzed for the Mississippi River basin and a model
of random directed networks. Numbers of side streams are found to follow
exponential distributions as are stream lengths and inter-tributary distances
along streams. Finally, we derive the joint variation of side stream abundance
with stream length, affording a full description of fluctuations in network
structure. Fluctuations in side stream numbers are shown to be a direct result
of fluctuations in stream lengths. This is the last paper in a series of three
on the geometry of river networks.
