Geometry of River Networks III: Characterization of Component   Connectivity
  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. 
Peter Sheridan Dodds, Daniel H. Rothman