Optimal Form of Branching Supply and Collection Networks

For the problem of efficiently supplying material to a spatial region from a single source, we present a simple scaling argument based on branching network volume minimization that identifies limits to the scaling of sink density. We discuss implications for two fundamental and unresolved problems in organismal biology and geomorphology: how basal metabolism scales with body size for homeotherms and the scaling of drainage basin shape on eroding landscapes.

Figure 1

(a) We consider families of $d$-dimensional spatial regions that scale allometrically with $L_i\propto V^{\gamma i}$, and exist in a $D$-dimensional space where $D\ge d$. For the $d=D=2$ example shown, ${\gamma }_{\max }={\gamma }_1>{\gamma }_2$, and $L_1$ grows faster than $L_2$. We require that each spatial region is star-convex, i.e., from at least one point all other points are directly observable, and the single source must be located at any one of these central points. (b) Distribution (or collection) networks can be thought of as a superposition of virtual vessels. In the example shown, the source (circle) supplies material to the three sinks (squares). (c) Allowing virtual vessels to expand as they move away from the source captures a potential decrease in speed in material flow. For scaling of branching network form to be affected, the radius $r$ of a virtual vessel must scale with vessel length $s$ (measured from the sink) as $s^{-ϵ}$.

Figure 2

The scaling of network volume versus basin area for four continental-scale river networks. The solid line indicates a scaling of $3/2$. Network volume is estimated for an idealized steady-state, uniform rainfall condition normalized such that $V_{\mathrm{net}}=\sum _{ij}{a}_{ij}$, where $a_{ij}$ is the area of the basin draining into the $ij$th cell on a coarse-grained version of a landscape, and the data are binned in log space [5].