  Food webs can be regarded as energy transporting networks in which the weight
of each edge denotes the energy flux between two species. By investigating 21
empirical weighted food webs as energy flow networks, we found several
ubiquitous scaling behaviors. Two random variables $A_i$ and $C_i$ defined for
each vertex $i$, representing the total flux (also called vertex intensity) and
total indirect effect or energy store of $i$, were found to follow power law
distributions with the exponents $\alpha\approx 1.32$ and $\beta\approx 1.33$,
respectively. Another scaling behavior is the power law relationship, $C_i\sim
A_i^\eta$, where $\eta\approx 1.02$. This is known as the allometric scaling
power law relationship because $A_i$ can be treated as metabolism and $C_i$ as
the body mass of the sub-network rooted from the vertex $i$, according to the
algorithm presented in this paper. Finally, a simple relationship among these
power law exponents, $\eta=(\alpha-1)/(\beta-1)$, was mathematically derived
and tested by the empirical food webs.
