Scaling Behaviors of Weighted Food Webs as Energy Transportation Networks

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.