Food web networks: Scaling relation revisited

Food webs seem to possess scale invariant attributes among which efficiency has been recently included. Considering food webs as transportation networks it has been shown that minimum spanning trees, topologies that minimize cost for delivering medium, satisfy a universal scaling relation. It is not clear, however, whether resource distribution follows the criterion of minimum cost, because longer, less efficient routes are used as well. Because of this, instead of focusing on minimum length spanning trees (MLST) we consider directed acyclic graphs (DAGs) as better descriptors of food web hierarchies. Twenty well known empirical food webs have been transformed into DAGs and a scaling relation has been observed between number of nodes and their level of effective connectivity. Although we derived the scaling relation for DAGs using topological arguments, the exponent of the equation C proportional to A(eta) shows same mathematical properties than its functional counterpart computed through flow analysis. This suggests that eta can be used as a proxy for efficiency in food webs. The values of this coefficient for DAGs are lower than the ones obtained for minimum spanning trees, suggesting that food webs lie in the range of medium-to-low efficiency networks. This challenges the idea that these systems would be more efficient than other types of networks.