View point Food web networks: Scaling relation revisited Stefano Allesina a, * , Antonio Bodini b a Dipartimento di Scienze Ambientali, Universita ` di Parma, Department of Fisheries and Wildlife, Michigan State University, 13 Natural Resources Building, East Lansing, MI 8824, USA b Dipartimento di Scienze Ambientali, Universita ` di Parma Received 15 January 2005; received in revised form 27 April 2005; accepted 9 May 2005 Available online 17 October 2005 Abstract 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 / A h shows same mathematical properties than its functional counterpart computed through flow analysis. This suggests that h 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. # 2005 Published by Elsevier B.V. Keywords: Directed acyclic graphs; Ecological networks; Food webs; Scale invariant relations; Spanning trees; Allometry 1. Introduction Scientists always look for invariant properties in the understanding that these attributes help to unveil organizing principles of nature (Rodriguez-Iturbe and Rinaldo, 1996; West et al., 1999). Ecologists are not exceptions in this respect. Food web ecology, in particular, has become a particularly fertile arena for discussion about invariant properties. Several food web features, such as chain length, predator/prey ratio, fraction of species that occupy definite trophic positions, and number of connections per species, have been documented to be scale-invariant (Briand and Cohen, 1984, 1987; Sugihara et al., 1989). In parallel, however, other authors have cast doubts on the validity of these findings, inflating a debate whose http://www.elsevier.com/locate/ecocom Ecological Complexity 2 (2005) 323–338 * Corresponding author. Tel.: +1 734 741 2370; fax: +1 734 741 2235. E-mail address: allesina@msu.edu (S. Allesina). 1476-945X/$ – see front matter # 2005 Published by Elsevier B.V. doi:10.1016/j.ecocom.2005.05.001