Food-web structure and network theory: The role of connectance and size

Abstract

Networks from a wide range of physical, biological, and social systems have been recently described as "small-world" and "scale-free." However, studies disagree whether ecological networks called food webs possess the characteristic path lengths, clustering coefficients, and degree distributions required for membership in these classes of networks. Our analysis suggests that the disagreements are based on selective use of relatively few food webs, as well as analytical decisions that obscure important variability in the data. We analyze a broad range of 16 high-quality food webs, with 25-172 nodes, from a variety of aquatic and terrestrial ecosystems. Food webs generally have much higher complexity, measured as connectance (the fraction of all possible links that are realized in a network), and much smaller size than other networks studied, which have important implications for network topology. Our results resolve prior conflicts by demonstrating that although some food webs have small-world and scale-free structure, most do not if they exceed a relatively low level of connectance. Although food-web degree distributions do not display a universal functional form, observed distributions are systematically related to network connectance and size. Also, although food webs often lack small-world structure because of low clustering, we identify a continuum of real-world networks including food webs whose ratios of observed to random clustering coefficients increase as a power-law function of network size over 7 orders of magnitude. Although food webs are generally not small-world, scale-free networks, food-web topology is consistent with patterns found within those classes of networks.

Figure 1

Log–log plot of the clustering coefficient ratios (empirical/random web values) as a function of size of the network. Open circles represent data from 16 trophic food webs from the current analysis. Dark circles represent data from previous studies of 18 scale-free small-world networks summarized in ref. : 2 taxonomic food webs (22); E. coli substrate and reaction graphs (40); C. elegans neural network, movie actors, and power grid (17); 4 science coauthorship data sets (41, 42); 2 math and science coathorship data sets (43); low and high estimates for Internet domains (44, 45); world wide web sites (46); and concurrence and synonomy of words (47, 44).

Figure 2

Linear-log plots of the cumulative distributions of links per species (both predator and prey links) in 16 food webs. Webs are ordered by increasing connectance (see Table 1). Lines and r² values show the fit to the data of the best of three simple models: power–law distribution (upward curved line), exponential decay (straight line), or uniform distribution (downward curved line). No food web is well fit by a Poissonian or Gaussian distribution.

Figure 3

Log–log overlay plot of the cumulative distributions of links per species in 16 food webs. The link data are normalized by the average number of links/species in each web. If the distributions followed a power law, the data would tend to follow a straight line. The overlay also displays a significant amount of scatter in the data.