Biology, methodology or chance? The degree distributions of bipartite ecological networks

Abstract

The distribution of the number of links per species, or degree distribution, is widely used as a summary of the topology of complex networks. Degree distributions have been studied in a range of ecological networks, including both mutualistic bipartite networks of plants and pollinators or seed dispersers and antagonistic bipartite networks of plants and their consumers. The shape of a degree distribution, for example whether it follows an exponential or power-law form, is typically taken to be indicative of the processes structuring the network. The skewed degree distributions of bipartite mutualistic and antagonistic networks are usually assumed to show that ecological or co-evolutionary processes constrain the relative numbers of specialists and generalists in the network. I show that a simple null model based on the principle of maximum entropy cannot be rejected as a model for the degree distributions in most of the 115 bipartite ecological networks tested here. The model requires knowledge of the number of nodes and links in the network, but needs no other ecological information. The model cannot be rejected for 159 (69%) of the 230 degree distributions of the 115 networks tested. It performed equally well on the plant and animal degree distributions, and cannot be rejected for 81 (70%) of the 115 plant distributions and 78 (68%) of the animal distributions. There are consistent differences between the degree distributions of mutualistic and antagonistic networks, suggesting that different processes are constraining these two classes of networks. Fit to the MaxEnt null model is consistently poor among the largest mutualistic networks. Potential ecological and methodological explanations for deviations from the model suggest that spatial and temporal heterogeneity are important drivers of the structure of these large networks.

Figure 1. Schematic showing (a) two bipartite networks coupled in (b) by making the most general animal species (marked with a dotted circle in (a)) be the only node shared across the two subwebs.

Figure 2. Relative width W₉₅ versus goodness of fit f_G of the MaxEnt model for (a) plant distributions and (b) animal distributions of 98 networks with S<135.

Shading of the data points shows the number of species in the networks.

Figure 3. Relative width W₉₅ versus number of species S for the animal distributions of (a) antagonistic networks and (b) mutualistic networks with S<135.

Solid line is linear regression, dotted lines are upper and lower confidence intervals. In (a) R² = 0.25, p<0.001, in (b) R² = 0.10, p = 0.015.

Figure 4. Relative width W₉₅ versus number of species S of (a) plant distributions and (b) animal distributions for 17 networks with S>140.

Shading of data points shows the goodness of fit f_G of the MaxEnt model.