Fits functions to cumulative degree distributions of both trophic levels of a network.
This function first calculates degrees for each species, then constructs a cumulative distribution with them, and finally fits three different functions to these distributions: exponential, power law and truncated power law. Coefficients and fits are returned.
degreedistr(web, plot.it=TRUE, pure.call=TRUE, silent=TRUE, level="both", ...)
web |
A bipartite network matrix. |
plot.it |
Logical; returns graphs of fits when set to TRUE (default). Dark, median and light grey lines refer to exponential, power and truncated power law, respectively. |
pure.call |
Logical; adjusts par for two panels (for TRUE) or leaves this to the wrapper function (FALSE). |
silent |
Logical; suppresses error reporting in the |
level |
For which level shall the degree distributions be computed: both, lower or higher? Defaults to both. |
... |
Arguments passed on to the plot function (e.g. las=1). |
Jordano et al. (2003) proposed that plant-animal networks may show scale invariance, as indicated by the presence of a power law in species degrees. They report on consistently better fits of the truncated power law, hypothesising that such patterns may arise from morphological mismatch or phenological uncoupling.
Most problematic with the use of this particular approach is the extreme demand for data. The example web Safariland in this package is large (1130 interactions), but it provides only 5 different degree levels (for plants, only 4 for pollinators). Hence fitting three different non-linear functions to these few points is stretching it a bit.
Furthermore, the least-square-fit to the cumulative distribution is not ideal. While the most common approach, it has a bias (albeit much less so than a fit to the probability density function: see Clauset et al. 2009). In an ideal world, we would want to fit the power law properly. This would require a) an estimation of the lower bound of the power law (xmin) and b) the maximum likelihood fit to the remaining data (x > xmin). The data demand is however such that is unlikely that any ecological bipartite network in the near future will match it. Clauset et al. state that hundreds to thousands of data points are required to yield satisfactory estimates for xmin and the slope itself. If you happen to have this much data, please consult the software they provide (even in R!).
For both trophic levels, a table:
... trophic level dd fits |
Contains coefficient estimates, estimate's standard error and P-value, R2 and AIC for each of the three model fits, for the respective trophic level. |
If plots are returned, exponential, power law and truncated power law are given in black, dark grey and light grey, respectively.
The truncated power law fits two coefficients: slope and cut-off. The function only returns the slope. R2-values for non-linear fits are not well liked among statisticians! See the discussion the R-help list (e.g. https://stat.ethz.ch/pipermail/r-help/2002-July/023461.html). Finally, often data are too few to yield any fit. In this case the error message “singular gradient” is returned to signal this problem!
Post finally, yes, I am aware that degrees are integers and unlikely to be normally distributed, and that thus the nls procedure is not really a good idea. My (poor) excuse: I followed the implementation of the above-cited paper and do not believe enough in degree distributions (and power laws, for that matter) to implement a proper likelihood-based approach. Check out the statnet bundle for alternative approaches to this problem.
Carsten F. Dormann carsten.dormann@biom.uni-freiburg.de
Clauset, A., Shalizi, C. R., & Newman, M. E. (2009). Power-law distributions in empirical data. SIAM Review 51, 661–703
Jordano, P., Bascompte, J. and Olesen, J. M. (2003) Invariant properties in coevolutionary networks of plant-animal interactions. Ecology Letters 6, 69–81
networklevel, where degreedistr is called (without picturing the results)
data(Safariland) degreedistr(Safariland)
Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.