Function to generate null model webs under the following constraints: 1. marginal
totals are identical to those observed (as in r2dtable), 2. connectance is as observed (as in shuffle.web.)
swap.web(N, web, verbose=FALSE, c.crit=1e4)A list of N randomised matrices with the same dimensions as the initial web.
Number of desired null model matrices.
An interaction matrix.
Should various verbal outputs of this function be shown? Defaults to FALSE, since it was mainly used during the debugging period.
Sometimes the algorithm gets stuck in a very sparse matrix. Then c.crit sets the number of swaps it shall attempt before giving up and starting over on a new random matrix. Defaults to 10000.
Carsten F. Dormann <carsten.dormann@biom.uni-freiburg.de>
This function is designed to behave similar to r2dtable, i.e. it returns a list of randomised matrices. In addition to r2dtable is also keeps the connectance constant!
This function is thought of as a very constrained null model for the analysis of bipartite webs. It keeps two web properties constant: The marginal totals (as in r2dtable and the number of links (and hence connectance). A comparison of swap.web- and r2dtable-based webs allows to elucidate the effect of evolutionary specialisation, since the unrealised connections may represent “forbidden links”.
This null model is similar to the one employed by Vázquez et al. But while Vázquez starts by assigning 1s to the allowed connections and then fills the web, swap.web starts with an r2dtable-web and successively “empties” it. The two approaches should result in very similar null models, since both constrain marginal totals and connectance.
A few words on the way swap.web works. It starts with a random web created by r2dtable. Then, it finds, randomly, 2x2 submatrices with entries all larger than 0. Next, it subtracts the minimum value from the diagonal and adds it to the off-diagonal (minor diagonal). Thereby one cell becomes 0, but the column and row sums do not change. This idea is adapted from the swap-algorithm used in various binary null models by Nick Gotelli. If the random web has too few 0s (which is I have yet to encounter), then the opposite strategy is applied.
The algorithm in our implementation has some variations on finding the submatrix and constraining the number of unsuccessfull trials before starting on a new random matrix, but they are only for speeding up the process.
Artzy-Randrup, Y., and Stone, L. (2005) Generating uniformly distributed random networks. Physical Review E 72, 1--7
Miklós, I. and Podani, J. (2004) Randomization of presence-absence matrices: comments and new algorithms. Ecology 85, 86--92
Vázquez, D. P., and M. A. Aizen (2003) Null model analyses of specialization in plant-pollinator interactions. Ecology 84, 2493--2501
Vázquez, D. P., C. J. Melian, N. M. Williams, N. Blüthgen, B. R. Krasnov, and R. Poulin (2007) Species abundance and asymmetric interaction strength in ecological networks. Oikos 116, 1120--1127
For a very nice and thorough overview of null models in general see:
Gotelli, N. J., and G. R. Graves (1996) Null Models in Ecology. Smithsonian Institution Press, Washington D.C.
r2dtable, vaznull and shuffle.web