Ecological Archives E092-096-A2

José Rojas-Echenique and Stefano Allesina. 2011. Interaction rules affect species coexistence in intransitive networks. Ecology 92:1174–1180.

Appendix B. Description of the algorithm used to generate tournaments with an even distribution of relative intransitivity indices.

Random tournaments, where for each pair of species we draw a connection at random, do not yield uniform distribution for RI. For example, take a completely hierarchical tournament (RI = 0). If we have n species, there are exactly n! ways of drawing a perfectly hierarchical tournament. However, there are 22(n-1)/2 possible tournaments on n species so that the probability of sampling a perfectly hierarchical tournament decreases rapidly with the number of species. Similarly, the probability of finding a perfectly "non-hierarchical" tournament (i.e., with RI close to 1) becomes more and more improbable when n increases.

To avoid this problem, we generated tournaments in the following way: (a) we start from a matrix containing all 1s above the diagonal and all 0s below the diagonal. (b) we "perturb" the matrix reassigning at random a fraction f of the connections: for each pair of species (i, j) we draw a random number; if the number is lower than f, we assign the connection Tij = 1 with probability 1/2 and otherwise we assign Tji = 1.

For the results presented in the text, we generated, for each size n and interaction rule, 50 tournaments with f = {0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.4, 0.5, 0.75, 1.0}. In this way, we obtained a broad spectrum of RI allowing us to replicate the results of the original article.

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