The structure of probabilistic networks

O_LIThere is a growing realization among community ecologists that interactions between species vary across space and time, and that this variation needs be quantified. Our current numerical framework to analyze the structure of species interactions, based on graph-theoretical approaches, usually do not consider the variability of interactions. Since this variability has been show to hold valuable ecological information, there is a need to adapt the current measures of network structure so that they can exploit it.\nC_LIO_LIWe present analytical expressions of key measures of network structured, adapted so that they account for the variability of ecological interactions. We do so by modeling each interaction as a Bernoulli event; using basic calculus allows expressing the expected value, and when mathematically tractable, its variance. When applied to non-probabilistic data, the measures we present give the same results as their non-probabilistic formulations, meaning that they can be generally applied.\nC_LIO_LIWe present three case studies that highlight how these measures can be used, in re-analyzing data that experimentally measured the variability of interactions, to alleviate the computational demands of permutationbased approaches, and to use the frequency at which interactions are observed over several locations to infer the structure of local networks. We provide a free and open-source implementation of these measures.\nC_LIO_LIWe discuss how both sampling and data representation of ecological networks can be adapted to allow the application of a fully probabilistic numerical network approach.\nC_LI