The robustness of ecosystems to species losses is a central question in ecology given the current pace of extinctions and the many species threatened by human impacts. The simplest approach to robustness has relied on secondary extinctions resulting from the loss of all prey of a given predator. These "bottom-up" extinctions represent the most predictable subset of secondary losses and the best case scenario, since the addition of "top-down" effects related to dynamics can only result in additional losses. Prediction of "bottom-up" extinctions in complex networks is possible when a single species loss is considered. However, questions on multiple extinctions and extinction sequences rapidly become intractable. In addition, species' importance in networks has been traditionally measured using local properties such as the number of connections. Thus, "hubs", or species with a large number of links, are considered key determinants of robustness. Here we propose a different view based on consideration of extinction sequences and a new algorithm that ranks species according to their importance for robustness.
Results/Conclusions

One natural way to assign "importance" to species is to rank them in such a way that if removed in this order, they lead to the fastest collapse of the network.  We propose a numerical approach inspired by the famous PageRank(TM), the algorithm at the core of Google(R).  A recursive definition is at the core of the algorithm: a species is important if important species rely on it for their survival. Results show that when the algorithm is adapted to food webs by introducing basic ecological concepts such as the efficiency of transfer between prey and predators, it outperforms all other measures of species importance. Moreover, the algorithm performs as well as adaptive simulated annealing - an intensive search that can evaluate millions of different sequences - even if the "googling" of food webs evaluates a single possibility using simple metrics. This indicates that the algorithm closely approximates the best possible solution.  The proposed approach provides a useful tool for conservation biology from a multi-species perspective. Results for empirical food webs emphasize their sensitivity to the loss of important species: network collapse occurs on average after the removal of only 15% of the species.