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Background  Network analysis has become a relevant approach to analyze cascading species extinctions resulting from perturbations on mutualistic interactions as a result of environmental change. In this context, it is essential to be able to point out key species, whose stability would prevent cascading extinctions, and the consequent loss of ecosystem function. In this study, we aim to explain how the  k -core decomposition sheds light on the understanding the robustness of bipartite mutualistic networks.    Methods  We defined three  k -magnitudes based on the  k -core decomposition:  k -radius,  k -degree, and  k -risk. The first one,  k -radius, quantifies the distance from a node to the innermost shell of the partner guild, while  k -degree provides a measure of centrality in the  k -shell based decomposition.  k -risk is a way to measure the vulnerability of a network to the loss of a particular species. Using these magnitudes we analyzed 89 mutualistic networks involving plant pollinators or seed dispersers. Two static extinction procedures were implemented in which  k -degree and  k -risk were compared against other commonly used ranking indexes, as for example MusRank, explained in detail in Material and Methods.    Results  When extinctions take place in both guilds,  k -risk is the best ranking index if the goal is to identify the key species to preserve the giant component. When species are removed only in the primary class and cascading extinctions are measured in the secondary class, the most effective ranking index to identify the key species to preserve the giant component is  k -degree. However, MusRank index was more effective when the goal is to identify the key species to preserve the greatest species richness in the second class.    Discussion  The  k -core decomposition offers a new topological view of the structure of mutualistic networks. The new  k -radius,  k -degree and  k -risk magnitudes take advantage of its properties and provide new insight into the structure of mutualistic networks. The  k -risk and  k -degree ranking indexes are especially effective approaches to identify key species to preserve when conservation practitioners focus on the preservation of ecosystem functionality over species richness.

Ranking of critical species to preserve the functionality of mutualistic networks using the k-core decomposition