Management targets for biodiversity preservation are shifting from individual species to an ecosystem-wide focus. Indeed, the perturbation analysis of interaction networks, such as food webs, better captures the response of biodiversity to environmental pressures than single-species considerations. Here we propose a framework that examines food web robustness to a given perturbation based on life history traits and the topology of the food web, at different scales: local (species), intermediate (species directly linked together by a trophic interaction), and global (food web). Applying this framework to the Celtic Sea, a historically exploited fishing ground, we showed that the species sensitive to fishing were not the most central (i.e. with many interaction links, estimated based on eigenvector centrality) and that there is no both highly sensitive and exposed species to fishing. We then investigated how the loss of central, sensitive and exposed species to fishing could impact the robustness of the food web. We showed that the food web was the least robust to the simulated loss of species with many predators (i.e. forage species) and most exposed to fishing pressure, indicating that conservation priority could be focused on these species. Estimating species’ sensitivity to fishing was insufficient to predict food web robustness since the simulated removal of the most sensitive species led to a robustness level similar to that of a random removal sequence. Unlike what is often documented, the network appeared relatively robust to the simulated loss of the most central species, due notably to their implication in redundant trophic interactions and the fact that their disappearance increases modularity. This suggests that species-level metrics such as centrality should be completed by analysis at the scale of the whole food web to prioritize species conservation.