Species extinctions are accelerating globally, yet the mechanisms that maintain local biodiversity remain poorly understood. The extinction of species that feed on or are fed on by many others (i.e. ‘hubs’) has traditionally been thought to cause the greatest threat of further biodiversity loss. Very little attention has been paid to the strength of those feeding links (i.e. link weight) and the prevalence of indirect interactions. Here, we used a dynamical model based on empirical energy budget data to assess changes in ecosystem stability after simulating the loss of species according to various extinction scenarios. Link weight and/or indirect effects had stronger effects on food-web stability than the simple removal of ‘hubs’, demonstrating that both quantitative fluxes and species dissipating their effects across many links should be of great concern in biodiversity conservation, and the potential for ‘hubs’ to act as keystone species may have been exaggerated to date.