The strength of interactions is crucial to the stability of ecological networks. However, the patterns of interaction strengths in mathematical models of ecosystems have not yet been based upon independent observations of balanced material fluxes. Here we analyse two Antarctic ecosystems for which the interaction strengths are obtained: (1) directly, from independently measured material fluxes, (2) for the complete ecosystem and (3) with a close match between species and 'trophic groups'. We analyse the role of recycling, predation and competition and find that ecosystem stability can be estimated by the strengths of the shortest positive and negative predator-prey feedbacks in the network. We show the generality of our explanation with another 21 observed food webs, comparing random-type parameterisations of interaction strengths with empirical ones. Our results show how functional relationships dominate over average-network topology. They make clear that the classic complexity-instability paradox is essentially an artificial interaction-strength result.