Interaction networks are basic descriptions of ecological communities and are at the core of community dynamics models. Knowledge of their structure should enable us to understand dynamical properties of ecological communities. However, the relationships between dynamical properties of communities and qualitative descriptors of network structure remain unclear. To improve our understanding of such relationships, we develop a framework based on the concept of strongly connected components, which are key structural components of networks necessary to explain stability properties such as persistence and robustness. We illustrate this framework for the analysis of qualitative empirical food webs and plant-plant interaction networks. Both types of networks exhibit high persistence (on average, 99% and 80% of species, respectively, are expected to persist) and robustness (only 0.2% and 2% of species are expected to disappear following the extinction of a species). Each of the networks is structured as a large group of interconnected species accompanied by much smaller groups that most often consist of a single species. This low-modularity configuration can be explained by a negative modularity-stability relationship. Our results suggest that ecological communities are not typically structured in multispecies compartments and that compartmentalization decreases robustness.