Ecological networks, or food webs, describe the feeding relationships between interacting species within an ecosystem. Understanding how the complexity of these networks influences their response to changing top-down control is a central challenge in ecology. Here, we provide a model-based investigation of trophic cascades - an oft-studied ecological phenomenon that occurs when changes in the biomass of top predators indirectly effect changes in the biomass of primary producers - in complex food webs that are representative of the structure of real ecosystems. Our results reveal that strong cascades occur primarily in low richness and weakly connected food webs, a result in agreement with some prior predictions. The primary mechanism underlying weak or absent cascades was a strong compensatory response; in most webs predators induced large population level cascades that were masked by changes in the opposite direction by other species in the same trophic guild. Thus, the search for a general theory of trophic cascades in food webs should focus on uncovering features of real ecosystems that promote biomass compensation within functional guilds or trophic levels.