Predicting the consequences of species loss requires extending our traditional understanding of simpler dynamic systems of few interacting species to the more complex ecological networks found in natural ecosystems. Especially important is the scaling up of our limited understanding of how and under what conditions loss of ‘keystone’ species causes large declines of many other species. Here we explore how these keystone effects vary among simulations progressively scaled up from simple to more complex systems. Simpler simulations of four to seven interacting species suggest that species up to four links away can strongly alter keystone effects and make the consequences of keystone loss potentially indeterminate in more realistically complex communities. Instead of indeterminacy, we find that more complex networks of up to 32 species generally buffer distant influences such that variation in keystone effects is well predicted by surprisingly local ‘top-down’, ‘bottom-up’, and ‘horizontal‘ constraints acting within two links of the keystone subsystem. These results demonstrate that: (1) strong suppression of the competitive dominant by the keystone may only weakly affect subordinate competitors; (2) the community context of the target species determines whether strong keystone effects are realized; (3) simple, measurable, and local attributes of complex communities may explain much of the empirically observed variation in keystone effects; and (4) increasing network complexity per se does not inherently make the prediction of strong keystone effects more complicated.