Online social media have been one of the greatest drivers of societal change of the past two decades, but are now being recognized as one of the major causes of opinion radicalization and one of the most effective tools for opinion manipulation. Starting from a class of stochastic models of opinion dynamics, and considering different structures of social networks with increasingly realistic features (including a snapshot of the Facebook friendship network), we develop a mathematical model of different forms of opinion manipulation. We then explore how network properties, and in particular degree distribution and community structure, interact with the attack to amplify or reduce its effect on the population, both globally and on specific subsets. We find, in particular, that degree heterogeneity is key to making online social media susceptible to very effective attacks, even with relatively little effort. Communities instead play a more complex role, acting both as barriers to the spread of manipulated opinions through the whole population and as amplifiers of manipulated opinions when the target of the attack is a community of the online social medium. The results of our study can help design effective strategies to prevent the manipulation of opinions through online social media.