A general framework for online learning with partial information is one where feedback graphs specify which losses can be observed by the learner. We study a challenging scenario where feedback graphs vary with time stochastically and,more importantly, where graphs and losses are stochastically dependent. This scenario appears in several applications that we describe. We give a new algorithm for this scenario that exploits the stochastic properties of the graphs and that benefits from favorable regret guarantees in this challenging setting. We present a detailed theoretical analysis of this algorithm, and also report the results of a series of experiments on real-world datasets, which show that our algorithm outperforms standard baselines for online learning with feedback graphs.