Social learning algorithms provide a model for the formation and propagation of opinions over social networks. However, most studies focus on the case in which agents share their information synchronously over regular intervals. In this work, we analyze belief convergence and steady-state learning performance for both traditional and adaptive formulations of social learning under asynchronous behavior by the agents, where some of the agents may decide to abstain from sharing any information with the network at some time instants. We also show how to recover the underlying graph topology from observations of the asynchronous network behavior.