Estimation of social influence in networks can be substantially biased in observational studies due to homophily and network correlation in exposure to exogenous events. Randomized experiments, in which the researcher intervenes in the social system and uses randomization to determine how to do so, provide a methodology for credibly estimating of causal effects of social behaviors. In addition to addressing questions central to the social sciences, these estimates can form the basis for effective marketing and public policy. In this review, we discuss the design space of experiments to measure social influence through combinations of interventions and randomizations. We define an experiment as combination of (1) a target population of individuals connected by an observed interaction network, (2) a set of treatments whereby the researcher will intervene in the social system, (3) a randomization strategy which maps individuals or edges to treatments, and (4) a measurement of an outcome of interest after treatment has been assigned. We review experiments that demonstrate potential experimental designs and we evaluate their advantages and tradeoffs for answering different types of causal questions about social influence. We show how randomization also provides a basis for statistical inference when analyzing these experiments.