Recent experiments show that public goods can be provided at high levels when mutual monitoring and costly punishment are allowed. All these experiments, however, study monitoring and punishment in a setting where all agents can monitor and punish each other (i.e., in a complete network). The architecture of social networks becomes important when individuals can only monitor and punish the other individuals to whom they are connected by the network. We study several non-trivial network architectures that give rise to their own distinctive patterns of behavior. Nevertheless, a number of simple, yet fundamental, properties in graph theory allow us to interpret the variation in the patterns of behavior that arise in the laboratory and to explain the impact of network architecture on the efficiency and dynamics of the experimental outcomes.