Common knowledge of intentions is crucial to basic social tasks ranging from cooperative hunting to oligopoly collusion, riots, revolutions, and the evolution of social norms and human culture. Yet little is known about how common knowledge leaves a trace on the dynamics of a social network. Here we show how an individual's network properties---primarily local clustering and betweenness centrality---provide strong signals of the ability to successfully participate in common knowledge tasks. These signals are distinct from those expected when practices are contagious, or when people use less-sophisticated heuristics that do not yield true coordination. This makes it possible to infer decision rules from observation. We also find that tasks that require common knowledge can yield significant inequalities in success, in contrast to the relative equality that results when practices spread by contagion alone.