Abstract This paper is about the diffusion of cooperation in an infinite population of networked individuals repeatedly playing a Prisoner's Dilemma. We formulate conditions on payoffs and network structure such that, starting from an initial seed group, imitative learning results in the overall adoption of cooperation—efficient contagion. Key to this result is the pattern of interaction among players who are at the same distance from the initial seed group. We find that the more these agents interact among themselves rather than with players who are closer to or further away from the initial seed group, the easier it is for efficient contagion to take place. We highlight the importance of cycles for efficient contagion, and show that the presence of critical edges prevents it. We also find that networks organized as dense clusters sparsely connected to one another tend to resist efficient contagion. Finally, we find that the likelihood of efficient contagion in a network increases when information neighborhoods extend beyond interaction neighborhoods.