We simulate the dynamics of diffusion and establishment of norms, variants adopted by the majority of agents, in a large social influence network with scale-free small-world properties. Diffusion is modeled as the probabilistic uptake of one of several competing variants by agents of unequal social standing. We find that novel variants diffuse following an S-curve and stabilize as norms when three conditions are simultaneously satisfied: the network comprises both extremely highly connected agents (centers) and very isolated members (peripheries), and agents pay proportionally more attention to better connected, more “popular”, neighbors. These findings shed light on little known dynamic properties of centers and peripheries in large influence networks. They show that centers, structural equivalents of highly influential leaders in empirical studies of social networks, are propagators of linguistic influence, while certain peripheral individuals, or loners, can act either as repositories of old forms or initiators of new variants depending on the current state of the rest of the population.