Tremendous progress in information and communication technologies in the last two decades has enabled the phenomenon of Internet-based groups and collectives, generally referred to as online communities. Many online communities have developed distinct cultures of their own, with accompanying norms. A particular research puzzle is the persistence and stability of such norms in online communities, even in the face of often exponential growth rates in uninitiated new users. We propose a network-theoretic approach to explain this persistence. Our approach consists of modelling the online community as a network of interactions, and representing cultural norms as transmissible ideas (or ‘memes’) propagating through this network. We argue that persistence of a norm over time depends, amongst other things, on the structure of the network through which it propagates. Using previous results from Network Science and Epidemiology, we show that certain structures are better than others to ensure persistence: namely, structures which have scale-free degree distributions and assortative mixing. We illustrate this theory using the case of the community of contributors at Wikipedia, a collaboratively generated online encyclopaedia.