Journal article: We study cultural diffusion in a random network where the transition probabilities are determined by a cultural transmission technology with endogenous vertical transmission rates (à la Bisin and Verdier, 2001). We present a two-way epidemic model in which both the infection and the recovery rates are endogenous and depend on the topology of the network. We characterize two balancing conditions satisfied by the network degree distribution and the vertical transmission rate distribution to ensure the sustainability of long run cultural heterogeneity. We discuss the effects of cultural homophily and paternalistic motivations for cultural transmission on the persistence of steady state cultural diversity.