  This note presents a simple overlapping-generations (OLG) model of the
transmission of a trait, such as a culture. Initially, some fraction of agents
carry the trait. In each time period, young agents are "born" and are
influenced by some older agents. Agents adopt the trait only if at least a
certain number of their influencers have the trait. This influence may occur
due to rational choice (e.g., because the young agents are playing a
coordination game with old agents who are already committed to a strategy), or
for some other reason. Our interest is in how the process of social influence
unfolds over time, and whether a trait will persist or die out. We characterize
the dynamics of the fraction of active agents and relate the analysis to
classic results on branching processes and random graphs.
