  Evolution of beliefs of a society are a product of interactions between
people (horizontal transmission) in the society over generations (vertical
transmission). Researchers have studied both horizontal and vertical
transmission separately. Extending prior work, we propose a new theoretical
framework which allows application of tools from Markov chain theory to the
analysis of belief evolution via horizontal and vertical transmission. We
analyze three cases: static network, randomly changing network, and
homophily-based dynamic network. Whereas the former two assume network
structure is independent of beliefs, the latter assumes that people tend to
communicate with those who have similar beliefs. We prove under general
conditions that both static and randomly changing networks converge to a single
set of beliefs among all individuals along with the rate of convergence. We
prove that homophily-based network structures do not in general converge to a
single set of beliefs shared by all and prove lower bounds on the number of
different limiting beliefs as a function of initial beliefs. We conclude by
discussing implications for prior theories and directions for future work.
