The distinct roles of initial transmission and retransmission in the persistence of knowledge in complex networks

The rate of successfully acquiring knowledge depends on whether the individual has previously held that knowledge. In an earlier work, we represented this phenomenon by dividing the dynamical process of knowledge transmission into initial and retransmission stages, and applied mean field theory to identify an approximate condition for knowledge survival on homogeneous networks. In this work we move beyond our earlier, approximate results for homogeneous networks to provide rigorous results applicable to complex networks of arbitrary topology - including heterogeneous real world social networks. Specifically, we extend the Intertwined Continuous Markov Chain (ICMC) and Probabilistic Discrete Markov Chain (PDMC) models to address the Naive-Evangelical-Agnostic-Evangelical (VEAE) knowledge transmission process in complex networks. We identify the corresponding basic reproduction number R0, the quantity which dictates whether or not knowledge survives, and deduce simple upper and lower bounds for this measure. Moreover, simulations are performed to verify both the theoretical results, and the mutual consistency of the ICMC, PDMC and Monte Carlo methods. The simulations demonstrate that the initial transmission process directly affects the initial rate of change of the number of evangelical individuals, but has no effect on evangelical density in the steady state. However, the retransmission process has a direct effect on the steady state density of evangelical individuals.