The relationship between transfer entropy and directed information

One challenging problem in the study of complex networks is the quantification of relationships between time series recorded across the network. Two information-theoretic measures, i.e., transfer entropy and directed information, have been extensively studied to capture the causality relationship between subsystems of a network. However, the relationship between these two measures have not been fully investigated to date. In this paper, we derive a formula to show the relationship between these two measures, in particular show that, transfer entropy is equal to the upper bound of directed information rate, and verify it through simulations.