Network inference using directed information: The deterministic limit

Consider a network comprised of many variables, which interact with each other following a causal stochastic dynamical system model. Given time-series measurements of these variables, an important task is the inference of the structure of the causal dynamical system. Recently, directed information has been proposed as a generalization of Granger causality to accurately infer the structure of the network. We observe a curious phenomenon: in the limit that the relationships become purely deterministic, this measure loses power completely. In this paper, we study this phenomenon, explore its connection to Taken's delay embedding theorem, propose a remedy called restricted directed information (RDI); and finally, demonstrate its efficacy in simulations. In particular we show that, RDI recovers the graph correctly in all instances, deterministic or stochastic, where there is enough information to recover the graph uniquely.