In the last few years there have been many efforts in econophysics studying how network theory can facilitate understanding of complex financial markets. These efforts consist mainly of the study of correlation-based hierarchical networks. This is somewhat surprising as the underlying assumptions of research looking at financial markets are that they are complex systems and thus behave in a nonlinear manner, which is confirmed by numerous studies, making the use of correlations which are inherently dealing with linear dependencies only baffling. In this paper we introduce a way to incorporate nonlinear dynamics and dependencies into hierarchical networks to study financial markets using mutual information and its dynamical extension: the mutual information rate. We show that this approach leads to different results than the correlation-based approach used in most studies, on the basis of 91 companies listed on the New York Stock Exchange 100 between 2003 and 2013, using minimal spanning trees and planar maximally filtered graphs.