  In this paper we explore the concept of hierarchy as a quantifiable
descriptor of ordered structures, departing from the definition of three
conditions to be satisfied for a hierarchical structure: {\em order}, {\em
predictability} and {\em pyramidal structure}. According to these principles we
define a hierarchical index taking concepts from graph and information theory.
This estimator allows to quantify the hierarchical character of any system
susceptible to be abstracted in a feedforward causal graph, i.e., a directed
acyclic graph defined in a single connected structure. Our hierarchical index
is a balance between this predictability and pyramidal condition by the
definition of two entropies: one attending the onward flow and other for the
backward reversion. We show how this index allows to identify hierarchical,
anti-hierarchical and non hierarchical structures. Our formalism reveals that
departing from the defined conditions for a hierarchical structure, feedforward
trees and the inverted tree graphs emerge as the only causal structures of
maximal hierarchical and anti-hierarchical systems, respectively. Conversely,
null values of the hierarchical index are attributed to a number of different
configuration networks; from linear chains, due to their lack of pyramid
structure, to full-connected feedforward graphs where the diversity of onward
pathways is canceled by the uncertainty (lack of predictability) when going
backwards. Some illustrative examples are provided for the distinction among
these three types of hierarchical causal graphs.
