  Trophic coherence, a measure of a graph's hierarchical organisation, has been
shown to be linked to a graph's structural and dynamical aspects such as
cyclicity, stability and normality. Trophic levels of vertices can reveal their
functional properties and partition and rank the vertices accordingly. Yet
trophic levels and hence trophic coherence can only be defined on graphs with
basal vertices, vertices with zero in-degree. Consequently, trophic analysis of
graphs had been restricted until now. In this paper we introduce a novel
framework, a generalisation of trophic levels, which we call hierarchical
levels, that can be defined on any simple graph. Within this general framework,
we develop additional metrics named influence centrality, a measure of a
vertices ability to influence dynamics, and democracy coefficient, a measure of
overall feedback in the system, both of which have implications for the
controllability of complex systems. We discuss how our generalisation relates
to previous attempts and what new insights are illuminated on the topological
and dynamical aspects of graphs. Finally, we show how the hierarchical
structure of a network relates to the incidence rate in a SIS epidemic model.
