Graph hierarchy and spread of infections

Trophic levels and hence trophic coherence can be defined only on networks with well defined sources, trophic analysis of networks had been restricted to the ecological domain until now. Trophic coherence, a measure of a network's hierarchical organisation, has been shown to be linked to a network's structural and dynamical aspects. In this paper we introduce hierarchical levels, which is a generalisation of trophic levels, that can be defined on any simple graph and we interpret it as a network influence metric. We discuss how our generalisation relates to the previous definition and what new insights our generalisation shines on the topological and dynamical aspects of networks. We also show that the mean of hierarchical differences correlates strongly with the topology of the graph. Finally, we model an epidemiological dynamics and show how the statistical properties of hierarchical differences relate to the incidence rate and how it affects the spreading process in a SIS model.