  Growing attention has been brought to the fact that many real directed
networks exhibit hierarchy and directionality as measured through techniques
like Trophic Analysis and non-normality. We propose a simple growing network
model where the probability of connecting to a node is defined by a
preferential attachment mechanism based on degree and the difference in fitness
between nodes. In particular, we show how mechanisms such as degree-based
preferential attachment and node fitness interactions can lead to the emergence
of the spectrum of hierarchy and directionality observed in real networks. In
this work, we study various features of this model relating to network
hierarchy, as measured by Trophic Analysis. This includes (I) how preferential
attachment can lead to network hierarchy, (II) how scale-free degree
distributions and network hierarchy can coexist, (III) the correlation between
node fitness and trophic level, (IV) how the fitness parameters can predict
trophic incoherence and how the trophic level difference distribution compares
to the fitness difference distribution, (V) the relationship between trophic
level and degree imbalance and the unique role of nodes at the ends of the
fitness hierarchy and (VI) how fitness interactions and degree-based
preferential attachment can interplay to generate networks of varying coherence
and degree distribution. We also provide an example of the intuition this work
enables in the analysis of a real historical network. This work provides
insight into simple mechanisms which can give rise to hierarchy in directed
networks and quantifies the usefulness and limitations of using Trophic
Analysis as an analysis tool for real networks.
