  The trophic levels of nodes in directed networks can reveal their functional
properties. Moreover, the trophic coherence of a network, defined in terms of
trophic levels, is related to properties such as cycle structure, stability and
percolation. The standard definition of trophic levels, however, borrowed from
ecology, suffers from drawbacks such as requiring source nodes, which limit its
applicability. Here we propose a simple new definition of trophic level that
can be computed on any directed network. We demonstrate how the method can
identify node function in examples including ecosystems, supply chain networks,
gene expression, and global language networks. We also explore how trophic
levels and coherence relate to other topological properties, such as
non-normality and cycle structure, and show that our method reveals the extent
to which the edges in a directed network are aligned in a global direction.
