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 basal nodes, which limit its applicability. Here we propose simple improved definitions of trophic levels and coherence 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.
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Data accessibility. The data used to produce figure 1 were downloaded from  and can be accessed from: https:// datadryad.org/stash/dataset/doi:10.5061/dryad.1mv20r6. The data used to produce figures 2 and 3 are from the OECD Input-Output Tables described and available here: http://www.oecd.org/sti/ind/input-outputtables.htm, from the OECD website. The data for figure 4 were downloaded from  here: https://www.ncbi.nlm.nih.gov/ pmc/articles/PMC2736650/. The data for figure 5 published in  were downloaded from http://language.media. mit.edu. The supply network datasets presented in figures 7 and 8 and used for the analysis presented in figure 10 are based on supply-chain relationships compiled from Bloomberg L.P. supply chain function. Bloomberg’s database compiles information from a wide variety of sources to provide a view of global supply chains at the firm level. More information on these data can be obtained from Bloomberg L.P. or . To construct our networks of supplier–buyer relationships, starting from a focal firm of interest we then followed links identified by the Bloomberg database. These datasets could with Bloomberg’s permission be made available on request, or re-compiled from Bloomberg. The data used for figures 12 and 13 can be downloaded from https://www.samuel-johnson.org/data, along with a list with references to the original sources. All code used to make empirical and computational analysis of public data and data-files is available for download at this Github repository, where we also provide a Matlab toolbox for the easy implementation of the methods we have introduced and related analysis. Authors’ contributions. R.S.M. obtained the grant for the project, came up with the improved notions of trophic level and incoherence and proved most of the results about them. S.J. did tests on a variety of networks, the comparisons of the new notion of trophic incoherence with other quantifiers of network structure, the ensemble theory for them, and some of the other proofs. B.S. built the Matlab toolbox, and did many of the empirical tests reported here, including all those on supply networks and input-output networks. Each of us wrote parts of the text and contributed to its finalization. Competing interests. R.S.M. is an Associate Editor of RSOS but played no role in its assessment. Funding. The support of the Economic and Social Research Council (UK), through grant no. ES/R00787X/1, is gratefully acknowledged. This was awarded via a call from the Instability hub of the Rebuilding Macroeconomics programme of the National Institute for Economic and Social Research (NIESR). S.J. and R.S.M. acknowledge support of the Alan Turing Institute under EPSRC grant no. EP/N510129/1 and Fellowship grant no. TU/B/000101. Acknowledgements. We are grateful to other members of our NIESR project team for their comments, especially Nicholas Beale for asking for a quantification of coherence that does not depend on having basal nodes and has a clear maximum and minimum, and to other members of the Instability hub for their comments and interest. We are also grateful to Giannis Moutsinas and Choudhry Shuaib for sharing their approach to the subject, to Mark Pollicott for useful discussion about the zeta function, and to reviewers for insightful comments and pointers to the wider literature.
© 2020 The Authors.
- directed network
- trophic coherence
- trophic level
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