Abstract
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.
Highlights
Many complex systems have an underlying network, whose nodes represent units of the system and whose edges indicate connections between the units [1]
In many domains of science, one is faced with a directed network and wishes to determine (i) to what extent the edges line up in an overall direction, and (ii) the relative position of individual nodes within any directional flow on the network
The notion of ‘trophic coherence’ introduced in [7], based on the old notion of trophic levels, provides a way to quantify the extent to which edges align in an overall direction and was shown to be connected with network properties such as cycles and spectral radius, but it lacks a natural notion of maximal incoherence
Summary
Many complex systems have an underlying network, whose nodes represent units of the system and whose edges indicate connections between the units [1]. The concept was reinvented in economics [4], where it is called ‘upstreamness’, though [5] trace it back to Leontief and the ‘output multiplier’ It is an ingredient in the construction of SinkRank, a measure of contribution to systemic risk [6]. In some contexts, like production networks indicating the flows of goods and services between firms or sectors, the reverse flow plays an equivalent role, representing the financial payments, but the standard concept of trophic level does not treat these symmetrically. We show that the new levels continue to be a useful indicator of function in the network and that the new incoherence measure continues to be related to stability, cycles and normality.
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