The connection between non-normality and trophic coherence in directed graphs

TL;DR

This paper investigates the relationship between trophic coherence and non-normality in directed graphs, examining their impact on epidemic persistence and linear dynamics, and proposes extending trophic coherence to matrices beyond graphs.

Contribution

It establishes a connection between trophic coherence and non-normality, linking graph structure to dynamic processes and suggesting broader applications of trophic coherence.

Findings

Higher trophic coherence correlates with increased epidemic persistence.

Non-normality influences the dynamics of linear operators in networks.

Extending trophic coherence to matrices may enhance understanding of complex systems.

Abstract

Trophic coherence and non-normality are both ways of describing the overall directionality of directed graphs, or networks. Trophic coherence can be regarded as a measure of how neatly a graph can be divided into distinct layers, whereas non-normality is a measure of how unlike a matrix is with its transpose. We explore the relationship between trophic coherence and non-normality by first considering the connections that exist in the literature and calculating the trophic coherence and non-normality for some toy networks. We then explore how persistence of an epidemic in an SIS model depends on coherence, and how this relates to the non-normality. A similar effect on dynamics governed by a linear operator suggests that it may be useful to extend the concept of trophic coherence to matrices which do not necessarily represent graphs.