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

Catherine Drysdale; Samuel Johnson

REVIEW 4 major objections 5 minor 35 references

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

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper argues that trophic coherence and non-normality are two views of the same directedness in networks, and that a single scalar, the trophic incoherence F, predicts both spectral and dynamical behavior.

desk verdict A clear synthesis of the authors' own prior work on trophic coherence and non-normality, but the GPPM numerics do not separate the new scalar from spectral radius and SCC size, so the causal claim is overstated. read the letter →

arxiv 2412.01847 v1 pith:IAOY2W4I submitted 2024-11-27 physics.soc-ph cond-mat.stat-mech

Catherine Drysdale , Samuel Johnson This is my paper · ORCID

classification physics.soc-phcond-mat.stat-mech MSC 05C2005C8215A1815A60

keywords directedgraphstrophiccoherencenon-normalitypseudospectralevelsSISepidemicmodellinearoperatorsstronglyconnectedcomponents

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that the trophic coherence of a directed graph — how neatly its edges can be arranged into layers all pointing in one direction — and the non-normality of its adjacency matrix — how far the matrix is from commuting with its transpose — are not separate properties but two views of the same underlying directedness. It assembles existing bounds showing that as the trophic incoherence $F$ goes to $0$, the non-normality $d_F$ goes to $1$, and it reproduces this numerically in networks generated with the Generalised Preferential Preying Model. In both an SIS epidemic model and the linear iteration $x(t+1) = cAx(t)$, trophically coherent graphs produce transient activity that dies out, while incoherent graphs sustain activity, and the spectral radius and strongly connected component size move with $F$. The payoff is that a single scalar, the trophic incoherence $F$, may summarize the spectral and dynamical behaviour of a directed matrix, and that the concept can be extended to matrices that do not represent graphs.

What carries the argument

The load-bearing object is the trophic incoherence $F = \sum_{ij} A_{ij}(h_j - h_i - 1)^2 / \sum_{ij} A_{ij}$, defined from trophic levels $h$ that solve $\Lambda h = v$ with $\Lambda = \mathrm{diag}(u) - A - A^{T}$ and $v = k^{\mathrm{in}} - k^{\mathrm{out}}$; $F$ runs from $0$ for perfect layering to $1$ for a balanced graph, and $z = 1 - F$ is the directedness. The non-normality side is the normalized Henrici deviation $d_F = \sqrt{1 - \nu}$, with $\nu = \sum_j |\lambda_j|^2 / \|A\|_F^2$. The two are connected through the coherence ensemble, in which the expected spectral radius is $\rho = e^{\tau}$ with loop exponent $\tau = \ln \alpha + L_B / (2(L - L_B)) - (1 - F)/(2F)$, giving the non-normality bound $d_F \ge \sqrt{1 - e^{2\tau}/\langle k \rangle}$ and the approximate relation $d_F \simeq \sqrt{1 - \exp(1 - 1/F)}$. The Generalised Preferential Preying Model, a network generator whose parameter $T$ tunes the coherence, provides the graphs used to test the link in the spreading simulations.

What would settle it

Build two directed graph ensembles with the same degree sequence and the same spectral radius but different values of $F$ (for instance by reconnecting edges to preserve degrees while changing layer alignment), and run the SIS model on each; if the stationary infected proportion is the same in both ensembles, then $F$ is not the causal variable and the claimed coherence-driven dynamical difference collapses.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that trophically coherent directed graphs are strongly non-normal, and conversely that incoherent graphs are comparatively normal: as $F \to 0$ (equivalently $\tau \to -\infty$) one has $d_F \to 1$, so the two quantities are near-interchangeable descriptions of directionality. The link is made by collecting the bound $d_F \ge \sqrt{1 - e^{2\tau}/\langle k \rangle}$ and the approximation $d_F \simeq \sqrt{1 - \exp(1 - 1/F)}$, and by showing in the Generalised Preferential Preying Model that $F$, $d_F$, the spectral radius $\rho$, and the size of the strongly connected component all shift together as the generation parameter $T$ is tuned. Two spreading processes demonstrate the dynamical consequence: coherent networks show a transient bump of activity followed by extinction, while incoherent networks keep activity alive through feedback. The paper concludes that trophic coherence is not confined to graphs, and that the same layer-based measure can be used for non-negative matrices and linear operators more broadly.

Load-bearing premise

The numerical argument assumes that tuning the parameter $T$ in the Generalised Preferential Preying Model changes trophic coherence without independently changing other structural features, such as degree sequence, spectral radius, or strongly connected component size, that could themselves be driving the observed dynamics.

Editorial extensions

If this is right

If the connection holds, computing $F$ on a directed network gives a cheap scalar estimate of its non-normality and of the transient amplification that non-normal operators are known to exhibit.
Epidemic persistence in the SIS model should be predictable from $F$: coherent networks ($F$ near $0$) extinguish outbreaks, while incoherent networks sustain them, because $F$ controls the spectral radius and the size of the strongly connected component.
The correspondence extends beyond graphs, so the layered-coherence formalism can be applied to any non-negative square matrix or linear operator, giving a notion of 'directedness' to matrices that have no network interpretation.
The loopful/loopless dichotomy ($\tau > 0$ vs $\tau < 0$) becomes a dynamical dichotomy between networks that sustain feedback and networks that only allow transient activity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Our inference: if $F$ truly determines $d_F$ and the dynamics, then $F$ could serve as a practical stand-in for the pseudospectral abscissa in large directed networks, sidestepping the computational cost of full pseudospectra.
Our inference: the same logic predicts that transient growth in $x(t+1) = cAx(t)$ should peak at intermediate coherence rather than simply at maximal non-normality, a pattern one could test directly on random matrix ensembles with controlled $F$.
Our inference: the toy-network table suggests that different non-normality measures disagree about which edge perturbation is 'more non-normal'; systematically comparing $F$ with both measures under edge addition and deletion would clarify which one tracks the layer structure.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is a perspective piece that connects trophic coherence and non-normality, mostly by synthesizing the authors' own earlier results. The genuinely new bits are the small toy graphs in Table 1 and the joint GPPM numerics in Fig. 1. The toy examples are actually useful: they show that two common non-normality measures—the commutator norm and Henrici's deviation—can disagree, which is a caution worth having. The writing is clear and the literature tour is helpful for someone entering the area.

The soft spot is the numerical core. Fig. 1 varies the GPPM parameter T and shows F, dF, rho, and SCC size all moving together. The text then says that nodes begin to sustain the epidemic once F > 0 or dF < 1. But the SIS persistence is, as the paper itself notes, governed by directed cycles and the spectral radius, and the linear growth is governed by c*rho with non-normality contributing only transients. Because all four quantities co-vary, Fig. 1 does not show that F or dF is the causal variable—it could just be a collinear proxy for rho and Phi. That is a real gap, and it is the load-bearing claim of the paper. The "stationary" infected fraction is also a finite-time average from t=90 to 100, which may include long-lived transients, and the linear operator is evaluated at a single time t=100. No code or data are provided, so the numerics are not reproducible as presented. The imported equations from the coherence ensemble are also assumed to hold for GPPM samples without a derivation here.

That said, the central qualitative message—that coherence and non-normality co-move and both correlate with dynamical persistence—is probably correct and consistent with the prior literature. The issue is the strength of the claim, not the direction. The paper would be strengthened by softer language ("correlates with" rather than "sustains"), by providing artifacts, and by at least discussing the collinearity problem and the ensemble-transfer assumption.

I would take this to a reading group if the topic is directed networks, but I would not cite it as a source for the causal relationship. For peer review: it deserves a serious referee—it is a well-written synthesis with a plausible unifying idea—but it needs major revision, mostly to fix the causal overreach and add reproducibility. Not a desk reject.

Referee Report

4 major / 5 minor

Summary. The paper argues that trophic coherence and non-normality are two complementary measures of the same underlying property of directedness in directed graphs. It reviews definitions of trophic levels, trophic incoherence F, and non-normality dF, presents toy examples that illustrate how these quantities respond to edge deletion and addition, and then reports numerical experiments using the Generalised Preferential Preying Model (GPPM). In the first experiment, an SIS model is simulated on generated networks with tunable coherence parameter T, and the stationary infected proportion is plotted against T. In the second, the same networks are used to iterate a linear operator x(t+1)=cAx(t), and the activity at t=100 is measured. The paper reports that F and dF, as well as the spectral radius rho and the strongly connected component size Phi, all vary with T, and connects the dynamical transitions to the coherence and non-normality measures. A discussion of possible generalizations of trophic coherence to arbitrary matrices closes the paper.

Significance. If the central claim is upheld, the paper would strengthen the case for using trophic incoherence F as a single scalar summary of directionality that predicts spectral and dynamical properties of directed networks, complementing non-normality. The paper is clearly written, the definitions are standard, and the toy examples are instructive. It also usefully points to a potential generalization of trophic coherence to linear operators and matrices beyond graphs. However, the supporting numerical evidence is currently under-controlled: the GPPM parameter T simultaneously changes F, dF, rho, and Phi, so the dynamical transitions shown in Fig. 1 cannot be attributed to F or dF alone. The imported ensemble formulas (Eq. (5) and the dF bounds from [19,21,27]) are not shown to apply to GPPM samples, leaving a gap between theory and simulation. The paper's significance therefore rests on a plausible but not yet demonstrated connection; the experiments as presented are illustrative rather than decisive.

major comments (4)

[Spreading processes with graphs and operators, Fig. 1] This is a complete sentence.
[Measuring trophic coherence and non-normality, Eq. (5) and following bounds] This is a complete sentence.
[Spreading processes with graphs and operators, Fig. 1 caption] This is a complete sentence.
[Spreading processes with graphs and operators, middle-panel discussion] This is a complete sentence.

minor comments (5)

[Introduction] This is a complete sentence.
[Spreading processes with graphs and operators] This is a complete sentence.
[Spreading processes with graphs and operators] This is a complete sentence.
[Figure 1 caption] This is a complete sentence.
[General] This is a complete sentence.

Circularity Check

1 steps flagged · score 3.0 of 10

The titular F–dF connection is asserted via 'Hence' from formulas imported from the authors' own refs [21,27], un-derived and untested here; the SIS and linear-operator simulations are honest independent experiments, so the paper is partially self-citation-dependent but not definitionally circular.

self citation load bearing [Section 'Measuring trophic coherence and non-normality', immediately after Eq. (5)]
"This approach has been used to show that the expected non-normality is bounded, dF ≥ p 1 − e^{2τ}/⟨k⟩ [21], and approximated by dF ≃ p 1 − exp(1 − 1/F ) [27]. Hence, trophically coherent graphs (F → 0 or τ → −∞) are non-normal (dF → 1)."

The paper's titular claim — that trophic coherence and non-normality describe the same underlying directedness — is discharged by the word 'Hence' from formulas imported wholesale from the authors' own prior papers [21] and [27] (both co-authored by S. Johnson), with the ensemble machinery of [19]. No derivation of dF(F), or of the bound, appears in this paper, and Fig. 1 only plots F and dF separately against the GPPM parameter T, never testing the imported relation. Within this paper's derivation chain, the central connection is therefore inherited from self-citations, not established. It is load-bearing: the SIS and linear-operator interpretations lean on it.

full rationale

The paper is best read as a perspective that imports its entire theoretical core — Eq. (5) for the expected spectral radius, the bound dF ≥ sqrt(1 − e^{2τ}/⟨k⟩), the approximation dF ≃ sqrt(1 − exp(1 − 1/F)), and the GPPM generator — from the authors' own earlier work (refs [19], [21], [24], [27], all authored or co-authored by S. Johnson). The headline assertion that trophically coherent graphs are non-normal is stated as a 'Hence' consequence of those imported formulas rather than derived or independently verified here; Fig. 1 plots F and dF separately against the same GPPM parameter T and never checks the imported dF(F) relationship. This is load-bearing self-citation, which raises the score above 2. It is not a wholly circular derivation, for three reasons: [19], [21], and [27] are published results with their own derivations; the SIS and linear-operator outcomes in Fig. 1 are genuine numerical experiments computed by direct simulation rather than by evaluating Eq. (5) or the dF bounds; and no equation in this paper reduces to another by construction, nor is any fitted parameter renamed as a prediction. The skeptic's objection that GPPM moves F, dF, ρ, and Φ collinearly (bottom panels) is a confound about causal attribution — the paper never claims to isolate F — and belongs to correctness risk, not circularity. Accordingly, one self-citation-load-bearing step is flagged and the paper scores 3.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

No new free parameters are fitted to data; the analytical scaffolding comes from prior work by the same group, and the simulations depend on the GPPM. The main unstated assumptions are the transferability of coherence-ensemble formulas to the specific generated networks and the sufficiency of F for the observed dynamics.

free parameters (2)

T (GPPM coherence parameter) = 0 to 2 (varied)
Control parameter in the Generalised Preferential Preying Model used to sweep trophic coherence; not fitted to data.
c (linear operator gain) = 0.5, 1.0, 2.0
Chosen constants for the linear iteration Eq. (8); not fitted.

assumptions (5)

standard math Equation (1), Lambda h = v, always has a solution, unique up to additive constant.
Used to define trophic levels in Def. 0.3; presented as fact.
standard math F is bounded between 0 and 1, with F=0 for perfectly layered graphs and F=1 for balanced graphs.
Stated after Def. 0.4 and used to interpret all subsequent results.
domain assumption The coherence-ensemble formula for the expected spectral radius (Eq. 5) and the non-normality bounds hold for the GPPM-generated graphs.
Imported from refs [19,21,27] without derivation; the paper uses them to explain the numerical curves.
domain assumption The Generalised Preferential Preying Model generates networks with tunable trophic coherence approximating the coherence ensemble.
All simulations rely on this model from ref [24]; the model is not described in the manuscript.
domain assumption Epidemic survival is governed by the spectral radius and by the presence of a strongly-connected component.
Used in the spreading processes section; the critical rate lower bound is cited to [33] and the SCC relation to [22].

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Cite this review

Pith. "Pith review of The connection between non-normality and trophic coherence in directed graphs." pith.science (2026). https://pith.science/paper/IAOY2W4I

@misc{pith2026241201847,
  author       = {Pith},
  title        = {Pith review of: The connection between non-normality and trophic coherence in directed graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IAOY2W4I}},
  note         = {Machine review of arXiv:2412.01847}
}

read the original 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.

Figures

Figures reproduced from arXiv: 2412.01847 by the authors.

**Figure 1.** Top left: Stationary proportion of infected nodes against parameter T for networks generated with the Generalised Preferential Preying Model [24]. Number of nodes and edges: N = 100 and L = 500. Averages over 100 networks; error bars are standard deviations. Stationary values computed as the average from t = 90 to t = 100. Inset: Time series of the proportion of infected nodes for three networks, generated with T = … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Global measures of local convective insta- bilities,

C. Cossu and J.-M. Chomaz, “Global measures of local convective insta- bilities,” Physical review letters , vol. 78, no. 23, p. 4387, 1997

work page 1997

[2] [2]

Global instabilities in spatially developing flows: non- normality and nonlinearity,

J.-M. Chomaz, “Global instabilities in spatially developing flows: non- normality and nonlinearity,” Annu. Rev. Fluid Mech. , vol. 37, no. 1, pp. 357–392, 2005

work page 2005

[3] [3]

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R. Sujith, M. Juniper, and P. Schmid, “Non-normality and nonlinearity in thermoacoustic instabilities,” International Journal of Spray and Combus- tion Dynamics , vol. 8, no. 2, pp. 119–146, 2016

work page 2016

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P. J. Schmid, D. S. Henningson, and D. Jankowski, “Stability and transition in shear flows. applied mathematical sciences, vol. 142,” Appl. Mech. Rev., vol. 55, no. 3, pp. B57–B59, 2002

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C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quan- tum mechanics,” Phys. Rev. Lett. , vol. 89, no. 27, p. 270401, 2002

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C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys., vol. 70, no. 6, pp. 947–1018, 2007

work page 2007

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Real spectra in non-Hermitian Hamiltoni- ans having PT symmetry,

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