{"id":"99cf2f0b-819a-4647-851f-fd7c97aeebc0","arxiv_id":"2412.01847","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Trophic coherence and non-normality are shown to be linked measures of directedness, and both affect epidemic persistence and linear transient growth in simulated networks.","lead":"This paper argues that two ways of measuring direction in networks, trophic coherence and non-normality, are closely related and both predict whether a spreading process survives or fades. It suggests the same idea may apply to any matrix, not just graphs.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The GPPM experiment in Fig. 1 cannot separate trophic coherence from spectral radius, SCC size, and degree sequence, so the central dynamical claim rests on uncontrolled collinear variables.","rationale":"The central assertion—that F and dF capture the same directedness and hence the same dynamical phenomenology—requires that the observed dynamics be attributable to these measures rather than to standard determinants. The paper's own text concedes that SIS persistence requires a strongly connected component and cites the spectral-radius threshold, and the linear iteration is governed by cρ. Fig. 1 shows ρ and Φ undergoing the same transition as the dynamics, and GPPM is known to alter more than F as T changes. Consequently, the most parsimonious reading is that F (and dF, via its correlation with F) is a proxy for cycle structure rather than an independent causal summary. This is an identification problem, not a claim of misconduct. The paper provides no code or data, and it imports Eq. (5) and the non-normality bounds from prior work without verifying them on GPPM, so the ensemble-transfer assumption is also unchecked. The analytic direction F→0 ⇒ dF→1 is sound (perfect layering makes A nilpotent), so the concern is not about the existence of the connection but about whether the dynamical claim is independent of ρ and Φ. The proposed test—degree-preserving coherence-ensemble sampling with ρ/Φ stratification—would settle this cleanly. If F/dF has no residual predictive power, the perspective's central contribution reduces to a restatement of known spectral/SCC results; if it does, the connection is real and the CONDITIONAL verdict can be upgraded.","tokens_in":10332,"tokens_out":18193,"duration_ms":175309,"concrete_test":"Re-analyze Fig. 1 with confound control: for fixed degree sequences (coherence ensemble of [19,27]), sample graphs spanning F ∈ [0,1], measure ρ, Φ, dF, the SIS stationary fraction, and m(100); then stratify by ρ and Φ and test whether F/dF retains predictive power within strata. If the F–dynamics relation disappears after matching ρ and Φ, the GPPM result is epiphenomenal; if it survives, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The numerical core is Fig. 1, where the GPPM parameter T is varied and F, dF, ρ and Φ all move together. The text infers that trophic coherence and non-normality drive the SIS and linear-operator results ('some nodes begin to sustain the epidemic once F > 0 or dF < 1'). But SIS persistence in this deterministic model is governed by the existence of directed cycles, equivalently by ρ ≥ 1 and a nonzero SCC, and linear growth is governed by cρ, with non-normality contributing only transient corrections. Because GPPM changes degree sequence, connectivity, and cycle structure simultaneously with F, the plots cannot distinguish F/dF as causal from ρ/Φ as causal. The imported ensemble formulae (Eq. 5 and the dF bounds from [19,21,27]) are for the coherence ensemble and are not shown to hold for the GPPM samples, so they do not bridge this gap. If F is merely a collinear proxy for ρ and SCC size, the central claim that F and dF are interchangeable summaries of directedness and dynamics is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10518,"tokens_out":3904,"duration_ms":35137,"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":[{"comment":"This is a complete sentence.","section":"Spreading processes with graphs and operators, Fig. 1"},{"comment":"This is a complete sentence.","section":"Measuring trophic coherence and non-normality, Eq. (5) and following bounds"},{"comment":"This is a complete sentence.","section":"Spreading processes with graphs and operators, Fig. 1 caption"},{"comment":"This is a complete sentence.","section":"Spreading processes with graphs and operators, middle-panel discussion"}],"minor_comments":[{"comment":"This is a complete sentence.","section":"Introduction"},{"comment":"This is a complete sentence.","section":"Spreading processes with graphs and operators"},{"comment":"This is a complete sentence.","section":"Spreading processes with graphs and operators"},{"comment":"This is a complete sentence.","section":"Figure 1 caption"},{"comment":"This is a complete sentence.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper is more of a perspective than a fully controlled research article, and its central claim currently rests on confounded numerical experiments. I would recommend a revision that either adds control experiments to isolate trophic coherence from spectral radius and SCC size or substantially tempers the causal language. The paper could then be suitable for publication as a perspective, but in its present form the evidence is not strong enough to support the strong claims on the title page and in the abstract."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThat 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.\n\nI 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.","headline":"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.","tokens_in":11052,"tokens_out":2103,"would_cite":false,"duration_ms":20501,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C20","05C82","15A18","15A60"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["directed graphs","trophic coherence","non-normality","pseudospectra","trophic levels","SIS epidemic model","linear operators","strongly connected components"],"falsifier":"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.","tokens_in":10107,"feed_emoji":"🕸️","tokens_out":8809,"duration_ms":68201,"temperature":0.7,"pith_summary":"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.","feed_headline":"Trophic coherence and non-normality are two faces of directedness","feed_subtitle":"A single scalar, the trophic incoherence F, predicts transient vs sustained spreading in networks and matrices.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the coherence-ensemble formula for the expected spectral radius $\\rho = e^{\\tau}$ and the loop exponent $\\tau$ used throughout.","marker":"[19]"},{"why":"Gives the lower bound $d_F \\ge \\sqrt{1 - e^{2\\tau}/\\langle k \\rangle}$ connecting non-normality to trophic coherence.","marker":"[21]"},{"why":"Defines the trophic levels and incoherence $F$ via $\\Lambda h = v$, states the bounds $0 \\le F \\le 1$, and introduces the normality parameter $\\nu$ and the approximation $d_F \\simeq \\sqrt{1 - \\exp(1 - 1/F)}$.","marker":"[27]"},{"why":"Supplies the Generalised Preferential Preying Model used to generate networks with tunable trophic coherence in all numerical experiments.","marker":"[24]"},{"why":"Provides the pseudospectra and transient-phenomena framework that explains why non-normal matrices show early-time activity before decay or growth.","marker":"[16]"},{"why":"Supports the claim that a strongly connected component is required to sustain an epidemic and that its size is controlled by trophic coherence.","marker":"[22]"},{"why":"Gives the lower bound on the SIS epidemic threshold in terms of the inverse spectral radius, linking coherence to epidemic persistence.","marker":"[33]"}],"fun_headline_variants":["Coherent graphs are strongly non-normal","Trophic incoherence marks normal networks","One scalar links graph layering to non-normality","Non-normality is trophic coherence in disguise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Coherent graphs are strongly non-normal","Trophic incoherence marks normal networks","One scalar links graph layering to non-normality","Non-normality is trophic coherence in disguise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0012,"raw_usage":{"total_tokens":4927,"prompt_tokens":906,"completion_tokens":4021,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":522,"completion_tokens_details":{"reasoning_tokens":3964}},"tokens_in":522,"tokens_out":4021,"duration_ms":28545,"temperature":1.0,"reasoning_tokens":3964,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:36:53.713979+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Looplessness in networks is linked to trophic coherence,","cited_arxiv_id":null,"evidence_quote":"Supplies the coherence-ensemble formula for the expected spectral radius $\\rho = e^{\\tau}$ and the loop exponent $\\tau$ used throughout."},{"cited_title":"Digraphs are different: Why directionality matters in complex systems,","cited_arxiv_id":null,"evidence_quote":"Gives the lower bound $d_F \\ge \\sqrt{1 - e^{2\\tau}/\\langle k \\rangle}$ connecting non-normality to trophic coherence."},{"cited_title":"How directed is a directed network?,","cited_arxiv_id":null,"evidence_quote":"Defines the trophic levels and incoherence $F$ via $\\Lambda h = v$, states the bounds $0 \\le F \\le 1$, and introduces the normality parameter $\\nu$ and the approximation $d_F \\simeq \\sqrt{1 - \\exp(1 - 1/F)}$."},{"cited_title":"From neurons to epidemics: How trophic coher- ence affects spreading processes,","cited_arxiv_id":null,"evidence_quote":"Supplies the Generalised Preferential Preying Model used to generate networks with tunable trophic coherence in all numerical experiments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the pseudospectra and transient-phenomena framework that explains why non-normal matrices show early-time activity before decay or growth."},{"cited_title":"Strong connectivity in real directed networks,","cited_arxiv_id":null,"evidence_quote":"Supports the claim that a strongly connected component is required to sustain an epidemic and that its size is controlled by trophic coherence."},{"cited_title":"Non-markovian infection spread dramatically alters the susceptible-infected-susceptible epidemic threshold in networks,","cited_arxiv_id":null,"evidence_quote":"Gives the lower bound on the SIS epidemic threshold in terms of the inverse spectral radius, linking coherence to epidemic persistence."}],"review_version":1}