{"id":"e5da77ea-57f9-4234-a1c9-34a380bb8700","arxiv_id":"1908.04358","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Hierarchical levels, defined as the minimum-norm least-squares solution of a graph Laplacian equation, extend trophic analysis to arbitrary simple directed graphs and yield metrics that track SIS epidemic incidence.","lead":"This paper introduces a generalized measure of hierarchy, called hierarchical levels, that works on any directed network, not just food webs with clear starting points. It also defines new metrics for influence and feedback, and shows the hierarchy measure correlates with how infections spread in simulated networks.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Epidemic incidence result may be an artifact of excluding low-democracy NSPPM graphs and timed-out runs; Section 4.2 needs a no-exclusion reanalysis before the structural-predictor claim can be accepted.","rationale":"The mathematical core of the paper, hierarchical levels via Moore-Penrose inverses and the proven Lemmas 3.8 and 3.9, is well-posed and internally consistent; the generalization of trophic levels is a genuine contribution. The load-bearing weakness is not the mathematics but the paper's use of the framework as a structural predictor of epidemic incidence. That use is conditional on exclusions that are both data-dependent and generator-specific. The reader identified the 20/2500 democracy-coefficient threshold; I agree with that concern and add that the separate exclusion of timed-out runs (Appendix D) is a censoring bias: if non-terminating high-rho_f runs have high incidence, averaging only over terminated runs will systematically lower the high-rho_f end of Figure 3 and can create the claimed negative relationship even when no true relationship exists. The proposed re-analysis with no exclusions and with incidence imputed at the cutoff directly tests whether the relationship is robust. Until that is done, the paper should not be accepted for its epidemic claim, although the theoretical hierarchy metrics can stand on their own. The reader's CONDITIONAL verdict is therefore appropriate, and my read does not change it.","tokens_in":19692,"tokens_out":16882,"duration_ms":179060,"concrete_test":"Re-run the Section 4.2 protocol with no exclusions: for every NSPPM graph (including democracy coefficient <= 20/2500) and every alpha in (1, 1.7], run the SIS dynamics and, for runs that do not terminate by t=1000, record the incidence at t=1000 instead of discarding them. Then plot average incidence against rho_f for (a) the full sample and (b) the previously included subsample. If the negative trend is absent or reversed in the full sample, the Section 4.2 claim is not supported; if it persists, the exclusions are not the driver. Additionally, vary the 20/2500 cutoff over a grid such as 10/2500, 30/2500, and 50/2500 to check that the trend is insensitive to the empirically chosen value.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4.2's central applied claim, that hierarchical incoherence predicts SIS incidence, is established only after two data-dependent exclusions: (1) NSPPM graphs with democracy coefficient at most 20/2500 are dropped because they behave differently, and the threshold is 'chosen empirically based on the results' (Appendix C.2); (2) runs that reach the 1000-step timeout are discarded, with averages taken 'out of 1000 non timed out simulations' (Appendix D). Both exclusions can create the reported negative trend. Low-democracy graphs include many low-temperature, highly hierarchical realizations; if these mostly end with incidence 1 or time out, removing them changes the population over which the rho_f-incidence relation is computed. Similarly, if high-rho_f runs are more likely to time out with substantial incidence, discarding them depresses the average incidence at the high-rho_f end of Figure 3, manufacturing the apparent correlation. The threshold's theoretical grounding relies only on unproven Conjecture 3.6, so it does not establish that the excluded set is a natural class of graphs rather than a subset selected to make the trend visible. For the claim that hierarchical structure predicts incidence to hold, the relation must survive on the full generated population or after an outcome-independent treatment of non-terminating runs.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a generalisation of trophic levels to arbitrary positively weighted simple directed graphs, defining forward and backward hierarchical levels as the minimum-norm least-squares solutions g = M^+ d and gamma = L^+ delta, and the vector of hierarchical levels as h = (g - gamma)/2. It then defines edge-level hierarchical differences, the democracy coefficient, the hierarchical incoherence parameter, and influence centrality, and proves several structural lemmas in the appendices, including a characterization of influencer vertices and a proportionality between influence centrality and the stationary distribution of a random walk on strongly connected graphs. The final section applies the framework to SIS contagion dynamics on a non-source variant of the preferential preying model (NSPPM) and claims that forward hierarchical incoherence predicts incidence.","tokens_in":19958,"tokens_out":8762,"duration_ms":83811,"significance":"The theoretical core is a clean and genuinely useful extension: it removes the basal-vertex restriction of trophic analysis, gives explicit pseudoinverse formulas, and supplies real proofs for the main structural lemmas (the democracy-coefficient formula, the influence-centrality characterization, and the random-walk relation). The proposed metrics are computationally attractive because the underlying algorithm avoids explicit pseudoinverse construction. If the epidemic claim were robust, the paper would offer a practical structural predictor of spreading dynamics. However, as presented, that applied claim is not yet established, because the supporting simulations rely on data-dependent exclusions that are not justified on independent grounds.","major_comments":[{"comment":"The central applied claim that hierarchical incoherence predicts SIS incidence is obtained only after two data-dependent exclusions. First, NSPPM graphs with democracy coefficient at most 20/2500 are dropped, with the threshold 'chosen empirically based on the results' (Appendix C.2); second, simulations reaching the 1000-step limit are discarded, with means computed 'out of 1000 non timed out simulations' (Appendix D). The first exclusion removes a class that the paper itself describes as behaving differently (immediate full incidence or timeout), so the negative trend in Figure 3 may reflect a change of population rather than a structural effect; the second exclusion can bias the average incidence downward at the high-incoherence end if high-rho_f runs are more likely to time out with intermediate incidence. A no-exclusion reanalysis, or an outcome-independent treatment of non-terminating runs (for example, imputing their incidence or reporting worst-case bounds), is required before the claim that hierarchical structure predicts incidence can be accepted.","section":"Section 4.2 / Appendices C.2 and D"},{"comment":"The classification of NSPPM graphs into 'small' and 'large' democracy-coefficient categories is given theoretical meaning through the unproved Conjecture 3.6: the text in Appendix C.2 says that if the conjecture is true, then the first category has 20 or fewer edges in its source subgraphs. Since the epidemic result depends on excluding this category, the manuscript should either prove the conjecture, or the specific implication used, or replace the exclusion criterion by a directly computed structural quantity such as the size or number of source subgraphs. As it stands, the threshold is purely empirical, and the reader cannot distinguish a natural class of graphs from a subset selected to make the trend visible.","section":"Section 3.2, Conjecture 3.6, and Appendix C.2"}],"minor_comments":[{"comment":"Definition 2.3 contains the typos 'hierarchically deccomposable' and 'decompostition'; 'decomposable' and 'decomposition' are intended.","section":"Definition 2.3"},{"comment":"In Step 5 of the PPM algorithm, the expression 'L−N+B' uses an undefined symbol L; it should presumably be 'E−N+B', where E is the number of edges.","section":"Appendix C.1"},{"comment":"The sentence 'In Figure 3 we show the average incidence for graphs with democracy coefficient 20/2500' is missing a comparison; from the preceding discussion it should read 'greater than 20/2500'.","section":"Section 4.2"},{"comment":"In the proof of Lemma 3.5, 'we get ηf(g) = 1' should be 'ηf(G) = 1'.","section":"Proof of Lemma 3.5"},{"comment":"In the proof of Lemma 3.9, 'Let D be G's weighted in-degree diagonal matrix' should be 'weighted out-degree diagonal matrix' to match the statement of the lemma, and the phrase 'the matrices D and D are invertible' contains a duplicated symbol.","section":"Proof of Lemma 3.9"},{"comment":"The Figure 9 caption says 'The average is taken over 1000 runs', while Appendix D says the average is taken 'out of 1000 non timed out simulations'; these statements should be reconciled.","section":"Figure 9 caption and Appendix D"}],"recommendation":"major_revision","confidential_remarks":"The theoretical part is sound and publishable; the main risk is confined to Section 4.2. I would be willing to accept after the authors provide a no-exclusion or outcome-independent reanalysis of the epidemic simulations, and either prove Conjecture 3.6 or remove the reliance on it in the empirical classification."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know before you read it.\n\nThe mathematical core is genuinely new and mostly solid. Hierarchical levels are defined as the minimum-norm least-squares solution g = M⁺d (and γ = L⁺δ), which is a clean generalization of trophic levels to any positively weighted directed graph, including strongly connected graphs where previous methods assign everyone the same level. The appendix contains real proofs of the main lemmas — the democracy coefficient as a projection of the degree vector onto the Laplacian kernel (Lemma B.10), the influence centrality characterization, and the random-walk result cπ = D²ε (Lemma 3.9). That last one is a pleasant surprise: backward influence centrality is, up to scaling by out-degree squared, the stationary distribution of the random walk. The paper also ships open-source Python and Julia packages, which is more than most theory papers do.\n\nSecond, the epidemic application in Section 4.2 does not support its claim as written. The incidence-versus-incoherence trend is established only after two data-dependent exclusions: graphs with democracy coefficient at most 20/2500 are dropped, with the threshold 'chosen empirically based on the results' (Appendix C.2), and runs hitting the 1000-step timeout are discarded, with averages taken 'out of 1000 non timed out simulations' (Appendix D). Both cuts can manufacture the reported negative correlation. Low-democracy graphs are disproportionately low-temperature and either reach incidence 1 quickly or time out; and if high-incoherence runs are more likely to time out while still carrying substantial infection, removing them depresses the high-ρ end of Figure 3. The authors give a mechanistic story for why the excluded graphs behave differently, which is not nothing, but the structural-predictor claim needs a no-exclusion reanalysis or a principled treatment of non-terminating runs.\n\nWhere the paper is honest: Conjecture 3.6 (democracy coefficient ≤ 1, equality iff balanced) is labeled a conjecture and used only for derived bounds. That is a gap, but a visible and honestly flagged one. The controllability language in the abstract is framing, not result — I would not weight it in the verdict, and I would ask the authors to soften it.\n\nVerdict: the framework deserves serious refereeing. The math is worth the referees' time and the random-walk lemma alone is citable. The epidemic section is the soft spot and needs revision, not acceptance as-is.","headline":"A genuinely useful generalization of trophic levels with solid proofs in the core, but the epidemic application rests on exclusion thresholds that can manufacture the reported trend.","tokens_in":20446,"tokens_out":3748,"would_cite":true,"duration_ms":36103,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C82","05C20","15A09","92D30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper generalizes trophic levels to every positively weighted directed graph and uses the resulting hierarchy to predict epidemic spread.","keywords":["trophic coherence","hierarchical levels","influence centrality","democracy coefficient","hierarchical incoherence","directed graphs","SIS epidemic model","Moore-Penrose inverse"],"falsifier":"Run the same Monte Carlo SIS experiment on 500-vertex, 2500-edge NSPPM graphs across temperatures, and compare incidence with and without excluding graphs whose democracy coefficient is at most $20/2500$: if the monotone relationship between forward hierarchical incoherence and incidence survives when low-democracy graphs are included and the threshold is varied, the structural predictor is general; if the relationship appears only because of that exclusion, it is not.","tokens_in":19483,"feed_emoji":"🕸️","tokens_out":7944,"duration_ms":69471,"temperature":0.7,"pith_summary":"Trophic levels, the classic way to rank vertices in a food web, can only be defined when a graph has basal (zero in-degree) vertices, which rules out most real directed networks. This paper removes that restriction by defining forward and backward hierarchical levels as minimum-norm least-squares solutions $g = M^+ d$ and $\\gamma = \\Lambda^+ \\delta$, and taking $h = (g - \\gamma)/2$ as the graph's hierarchy. On any positively weighted simple directed graph this yields a vertex ranking plus two new metrics: influence centrality, which identifies the vertices that drive the dynamics, and the democracy coefficient, which measures how much the graph's influencers are themselves influenced. The paper proves that in strongly connected graphs backward influence centrality is proportional to the random-walk stationary distribution scaled by squared out-degree, and that forward hierarchical incoherence predicts incidence in an SIS epidemic model on source-less graphs. If correct, these metrics turn hierarchy from a property of specially structured networks into a general analysis tool for arbitrary directed graphs.","feed_headline":"Generalized trophic levels rank any directed network","feed_subtitle":"The metrics also identify who drives dynamics and predict how infections spread in source-free graphs.","key_machinery":"The load-bearing construction is the pair of Moore-Penrose solutions $g = M^+ d$ and $\\gamma = \\Lambda^+ \\delta$, with $h = (g - \\gamma)/2$, where $M$ and $\\Lambda$ are the transposes of the weighted in-degree and out-degree Laplacians and $d$, $\\delta$ are the weighted in- and out-degree vectors. Computing these levels via convex optimization avoids forming the pseudoinverse explicitly. The argument then works through hierarchical edge differences, whose weighted mean and variance define the democracy coefficient and the hierarchical incoherence, and through a projection identity that expresses influence centrality as the ratio of the projection of the degree vector onto the Laplacian kernel to the vertex degree. For strongly connected graphs the key result is $c\\pi = D^2\\epsilon$: backward influence centrality encodes the random-walk stationary distribution.","core_discovery":"The paper's central claim is that hierarchical levels, defined by $g = M^+ d$, $\\gamma = \\Lambda^+ \\delta$, and $h = (g - \\gamma)/2$, generalize trophic levels to every positively weighted simple directed graph, including strongly connected graphs with no basal vertices. Forward hierarchical levels grade vertices by distance from source subgraphs; backward levels grade them by distance from sink subgraphs; their difference combines the control and dependence perspectives into one ranking. On this foundation the paper defines influence centrality (a vertex is an influencer exactly when its influence centrality is positive) and the democracy coefficient (zero for simply forward influenced graphs, one for balanced graphs), and proves the identity $c\\pi = D^2\\epsilon$ relating backward influence centrality to the stationary distribution of a random walk on strongly connected graphs. The paper also shows that forward hierarchical incoherence matches the predictive behaviour of trophic incoherence on graphs where both are defined, and extends the prediction to source-less NSPPM graphs in an SIS model.","pith_inferences":["Editorial inference: the hierarchical level $h$ itself could be used as a combined control-and-dependence centrality for arbitrary directed networks, including the core-periphery detection the paper mentions only as initial exploration.","Editorial inference: if the democracy-coefficient conjecture (always at most 1, equality exactly for balanced graphs) is proven, the coefficient becomes a normalized, size-independent feedback measure for comparing graphs.","Editorial inference: the epidemic relationship could be tested on empirical source-less networks such as gene-regulatory or financial transaction graphs, to see whether the threshold $20/2500$ is a general feature or an artifact of the NSPPM generator.","Editorial inference: the identity $c\\pi = D^2\\epsilon$ suggests using backward influence centrality to target sampling or intervention in strongly connected networks before explicitly computing stationary distributions."],"forward_implications":["Trophic analysis is no longer restricted to graphs with basal vertices: hierarchical levels, influence centrality, and hierarchical incoherence are defined on any positively weighted simple directed graph, and on undirected graphs through the forward or backward version.","Influencer identification becomes a closed-form computation: a vertex drives the forward dynamics exactly when its forward influence centrality is positive, and in strongly connected graphs every vertex has positive forward and backward influence centrality.","The democracy coefficient gives a topological feedback measure that decomposes over minimal source and sink subgraphs, equals 1 on balanced graphs, and, if the paper's conjecture holds, always lies between 0 and 1.","Epidemic spread on source-less directed graphs can be read off from structure: lower forward hierarchical incoherence corresponds to higher SIS incidence, matching the known trophic-coherence effect in a broader setting.","Because $c\\pi = D^2\\epsilon$ holds in strongly connected graphs, backward influence centrality is a structural proxy for where a random walk spends its time."],"supporting_citations":[{"why":"Supplies the trophic coherence framework and the PPM graph generator that the paper generalizes and compares against.","marker":"[22]"},{"why":"Provides the SIS contagion model and the earlier result that trophic coherence affects spreading processes, the baseline the epidemic application extends.","marker":"[24]"},{"why":"Justifies $g = M^+d$ via the Moore-Penrose pseudoinverse as the minimum-norm least-squares solution.","marker":"[28]"},{"why":"Supplies the polynomial-time convex optimization algorithms used to compute hierarchical levels without forming the pseudoinverse.","marker":"[29]"},{"why":"A prior generalization of trophic analysis to more general directed graphs that the paper's hierarchical levels improve on.","marker":"[32]"},{"why":"Another prior trophic-style generalization based on circular flow, used as a comparison target for ranking strongly connected graphs.","marker":"[33]"},{"why":"Gives the original ecological definition of trophic levels that the framework generalizes.","marker":"[44]"},{"why":"Gives the kernel-dimension results for directed graph Laplacians used in the proofs of hierarchical decomposition and the democracy coefficient.","marker":"[45]"},{"why":"Provides the positive integer kernel vector for strongly connected graphs used in Lemma B.7.","marker":"[46]"},{"why":"Supplies the orthogonal-projector identity used to express influence centrality as the projection of the degree vector onto the Laplacian kernel.","marker":"[47]"}],"fun_headline_variants":["No source? No problem: new hierarchy ranking for any graph","Generalized hierarchy ranks any graph, no basal vertices needed","Trophic levels go universal: rank any directed graph","Hierarchy without sources: new levels rank all networks","New metrics reveal influence and democracy in any network"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the empirical threshold that source-less NSPPM graphs with democracy coefficient at most $20/2500$ can be excluded from the epidemic analysis; if that threshold is an artifact of the graph generator rather than a general property of source-less networks, the claimed link between hierarchical incoherence and epidemic incidence is not established.","fun_headline_variants_meta":{"raw":{"variants":["No source? No problem: new hierarchy ranking for any graph","Generalized hierarchy ranks any graph, no basal vertices needed","Trophic levels go universal: rank any directed graph","Hierarchy without sources: new levels rank all networks","New metrics reveal influence and democracy in any network"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00063,"raw_usage":{"total_tokens":2900,"prompt_tokens":926,"completion_tokens":1974,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":542,"completion_tokens_details":{"reasoning_tokens":1896}},"tokens_in":542,"tokens_out":1974,"duration_ms":15040,"temperature":1.0,"reasoning_tokens":1896,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:44:10.084768+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same Monte Carlo SIS experiment on 500-vertex, 2500-edge NSPPM graphs across temperatures, and compare incidence with and without excluding graphs whose democracy coefficient is at most $20/2500$: if the monotone relationship between forward hierarchical incoherence and incidence survives when low-democracy graphs are included and the threshold is varied, the structural predictor is general; if the relationship appears only because of that exclusion, it is not.","supporting_citations":[{"cited_title":"Trophic coherence determines food-web stability","cited_arxiv_id":null,"evidence_quote":"Supplies the trophic coherence framework and the PPM graph generator that the paper generalizes and compares against."},{"cited_title":"From neurons to epidemics: How trophic co- herence affects spreading processes","cited_arxiv_id":null,"evidence_quote":"Provides the SIS contagion model and the earlier result that trophic coherence affects spreading processes, the baseline the epidemic application extends."},{"cited_title":"On best approximate solutions of linear matrix equations","cited_arxiv_id":null,"evidence_quote":"Justifies $g = M^+d$ via the Moore-Penrose pseudoinverse as the minimum-norm least-squares solution."},{"cited_title":"How directed is a directed network?","cited_arxiv_id":"2001.05173","evidence_quote":"A prior generalization of trophic analysis to more general directed graphs that the paper's hierarchical levels improve on."},{"cited_title":"Community structure based on circular flow in a large- scale transaction network","cited_arxiv_id":null,"evidence_quote":"Another prior trophic-style generalization based on circular flow, used as a comparison target for ranking strongly connected graphs."},{"cited_title":"The trophic-dynamic aspect of ecology","cited_arxiv_id":null,"evidence_quote":"Gives the original ecological definition of trophic levels that the framework generalizes."},{"cited_title":"Kernels of directed graph Laplacians","cited_arxiv_id":null,"evidence_quote":"Gives the kernel-dimension results for directed graph Laplacians used in the proofs of hierarchical decomposition and the democracy coefficient."},{"cited_title":"Chip-firing games on directed graphs","cited_arxiv_id":null,"evidence_quote":"Provides the positive integer kernel vector for strongly connected graphs used in Lemma B.7."},{"cited_title":"Matrix Computations, Johns Hopkins U","cited_arxiv_id":null,"evidence_quote":"Supplies the orthogonal-projector identity used to express influence centrality as the projection of the degree vector onto the Laplacian kernel."}],"review_version":1}