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Graph Hierarchy: A novel approach to understanding hierarchical structures in complex networks

T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper generalizes trophic levels to every positively weighted directed graph and uses the resulting hierarchy to predict epidemic spread.

desk verdict 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. read the letter →

arxiv 1908.04358 v4 pith:ERLBGMZD submitted 2019-08-12 physics.soc-ph math.CO

Giannis Moutsinas , Choudhry Shuaib , Weisi Guo , Stephen Jarvis This is my paper · ORCID
classification physics.soc-phmath.CO MSC 05C8205C2015A0992D30
keywords trophiccoherencehierarchicallevelsinfluencecentralitydemocracycoefficientincoherencedirectedgraphsSISepidemicmodelMoore-Penroseinverse
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

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.

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 (2)
  1. [Section 4.2 / Appendices C.2 and D] 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.
  2. [Section 3.2, Conjecture 3.6, and Appendix C.2] 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.
minor comments (6)
  1. [Definition 2.3] Definition 2.3 contains the typos 'hierarchically deccomposable' and 'decompostition'; 'decomposable' and 'decomposition' are intended.
  2. [Appendix C.1] 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.
  3. [Section 4.2] 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'.
  4. [Proof of Lemma 3.5] In the proof of Lemma 3.5, 'we get ηf(g) = 1' should be 'ηf(G) = 1'.
  5. [Proof of Lemma 3.9] 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.
  6. [Figure 9 caption and Appendix D] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the hierarchical-level formalism is a self-contained algebraic extension; the epidemic section's exclusions are data-selection limitations, not circular reductions.

full rationale

The derivation chain is self-contained rather than circular. The core objects (Definition 3.1) are explicit least-squares/pseudoinverse definitions: g = M^+d, gamma = L^+delta, h = (g - gamma)/2, and the structural lemmas (Lemma 3.4, 3.5, 3.8, 3.9) are proved in Appendix B using elementary linear algebra, the projection b = (I - M M^+)d, and the standard Markov-chain fact that the kernel of I - P^T is spanned by pi; none of these proofs assumes the incidence results or the democracy-coefficient split. The only passages that could look like fitted inputs are empirical selection choices in the epidemic study: Appendix C.2 states that the 20/2500 split 'was chosen empirically based on the results,' and Appendix D excludes timed-out runs and averages 'out of 1000 non timed out simulations.' These are legitimate robustness/selection concerns about the applied claim, but they are not circular reductions: the threshold is not a parameter used to define or compute the predicted incidence, and the theoretical metrics and lemmas do not depend on it. Per the hard rule requiring an exhibited equation-level reduction, I do not classify these as circularity. External citations used in proofs are independent published results, not self-citations by these authors, so there is no self-citation load-bearing issue. Score 0.

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

The central framework rests on the explicit choice of minimum-norm least-squares levels, on the forward influence dynamics as a model of influence, and on the NSPPM graph family for the empirical application. The only fitted constant is the democracy-coefficient exclusion threshold; the simulation parameters (N, B, E, T, alpha) are scanned, not fitted.

free parameters (2)
  • democracy coefficient exclusion threshold = 20/2500
    Chosen empirically in Section 4.2 and Appendix C.2 to separate NSPPM graphs into two dynamical categories; graphs below this threshold are excluded from the epidemic incidence results.
  • number of initially infected vertices = 25
    Section 4.2: simulations infect the 25 vertices with the lowest hierarchical levels; this choice is by hand and the incidence measure depends on it.
assumptions (5)
  • domain assumption The minimum-norm least-squares solution in Definition 3.1 is the correct generalization of trophic levels.
    The entire framework rests on this modeling choice; no external criterion justifies it over other normalizations such as fixing source levels to 1.
  • domain assumption Forward influence dynamics (Section 2) is a faithful model of influence propagation.
    Used to define influencer and influenced vertices and to motivate influence centrality; no empirical validation is provided.
  • ad hoc to paper NSPPM graphs are representative of directed graphs without basal vertices.
    Appendix C.2 introduces this model for the epidemic study; all empirical claims about source-less graphs are tested only on this family.
  • ad hoc to paper Conjecture 3.6 (democracy coefficient at most 1, equality iff balanced).
    Used to derive the bound m <= n(1-eta) and to justify the democracy coefficient interpretation; stated as a conjecture, not proven.
  • standard math Strongly connected Laplacian kernel is spanned by a positive vector (Lemma B.7).
    Taken from [45] and [46] to prove influence centrality and random walk results.

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Cite this review

Pith. "Pith review of Graph Hierarchy: A novel approach to understanding hierarchical structures in complex networks." pith.science (2026). https://pith.science/paper/ERLBGMZD

@misc{pith2026190804358,
  author       = {Pith},
  title        = {Pith review of: Graph Hierarchy: A novel approach to understanding hierarchical structures in complex networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ERLBGMZD}},
  note         = {Machine review of arXiv:1908.04358}
}
read the original abstract

Trophic coherence, a measure of a graph's hierarchical organisation, has been shown to be linked to a graph's structural and dynamical aspects such as cyclicity, stability and normality. Trophic levels of vertices can reveal their functional properties and partition and rank the vertices accordingly. Yet trophic levels and hence trophic coherence can only be defined on graphs with basal vertices, vertices with zero in-degree. Consequently, trophic analysis of graphs had been restricted until now. In this paper we introduce a novel framework, a generalisation of trophic levels, which we call hierarchical levels, that can be defined on any simple graph. Within this general framework, we develop additional metrics named influence centrality, a measure of a vertices ability to influence dynamics, and democracy coefficient, a measure of overall feedback in the system, both of which have implications for the controllability of complex systems. We discuss how our generalisation relates to previous attempts and what new insights are illuminated on the topological and dynamical aspects of graphs. Finally, we show how the hierarchical structure of a network relates to the incidence rate in a SIS epidemic model.

Figures

Figures reproduced from arXiv: 1908.04358 by the authors.

Figure 1
Figure 1. An example of a hierarchically decomposable graph. (a) The original graph. Its source subgraph is marked blue and its sink subgraph is marked green. If we apply the forward influence dynamics on the graph, then in finite time all vertices will become blue. On the other hand if we apply the backward influence dynamics, then in finite time all vertices will become green. In both case, red vertices do not affect the as… view at source ↗
Figure 2
Figure 2. Examples of hierarchical layout of graphs: The y coordinate of a vertex cor￾responds to its hierarchical level. In (b) the edge 3 → 1 has smaller weight that the other two edges. this case vertex 1 should be lower than the rest. We find that the forward and backward hierarchical levels of the vertices are (−0.5, 0, 0.5) and (0.5, 0, −0.5) respectively, which implies that the hierarchical levels are (−0.5, 0, 0.5). T… view at source ↗
Figure 3
Figure 3. Scatter plot of average incidence values from Monte Carlo simulations of the infection spreading with varying hierarchical incoherence ρf (G) and infection parameter α. The average is taken over an interval of hierarchical incoherence values. (a) Incidence against α for different values of ρf (G). (b) Incidence against ρf (G) for different values of α. 4.2. Monte Carlo simulations We ran Monte Carlo simulations usin… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Two graphs representing two different food webs. Trophic levels are printed in black and trophic differences in red. (a) A totally coherent graph with integer trophic levels and trophic incoherence 0. (b) A less coherent graph with non-integer trophic levels and trophi…
Figure 5
Figure 5. Figure 5: Hierarchical levels and differences on the same graphs as in [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]
Figure 6
Figure 6. Figure 6: The correlation between temperature, trophic incoherence and hierarchical incoherence in PPM graphs. (a) Scatter plot of trophic incoherence over temperature. (b) Scatter plot of forward hierarchical incoherence over temperature. (c) Scatter plot of forward hierarchica…
Figure 7
Figure 7. Figure 7: The scatter plots of hierarchical incoherence over democracy coefficient for NSPPM graphs. Two different regions are visible, one with democracy coefficient be￾tween 0 and 20/2500 and hierarchical incoherence that go up to 25 and another band with hierarchical incohere…
Figure 8
Figure 8. Figure 8: Scatter plots of hierarchical incoherence and democracy coefficient over tem￾perature for NSPPM graphs. (a) Democracy coefficient over temperature. The graphs with democracy coefficient less or equal than 20/2500 form a very tight band on the bottom of the figure. The …
Figure 9
Figure 9. Figure 9: Scatter plot of average incidence values from Monte Carlo simulations of the infection spreading with varying temperature T and infection parameter α. The average is taken over 1000 runs. (a) Incidence against α for different values of T. (b) Incidence against T for di…
Figure 3
Figure 3. Figure 3: Incoherence varied between the interval (0 [PITH_FULL_IMAGE:figures/full_fig_p031_3.png]
Figure 10
Figure 10. Figure 10: Heat map of average incidence values from Monte Carlo simulations of the infection spreading. (a) Incidence against α and T. (b) Incidence against α and ρf (G). graph cannot be chosen, each average was taken over graphs with incoherence in a small interval. Because th…

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    ker(L) is spanned by the non-negative vectors ki = (0,..., 0,κi, 0,..., 0), where i∈{i,...,l } and the position of κi in ki corresponds to the position of Li in L

  42. [50]

    kid = 0 if Γi is just a single vertex and kid> 0 otherwise. Proof. 23

  43. [51]

    Since Γi is a minimal source subgraph, by Lemma B.4 it is strongly connected and by Lemma B.7, ker(Li) is spanned by a positive vector κi

  44. [52]

    By renaming the vertices, the Laplacian L can be brought to the form L =   L1 0

    Since there are l minimal source subgraphs, the dimension of ker( L) is l, see [45]. By renaming the vertices, the Laplacian L can be brought to the form L =   L1 0 ... 0 C1 0 L2 ... 0 C2 ... ... ... ... ... 0 0 ... L l Cl 0 0 ... 0 Cl+1   . It is straightforward...

  45. [53]

    temperature

    Without loss of generality we we will consider Γ 1. If Γ 1 is a single vertex thenL1 is just the 1×1 zero matrix. This means thatk1 = (1, 0,..., 0) andd = (0,d 2,...,d n), thus k1d = 0. If Γ 1 is a strongly connected graph with m vertices, then κ1 is a positive m-vector and th...

  46. [58]

    temperature

    From all possible edges i → j such that j is not an source vertex, we choose L−N +B with probability proportional to P(aij = 1)∝ exp ( −(sj−si− 1)2 2T 2 ) . All PPM graphs are simply influenced graphs, so the democracy coefficient is 0. In Figure 6 we see that the trophic incoher...

  47. [59]

    We introduce B source vertices and no edges

  48. [60]

    We choose uniformly at random one of the existing vertices i and we add a new vertexj and the edge i→j

  49. [61]

    We repeat step 2 until we have N vertices in total

  50. [62]

    We assign each vertex i its trophic level si according to the graph we have up to this point

  51. [63]

    From all possible edges i → j such that j is not an source vertex, we choose L−N +B with probability proportional to P(aij = 1)∝ exp ( −(sj−si− 1)2 2T 2 )

  52. [64]

    28 (a) (b) Figure 7.: The scatter plots of hierarchical incoherence over democracy coefficient for NSPPM graphs

    We pick an source vertex i with in-degree 0, we pick another vertex j with proba- bility proportional to exp(−sj) and we add the edge j→i. 28 (a) (b) Figure 7.: The scatter plots of hierarchical incoherence over democracy coefficient for NSPPM graphs. Two different regions are vi...

  53. [65]

    We find that NSPPM graphs can be separated into two types

    We repeat step 6 until all source vertices have in-degree 1. We find that NSPPM graphs can be separated into two types. The ones with very small democracy coefficient (smaller than 20 /2500) and the one with a democracy coefficient bigger than 20/2500. We can see in Figures 7 and 8...

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Reviewed August 14, 2026 · model on record in the stance chip above.