Digraphs are different: Why directionality matters in complex systems

Samuel Johnson

REVIEW 4 major objections 5 minor 40 references

Digraphs are different: Why directionality matters in complex systems

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper argues that the most distinctive features of directed networks—trophic coherence, non-normality, spectral radius, strong connectivity, and cycle counts—share one origin: how strongly edge directions align with a global direction.

desk verdict The unifying story is appealing but the central theorem is not proven: Eq. (6) treats the trace as a Gelfand norm, and the one visible counterexample in the tables shows rho != e^tau for at least one network. read the letter →

arxiv 1908.07025 v1 pith:3XKN4RRX submitted 2019-08-19 physics.soc-ph math.DSnlin.AO

Samuel Johnson This is my paper · ORCID

classification physics.soc-phmath.DSnlin.AO MSC 05C2005C82 PACS 89.75.Hc89.75.-k

keywords directednetworkstrophiccoherencenon-normalityspectralradiusstronglyconnectedcomponentmajority-ruledynamicsnetworkensemblescomplexsystems

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

Directed networks are often treated as undirected graphs with arrows added at random. This paper tries to establish that the arrows are the story: in many real systems they align with a global direction, and that single ordering explains a cluster of otherwise unrelated anomalies. Using the coherence ensemble, the paper shows that trophic coherence and non-normality are two sides of the same coin, and that the sign of one loop exponent separates networks with large strongly connected components and abundant cycles from networks where both vanish. It also shows—on the worm neural network and on generated graphs—that the same ordering flips the stability of the simplest majority-rule dynamics, with a critical coherence at $q_c=1$. If this is right, a single measurable property of edge directions organizes much of directed-network structure and dynamics.

What carries the argument

The workhorse is the coherence ensemble: a random directed graph ensemble with fixed in- and out-degree sequences and a prescribed trophic coherence $q$, together with the basal ensemble in which each non-basal vertex receives the same fraction of basal in-edges. From that ensemble comes the identity $\mathrm{tr}(A^k)=\frac{\tilde\alpha \tilde q}{\alpha q}e^{\tau k}$, which turns the loop exponent $\tau$ into the spectral radius via Gelfand's formula. The sign of $\tau$ then acts as an order parameter: positive $\tau$ means exponentially growing counts of directed circuits and a giant strongly connected component; negative $\tau$ means exponentially suppressed circuits and a vanishing SCC. The non-normality theorem needs only this spectral-radius result plus the bound $|\lambda_i|\le\rho$, and the majority-rule stability calculation uses the basal-ensemble ratio $k_i^{\mathrm n}/k_i^{\mathrm b}=L/L_B-1\equiv\lambda$ to reduce flips to a binomial probability, yielding $q_c=1$.

What would settle it

Measure the spectral radius and SCC fraction on empirical directed networks with measured trophic coherence; any network with $\tau<0$ that still has a giant strongly connected component, or whose spectral radius stays far above $e^{\tau}$, would refute the regime split. In generated networks, pushing $q\to 0$ and finding $\rho$ not collapsing to $e^{\tau}$ would directly falsify Eq. (3).

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Extended reading notes

Core claim

The central claim is that a single number, the loop exponent $\tau = \ln\alpha + \frac{1}{2}\tilde q^2 - \frac{1}{2}q^2$, with branching factor $\alpha=\langle k^{\mathrm{in}}k^{\mathrm{out}}\rangle/\langle k\rangle$ and trophic incoherence $q$, governs whether a directed network lives in a loopful or loopless regime. In the coherence ensemble the expected trace of $A^k$ is a single exponential, so Gelfand's formula gives the spectral radius $\rho=e^{\tau}$. The paper proves that the normalized departure from normality tends to $1$ as $q\to 0$, and exceeds $\sqrt{1-1/\langle k\rangle}$ whenever $\tau<0$; coherent networks are therefore highly non-normal, with spectral radius and strongly connected component both collapsing. It further derives, from a mean-field basal-ensemble calculation, a critical value $q_c=1$ for majority-rule dynamics: below that coherence the mean activity flips sign persistently, above it the dynamics is stable. The empirical networks included in the paper line up with the predicted regime split, and the C. elegans connectome shows the predicted coherence-dependent bistability.

Load-bearing premise

The argument depends on Eq. (3)—that in the coherence ensemble closed walks of length $k$ are exponentially numerous at rate $e^{\tau k}$—an assumption imported from earlier work without derivation here, so if real directed networks deviate from that ensemble expectation the spectral-radius, non-normality, and strong-connectivity conclusions lose their quantitative grounding.

Editorial extensions

If this is right

If the coherence-ensemble description is correct, the sign of $\tau$ is a regime label: trophically coherent networks with $\tau<0$ should have near-zero spectral radii and no giant strongly connected component, while $\tau>0$ networks should be loopful and strongly connected.
Since $d_F\to 1$ as $q\to 0$, trophic coherence and non-normality are not independent properties; a mechanism that aligns edge directions automatically makes the adjacency matrix far from normal, so explanations of stability invoking one implicitly involve the other.
The majority-rule result implies that coherence can reverse the direction of stability: the same degree sequence is dynamically stable when incoherent and unstable when coherent, so degree statistics alone are insufficient to predict dynamical outcomes on directed networks.
The mean-field transition at $q_c=1$ gives a quantitative target: real networks with $q$ below 1 should exhibit persistent sign flips under majority-rule updates, while those above 1 should settle into a stable consensus.

Reading between the lines

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

A natural extension the author does not pursue: because the same exponent controls cycle counts and SCC size, one could use measured directed-cycle distributions as a proxy for trophic coherence, avoiding the need to solve for trophic levels in networks without basal vertices.
A natural extension the author does not pursue: because non-normality is known to cause transient amplification, coherent directed networks should show strong transient responses before reaching their stable state; this could be tested with linearized dynamics on the same empirical networks.
The basal-ensemble derivation assumes each non-basal vertex receives the same fraction of basal inputs; breaking that assumption in the preferential-preying model should shift or smear the predicted $q_c=1$, giving a controlled test of the mean-field logic.
The $q_c=1$ transition suggests a design rule for engineered information-flow networks: tuning coherence through rewiring should let one switch between stable consensus and sensitive novelty detection, a trade-off directly relevant to sensory and regulatory systems.

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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 worth a serious referee, but the central proof step does not hold. Eq. (6), rho = e^tau, is derived by "taking the norm in Eq. (5) to be the trace." Trace is not a matrix norm. Gelfand's formula applies to norms, and the trace functional does not satisfy the required properties. For a directed cycle, tr A^k vanishes except when the cycle length divides k, so the ordinary limit in Gelfand's formula does not even exist, although the spectral radius is well defined. You also need to justify exchanging the ensemble expectation in Eq. (3) with the limit in k before taking the kth root. Neither issue is addressed. This matters because the non-normality bounds (Eqs. 16-19) and the loopful/loopless spectral-radius classification all rest on rho = e^tau. The data tables make the problem visible: Bridge Brook Lake has tau = -0.53, so e^tau is about 0.59, yet Table S1 reports rho = 1. So the empirical support for the theorem is weaker than the text implies.

Now the credit. The paper does something genuinely useful: it shows that several apparently unrelated directed-network phenomena—spectral radius, strong connectivity, cycle growth, non-normality—all track a single scalar, the loop exponent tau. The figures are persuasive as empirical regularities. The new results are the non-normality bounds and the majority-rule phase transition at q_c = 1. The mean-field calculation for the basal ensemble is clearly laid out, and the simulation support on a different generative model is a nice touch. The author is also honest about the provenance of the coherence ensemble from Ref. [1] and about the fact that the same data set is used for validation, which is a mild circularity but not a fatal one given the prior work is published.

The soft spots, beyond the Gelfand issue: the link between cycle decay and SCC size is heuristic, and the majority-rule mean-field assumes a constant ratio k_i^n/k_i^b, which is only approximate for the preferential preying model. These are secondary.

Who should read this: network scientists and complex-systems people who want a compact synthesis of directionality effects. What it needs: a proper derivation of Eq. (6) or a recasting of the theorem as a numerically supported conjecture. I would send it to peer review with a request for major revision; the empirical sections can stand, but the theoretical claim needs to be made rigorous or honestly downgraded.

Referee Report

4 major / 5 minor

Summary. The paper argues that a single property of directed networks, trophic coherence, controls a cluster of apparently unrelated features: spectral radius, non-normality, the size of the strongly connected component, cycle abundance, and the stability of majority-rule dynamics. The central theoretical result is a theorem stating that the normalized departure from normality tends to 1 with increasing trophic coherence and exceeds sqrt(1 - 1/<k>) in the loopless regime. The proof proceeds by combining the coherence-ensemble formula for tr(A^k), Eq. (3), with Gelfand's formula to obtain the spectral radius rho = e^tau, and then bounding the sum of squared eigenvalue moduli. The author validates the predictions on 62 empirical networks and with simulations of a preferential preying model, and illustrates the dynamical consequences on the C. elegans neural network.

Significance. If the central derivation were correct, the paper would provide a valuable unification of several directed-network phenomena and strengthen the case that edge-direction alignment is a fundamental organizing principle. The manuscript is clearly written, the data set is diverse, and the stated data/code availability is a strength. However, the key mathematical step linking Eq. (3) to the spectral radius is invalid as stated, so the theorem and its bounds are not currently proven. The empirical observations remain suggestive, but the main quantitative claim needs a rigorous derivation or a clear rephrasing as a conjecture supported by numerics.

major comments (4)

[Results, 'Graph ensembles' and 'Non-normality', Eqs. (5)-(6)] The step 'taking the norm in Eq. (5) to be the trace' is not a legitimate application of Gelfand's formula. The trace functional is not a matrix norm: it is not submultiplicative and does not satisfy the axioms required for the formula. For a directed cycle of length n, tr(A^k) is n when n divides k and 0 otherwise, so the limit in Eq. (5) does not exist even though rho = 1. Moreover, Eq. (3) supplies only the ensemble expectation of tr(A^k), and no argument justifies interchanging the expectation with the limit and the k-th root before taking k to infinity. Since the proof of the theorem and the bounds in Eqs. (16)-(19) rely on rho = e^tau, the central theoretical result is not established.
[Results, 'Graph ensembles', Eq. (3) and its citation] The theorem's proof hinges entirely on Eq. (3), which is imported from the author's prior work without derivation or a statement of the conditions under which it holds. The main text attributes this result to 'Ref. [1]', but in the reference list [1] is Newman's review article; the intended source appears to be Johnson and Jones (2017), listed as [9]. As it stands, the proof is not self-contained, and the reader cannot verify the key formula on which all subsequent claims rest.
[Results, 'Strong connectivity' and 'Non-normality', Figures 2-3] The empirical validation in Figures 2 and 3 uses the same 62 networks on which the coherence-ensemble machinery was developed and tested in the prior work. This does not invalidate the theorem, but it weakens the claim that the bounds 'hold for these empirical cases too' as an independent confirmation. The preferential preying model simulations provide some independent support, but the main theoretical claim would benefit from out-of-sample tests on networks not used to calibrate the ensemble.
[Methods 0.2, 'Majority rule dynamics on basal ensemble networks'] The derivation of the critical value q_c = 1 assumes that the ratio k_i^n/k_i^b equals the constant lambda for every non-basal vertex, which is a property of the basal ensemble but only approximately true for the preferential preying model used in the simulations. The text appropriately notes that the mean-field analysis is for the basal ensemble and that the extension to other ensembles is heuristic, but the phrasing 'the same critical value may also apply' makes the status of q_c = 1 for general networks less definite than the abstract suggests.

minor comments (5)

[References throughout] The main text repeatedly cites 'Ref. [1]' for the coherence ensemble and the basal ensemble, but reference [1] in the main-text list is Newman (2003). The intended reference is presumably Johnson and Jones (2017), which is listed as [9]. Please correct all such citations.
[Section 'Strong connectivity'] The abbreviation 'SSC' is introduced for the strongly connected component, but 'SCC' is used everywhere else, including in the tables and figure captions. Please standardize.
[Figure 1 caption] The caption contains a repeated article: 'the the C. elegans neural network' should be 'the C. elegans neural network'.
[Author affiliation] The affiliation line contains spacing errors: 'Edgbasto n' and 'Lo ndon' should be 'Edgbaston' and 'London'.
[Table S1] For Bridge Brook Lake, tau = -0.53 gives e^tau approx 0.59, but the table reports rho = 1. The paper does not comment on this discrepancy between the actual spectral radius and the coherence-ensemble expectation, which is relevant because Figures 2 and 3 use tau to place networks on the horizontal axis.

Circularity Check

1 steps flagged · score 4.0 of 10

Central non-normality theorem rests on the author's prior coherence-ensemble expectation rather than an independent derivation; empirical checks reuse the same networks.

self citation load bearing [Results: Graph ensembles, Eq. (3); Results: Trophic coherence and non-normality, Eqs. (6), (16)-(19); Supplementary Material data note]
"In the coherence ensemble, we have shown that, in expectation, tr(Ak) = ˜α ˜q / α q e^{τ k}, (3) ... Taking the norm in Eq. (5) to be the trace, we can use Eq. (3) to find the expected value of the spectral radius in the coherence ensemble: ρ = e^τ . (6)"

The theorem's bounds (Eqs. 16-19) are obtained by inserting Eq. (6) into the definition of dF. Eq. (6) is not derived from first principles in this paper; it is obtained by substituting Eq. (3), a same-author coherence-ensemble expectation, into Gelfand's formula via the trace. Thus the theorem's content reduces to the prior self-cited result plus algebra, and the empirical validation reuses the same 62 networks from that prior work instead of providing independent evidence.

full rationale

The paper's new theorem on non-normality is not itself a fitted prediction: the dF bounds are algebraic consequences once Eq. (6) is granted, and no free parameters are adjusted to the data. The circularity burden is that Eq. (6) is not independently derived here. It is obtained by applying Gelfand's formula to Eq. (3), a same-author coherence-ensemble result that is cited ('In Ref. [1] we present results...') rather than proved in this paper. All subsequent claims about spectral radius, strong connectivity, and non-normality inherit that citation, and the empirical plots use the same 62 networks on which the prior machinery was developed, so they do not provide an external test. The majority-rule q_c = 1 result is a self-contained mean-field calculation from the basal-ensemble definition, so it is not circular. The trace-as-Gelfand step is mathematically suspect (the trace is not a matrix norm, and the limit may not exist as written), but that is a correctness problem rather than a circularity and is not counted in the score. Overall, the central claim has independent content, but it is partly load-bearing on a self-citation; hence score 4.

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

The paper introduces no new physical entities. The loop exponent τ is a derived scalar, not a postulated entity. The main conceptual input is the coherence-ensemble formula (Eq. 3), carried over from the author's previous work; no new free parameters are fitted to data in this paper.

assumptions (5)

domain assumption At least one basal vertex is required for trophic levels (Eq. 1)
The definition of trophic levels via Eq. (1) requires Λ invertible, which fails if there are no basal vertices; the paper notes this.
domain assumption Coherence-ensemble expectation Eq. (3): tr(A^k) = (\tilde α \tilde q)/(α q) e^{τ k}
Imported from Ref. [1] without proof in this paper; it is the load-bearing input for the spectral radius formula and the non-normality bounds.
standard math Gelfand's spectral radius formula and eigenvalue bounds
Used in Eqs. (5)-(15) to bound d_F.
domain assumption Basal ensemble homogeneity: all non-basal vertices have the same ratio k_i^n/k_i^b = λ
Used in mean-field majority-rule analysis (Methods 0.2); only approximate for empirical or preferential-preying networks.
domain assumption Mean-field approximation |m_n| ≈ 1
Used to derive Eq. (26); the paper states this approximation.

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Cite this review

Pith. "Pith review of Digraphs are different: Why directionality matters in complex systems." pith.science (2026). https://pith.science/paper/3XKN4RRX

@misc{pith2026190807025,
  author       = {Pith},
  title        = {Pith review of: Digraphs are different: Why directionality matters in complex systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3XKN4RRX}},
  note         = {Machine review of arXiv:1908.07025}
}

read the original abstract

Many networks describing complex systems are directed: the interactions between elements are not symmetric. Recent work has shown that these networks can display properties such as trophic coherence or non-normality, which in turn affect stability, percolation and other dynamical features. I show here that these topological properties have a common origin, in that the edges of directed networks can be aligned - or not - with a global direction. And I illustrate how this can lead to rich and unexpected dynamical behaviour even in the simplest of models.

Figures

Figures reproduced from arXiv: 1908.07025 by the authors.

**Figure 1.** Two directed networks: the Ythan Estuary food web (A and B) and the the C. elegans neural network (C and D). In panels A and C, the height of each vertex corresponds to its trophic level. In panels B and D, networks are plotted according to a standard energy minimisation algorithm. In all cases, vertices belonging to the largest strongly connected component are represented as diamonds, while the rest appear as circl… view at source ↗

**Figure 2.** Fraction of non-basal vertices in the strongly connected component, Φ, against loop exponent, τ, for several empirical networks. Inset: Φ against q for networks generated with the ‘preferential preying model’ [12]. ignored. A weakly connected graph may have strongly connected subgraphs, and the largest of these is the ‘strongly connected component’ (SSC). Directed cycles are strongly connected subgraphs, and large s… view at source ↗

**Figure 3.** Normalised deviation from normality, dF , against loop exponent, τ, for several empirical networks. The lower (blue) line is Eq. (16) for hki = 1.23; the upper (red) line is Eq. (17) for L = 15317 (these are the lowest hki and highest L, respectively, found in the set of networks). The inset shows Eq. (16) for various different mean degrees: hki = 2, 3, 10, and 20. Trophic coherence and non-normality Using results f… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: A: Time series from simulations of the majority rule dynamics (described in the main text) on the C. elegans neural network (blue) and a randomisation of this network which preserves degree sequences (red). B: Majority rule dynamics on two networks generated with the p…

**Figure 4.** Figure 4: D shows average results of such simulations on preferentia [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

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