Criticality in spreading processes without time-scale separation and the critical brain hypothesis

Daniel J. Korchinski; Javier G. Orlandi; J\"orn Davidsen; Seung-Woo Son

REVIEW 3 major objections 5 minor 109 references

Criticality in spreading processes without time-scale separation and the critical brain hypothesis

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Spontaneous activation moves spreading transitions from directed to undirected percolation.

desk verdict Solid analytical core and a novel universality-class claim, but the result is conditional on the causal-web merging rule, which is not independently validated and is ambiguously described in the Methods. read the letter →

arxiv 1908.08163 v2 pith:GQK3A2U3 submitted 2019-08-22 cond-mat.stat-mech cond-mat.dis-nn

Daniel J. Korchinski , Javier G. Orlandi , Seung-Woo Son , Jörn Davidsen This is my paper · ORCID

classification cond-mat.stat-mechcond-mat.dis-nn PACS 64.60.ah05.70.Jk89.75.Hc

keywords spreadingprocessesdirectedpercolationundirecteduniversalityclassspontaneousactivationcriticalbrainhypothesiscausalwebsavalanchemerging

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

This paper argues that any nonzero rate of spontaneous activation changes the universality class of a network spreading transition: instead of directed percolation, the transition belongs to undirected percolation, while directed-percolation exponents survive only below a crossover scale set by $p^{-2/3}$. The authors prove this for the mean-field $k$-regular network with a pair of coupled generating functions, give the critical line $0 = k(1-\sigma)^2 - (k-1)\sigma\sigma_m$, and support it numerically on small-world, power-law, and hierarchical modular networks. If right, common markers of criticality, such as a branching ratio of one and a diverging dynamic susceptibility, misidentify the transition whenever spontaneous events occur, which matters for epidemic thresholds and for the critical brain hypothesis.

What carries the argument

The central object is the causal-web avalanche definition: nodes with no active parents start new clusters, and overlapping streams of activity merge their cluster labels, mapping the dynamics onto percolation clusters. The analytic argument uses two coupled probability generating functions, $H_p$ and $H_d$, for cluster sizes reached along daughter and parent branches; their divergence condition yields the critical line $0 = k(1-\sigma)^2 - (k-1)\sigma\sigma_m$, where $\sigma$ is the branching ratio and $\sigma_m$ the merging number. Singly rooted, mergeless avalanches are counted by Fuss-Catalan trees, giving an exponential cutoff $s_m$; combining $s_m \sim (1/k - q)^{-2}$ with the critical-line scaling $(1/k - q)^3 \sim p$ produces the $s_m \sim p^{-2/3}$ crossover that organizes the universal curve collapses.

What would settle it

On a $k$-regular network with $p=10^{-4}$, measure the critical avalanche size distribution at $q=q_c(p)$ out to sizes well above $p^{-2/3}$: if the central claim is right, the tail must follow $s^{-5/2}$ after an $s^{-3/2}$ regime with a sharp crossover near $s_m \sim p^{-2/3}$; observing a single pure directed-percolation power law at all scales would falsify the universality shift.

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Extended reading notes

Core claim

In the $\varepsilon$-SIS model with spontaneous activation probability $p$ and spreading probability $q$, adding any $p>0$ destroys the pure directed-percolation transition; the genuine critical line, traced by the divergence of the causal-web susceptibility, belongs to the undirected percolation universality class, with giant-component exponent $\beta=1$, susceptibility exponent $\gamma=1$, and correlation-length exponent $\nu=1$ on $k$-regular networks. Along this line the avalanche size distribution exhibits two power laws: the directed-percolation exponent $\tau=3/2$ for small clusters and the undirected value $\tau=5/2$ above a merging scale $s_m \sim p^{-2/3}$. Only the singular point $p=0$, $q=1/k$ remains in the directed-percolation universality class. The crossover between the two regimes is governed by merging of initially independent cascades, and standard criticality measures such as the branching ratio and the dynamic susceptibility no longer mark the true transition once $p>0$.

Load-bearing premise

The load-bearing premise is that avalanches defined by the causal-web merging rule, where nodes with no active parents start new clusters and overlapping clusters merge, are the physically correct way to separate independent cascades; if that construction mislabels causal structure, the critical line and universality class describe the clustering algorithm rather than the spreading dynamics.

Editorial extensions

If this is right

For any nonzero spontaneous activation rate, the epidemic or activity threshold is lowered, and a local reproduction number below one can still coexist with a giant component.
Directed-percolation exponents appear only in avalanches smaller than $s_m \sim p^{-2/3}$; larger avalanches and the giant component follow undirected percolation exponents.
The branching-ratio line $\sigma=1$ and the dynamic-susceptibility Widom line scale differently near the directed-percolation point, so they cannot locate the true critical point once $p>0$.
Neuronal avalanche exponents in the range 1.2 to 2.5 can arise from the two-regime distribution near criticality, so fitting a single power-law exponent to data is an unreliable test of sub- or super-critical state.
A variant with permanent immunity (SIR-like) also shows the two-power-law structure, extending the merging mechanism to diseases with no reinfection.

Reading between the lines

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

If the causal-web observable is the right one, whole-brain recordings paired with tractography should show the $p^{-2/3}$ crossover directly: as the spontaneous rate rises, the tail exponent of the avalanche distribution should move from approximately $3/2$ toward $5/2$ at a size controlled by $p^{-2/3}$.
The merging mechanism is general: any driven branching process with a background initiation rate should exhibit the same shift, so the result likely extends beyond this model to rumor spreading, malware propagation, and neural cultures with external input.
A direct way to test the claim against alternative observables is to reanalyze the same simulated activity using avalanche definitions based on temporal bins or global quiet periods; if those definitions recover pure directed-percolation exponents at all scales for $p>0$, the universality shift is a property of the causal-web clustering rather than of the underlying dynamics.

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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 thing to know about this paper is that it makes a sharp, testable claim: introduce any spontaneous activation p>0 into a spreading process and the critical transition moves from directed-percolation to undirected-percolation universality, with directed-percolation exponents surviving only below a merging scale sm ~ p^{-2/3}. The second thing to know is that the claim is tied to the causal-web merging rule used to define avalanches, and that observable is load-bearing.

What is genuinely new: the analytical critical line for k-regular networks, Eq. (2), and the two-power-law structure with the p^{-2/3} crossover. The generating-function derivation in Appendix B is internally consistent and gives exact exponents (β=1, γ=1, ν=1) that match simulations on infinite and finite graphs. The finite-size scaling using the B-ratio is a solid, reproducible method, and the curve collapses are convincing. The paper also does useful service by showing that the branching ratio and dynamic susceptibility no longer identify the transition once p>0, and it distinguishes the Widom line, σ=1 line, and true critical line with different power laws near the directed-percolation limit. The Discussion is honest about needing network structure to recover the exponents.

The soft spots are real but not fatal. The main one is the merging rule itself. The Methods text is ambiguous about the fate of a node that activates spontaneously and also has active parents: it says such nodes initiate a new cluster only if they have no active parents, then says nodes with active parents that were not already spontaneously activated are checked for activation. That leaves a gap. The analytical PGFs implicitly merge such nodes into parent clusters, so if the simulations do something different, the theory-simulation agreement would break. Without code or data, this cannot be checked. The stress-test worry that the whole universality-class claim is an artifact of labeling is probably too strong—the q=0 endpoint is unaffected, and the agreement for p>0 is good—but the ambiguity should be fixed. Second, Table I reports several fitted exponents without error bars, and the hierarchical modular network shows a p-dependent Griffiths regime that complicates the universal p^{-2/3} scaling. These are minor-to-moderate issues.

Who gets value: people working on nonequilibrium percolation, epidemic reintroduction, and the critical brain hypothesis. It deserves a serious referee. The analytical core is solid, and the claim is important enough to warrant revision rather than rejection. I would recommend accepting it for review and pushing for a clarified merging rule, error bars on exponents, and code/data release.

Referee Report

3 major / 5 minor

Summary. The paper studies a discrete-time epsilon-SIS spreading model with spontaneous activation probability p and network-borne transmission probability q on directed networks. Because p>0 removes time-scale separation and produces overlapping independent cascades, the authors define avalanches through a causal-web merging rule: nodes without active parents are roots of new avalanches, and nodes with active parents inherit and merge cluster labels. Their central claim is that for any p>0 the transition is no longer in the directed-percolation universality class but instead in the undirected-percolation universality class, with directed-percolation exponents surviving only below a merging crossover scale s_m ~ p^{-2/3}. For k-regular networks they derive, via a pair of coupled probability generating functions, the critical line 0 = k(1-σ)^2 - (k-1)σσ_m (Eq. 2), the exponents β=1, γ=1, ν=1, and the crossover scaling. Numerical simulations on small-world, power-law, and hierarchical modular networks show two power-law regimes in the critical avalanche-size distribution and finite-size scaling of the giant component. The paper concludes that standard criticality markers such as the branching ratio and dynamic susceptibility fail in the absence of time-scale separation, and it draws implications for the critical brain hypothesis and for zoonotic disease spreading.

Significance. If the central claim holds, the paper is significant: it identifies a concrete mechanism—merging of independently initiated avalanches—by which a genuinely non-equilibrium directed-percolation transition is converted, for all p>0, into a transition with undirected-percolation exponents, and it makes a falsifiable prediction about two power-law regimes separated by a p-dependent crossover. The analytical generating-function derivation for k-regular networks is a genuine strength: Eq. (2) is derived from the model rather than fitted, and the resulting exponents are checked against simulations on infinite and finite networks. The paper also gives a clear, parameter-light prediction for the critical line and crossover, and the causal-web susceptibility offers a concrete observable for brain-avalanche analysis. The main limitation is that the universality-class claim is established for the particular avalanche-labeling observable defined by the causal-web merging rule, and the evidence for the generality of the undirected-percolation class across non-regular topologies rests on a small number of exponents, several reported without uncertainties.

major comments (3)

[Section II.A and Appendix A (Simulation of model on finite networks)] The causal-web merging rule is the load-bearing observable of the paper, but its relation to a direct physical separation of avalanches is not validated. The finite-network simulation paragraph specifies that spontaneous nodes with no active parents initiate a new cluster, while the general description in Section II.A states that nodes with active parents inherit labels; however, the label assignment for the case of a spontaneously activated node that also has active parents is not explicitly stated in the Methods paragraph. Since the q=0 endpoint, the critical line Eq. (2), and the p^{-2/3} crossover all depend on this labeling choice, the authors should either state the rule unambiguously and test its sensitivity (for example, by comparing with an alternative rule in which such nodes seed new clusters) or compare the resulting clusters directly with the causal-webs algorithm of reference [30]. The Discussion's admission that 'there is currently no way to recover the correct exponents without access to the network structure' makes the observable-dependence of the claim explicit but does not by itself justify it.
[Table I and Section II.B] The claim that the entire p>0 critical line belongs to the undirected-percolation universality class for each network topology rests on very few fitted exponents. For the small-world and hierarchical modular networks, Table I reports τ≈2.5 and τ≈2.1, β≈1.0 and β≈0.8, with no uncertainties or goodness-of-fit measures; only the k-regular network has a full analytical exponent set. A two-power-law fit of P(s) and a single β estimate are not sufficient to distinguish undirected percolation from an effective exponent, especially on networks where the directed-percolation and undirected-percolation exponents are close (e.g., small-world τ values). The authors should report exponent estimates with uncertainties and provide at least one additional independent exponent (such as γ or the finite-size exponent 1/ν) for at least one non-regular topology before claiming a full universality class.
[Section II.C, Eq. (3), Fig. 3, and Table II] The universal p^{-2/3} merging crossover is derived from the mean-field phase-line exponent a=3, but the numerically fitted phase lines in Table II are not all consistent with a=3. In particular, the hierarchical modular network has a≈1.94, and the same argument that gives s_m ~ p^{-2/3} from δ ~ p^{1/3} would give s_m ~ p^{-2/a} ≈ p^{-1} for this network, while the small-world network with rewire probability 10^{-3} is reported in Appendix F to collapse with s_m ~ p^{-0.75}. The statement in Section II.B that 'all critical-avalanche distributions exhibit a universal curve collapse ... by re-scaling the distribution by p^{-2/3}' is therefore internally inconsistent with the fitted phase lines for these topologies. The authors should either justify the p^{-2/3} collapse independently of Eq. (3), or restrict the p^{-2/3} statement to mean-field networks and state the expected topology-dependent exponents explicitly.

minor comments (5)

[Appendix B, Eq. (B11)] The sentence introducing complementary probabilities says 'using the notation p = 1− p', which is a typographical error; it should introduce a distinct symbol such as \bar p for the complementary probability, since p is already the spontaneous activation rate.
[References] Reference [64] duplicates reference [30] (Williams-Garcia, Beggs, and Ortiz, 'Unveiling causal activity of complex networks'); one of the two entries should be removed or replaced with the appropriate distinct citation.
[Table I and Table III] Several numerical exponents in Tables I and III are reported without error bars (for example, τ≈1.35 and β≈1.0 for the small-world network, α≈7 for durations), while the paper elsewhere reports uncertainties such as 0.36(2); a consistent policy of reporting uncertainties for all fitted exponents would strengthen the quantitative claims.
[Figure 4d and Appendix A] The caption of Fig. 4d says 'Solid line is g ∼ (q−qc)β for β=2', but the text in Section II.B reports β=2 for the power-law network while the finite-size scaling in panel d uses N^{0.50} and (q−q_c)N^{0.25}; clarifying whether the solid line is the infinite-size limit or a finite-size collapsed curve would avoid confusion.
[Section II.C, paragraph after Eq. (2)] The phrase 'the σ=1 line scales as p∼(1/k−q)^2' and the later statement that the Widom line scales as p∼(1/k−q) are derived only in the appendices; adding a one-sentence pointer to Appendix E at the first occurrence would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the universality-class claim is derived from the model and an external causal-web observable, not from fitted inputs or self-citation.

full rationale

The paper's central claim is derived from the dynamics in Eq. (1) together with the causal-web cluster rule imported from Williams-Garcia, Beggs and Ortiz (ref. [30]), none of whom are authors of the present paper. The generating-function calculation in Appendix B is a self-contained tree-level computation: it defines H0, Hp and Hd, derives the critical-line condition from the vanishing determinant in Eq. (B10), and obtains the exponents gamma=1, beta=1 and nu_perp=1 analytically. These results are not fitted to the quantities they are used to predict. The undirected-percolation endpoint q=0, p=1/(2k-1) is a known limiting case used as a consistency check, not an input that forces the entire critical line; the intermediate critical line, the p^{-2/3} crossover, and the two-power-law avalanche distribution follow from Eqs. (B27), (3), (C2)-(C4) and are verified by finite-size scaling and curve collapse. The merging rule is definitional, but it does not by itself determine the exponents; the PGF analysis is needed to obtain the critical condition and the distribution shapes. Whether causal webs are the physically appropriate avalanche definition is a modeling and validation question, not a circularity. No load-bearing self-citation is present, and the analytical derivation stands independently of the paper's own conclusions.

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

The analytical central claim for k-regular networks uses only model parameters (p, q, k) and standard generating function techniques; no free parameters are fitted in that derivation. The listed free parameters are numerical fits used to characterize the critical line and scaling collapses for specific networks. The key assumptions are the tree-like approximation and the causal-web cluster definition.

free parameters (2)

Merging size scaling exponent b (small-world, rewire 10^-3) = 0.75
Determined by best curve collapse of avalanche distributions (Fig. 12d); used to support the sm ~ p^{-b} scaling claim for this network.
Critical line exponent a (hierarchical modular network) = 1.94
Fit of numerical critical line p = c(q_c,DP - q)^a in Table II; used to characterize the phase line.

assumptions (2)

domain assumption Tree-like approximation (no loops and node equivalence) for the analytical generating function derivation.
Section II.C states 'the technique's key assumption is that there are no loops and that all nodes are equivalent'; this underlies the PGF equations and the derived critical line.
domain assumption Causal-web cluster merging is the correct observable for identifying independent avalanches without time-scale separation.
Section II.A defines avalanches via causal webs; the entire critical line and universality class claim are properties of this cluster definition.

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Pith. "Pith review of Criticality in spreading processes without time-scale separation and the critical brain hypothesis." pith.science (2026). https://pith.science/paper/GQK3A2U3

@misc{pith2026190808163,
  author       = {Pith},
  title        = {Pith review of: Criticality in spreading processes without time-scale separation and the critical brain hypothesis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GQK3A2U3}},
  note         = {Machine review of arXiv:1908.08163}
}

read the original abstract

Spreading processes on networks are ubiquitous in both human-made and natural systems. Understanding their behavior is of broad interest; from the control of epidemics to understanding brain dynamics. While in some cases there exists a clear separation of time scales between the propagation of a single spreading cascade and the initiation of the next -- such that spreading can be modelled as directed percolation or a branching process -- there are also processes for which this is not the case, such as zoonotic diseases or spiking cascades in neural networks. For a large class of relevant network topologies, we show here that in such a scenario the nature of the overall spreading fundamentally changes. This change manifests itself in a transition between different universality classes of critical spreading, which determines the onset and the properties of an avalanche turning epidemic or neural activity turning epileptic, for example. We present analytical results in the mean-field limit giving the critical line along which scale-free spreading behaviour can be observed. The two limits of this critical line correspond to the universality classes of directed and undirected percolation, respectively. Outside these two limits, this duality manifests itself in the appearance of critical exponents from the universality classes of both directed and undirected percolation. We find that the transition between these exponents is governed by a competition between merging and propagation of activity, and identify an appropriate scaling relationship for the transition point. Finally, we show that commonly used measures, such as the branching ratio and dynamic susceptibility, fail to establish criticality in the absence of time-scale separation calling for a reanalysis of criticality in the brain.

Figures

Figures reproduced from arXiv: 1908.08163 by the authors.

**Figure 2.** FIG. 2. Approach to the phase transition in directed perco [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** FIG. 4. Finite size scaling effects on giant component size [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Figures from the paper (11 more)

**Figure 5.** Figure 5: FIG. 5. Power-law transitions are governed by merging. [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 7.** Figure 7: FIG. 7. A firing pattern example represented both as the sum of [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 8.** Figure 8: FIG. 8. An example of the three size generating functions. To [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

**Figure 9.** Figure 9: FIG. 9. Average cluster size for [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]

**Figure 11.** Figure 11: FIG. 11. Power-law scaling of the critical and quasi-critic [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]

**Figure 12.** Figure 12: FIG. 12. Critical line and avalanche scaling of small-world [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]

**Figure 13.** Figure 13: FIG. 13. Perpendicular correlation length near criticalit [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]

**Figure 14.** Figure 14: FIG. 14. Isotropic correlation length near criticality for [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]

**Figure 15.** Figure 15: FIG. 15. Cluster size distributions on undirected 10-regul [PITH_FULL_IMAGE:figures/full_fig_p022_15.png]

**Figure 16.** Figure 16: FIG. 16. Unscaled critical avalanche distributions. [PITH_FULL_IMAGE:figures/full_fig_p023_16.png]

**Figure 17.** Figure 17: FIG. 17. Exponent transitions for critical avalanches, wit [PITH_FULL_IMAGE:figures/full_fig_p023_17.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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To proceed further, we must sup- ply Ai/o and Bi/o for a given network

Neighbour generating functions for k-regular networks So far, we’ve been quite generic in developing the gen- erating function H0. To proceed further, we must sup- ply Ai/o and Bi/o for a given network. For simplicity, we focus on the k-regular network. This will allow us to develop expressions for Φ, the active fraction, and Pd, the probability that the ...
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Pith tools

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

[1] [30]

Ponce-Alvarez, A

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work page 2018

[2] [1]

To proceed further, we must sup- ply Ai/o and Bi/o for a given network

Neighbour generating functions for k-regular networks So far, we’ve been quite generic in developing the gen- erating function H0. To proceed further, we must sup- ply Ai/o and Bi/o for a given network. For simplicity, we focus on the k-regular network. This will allow us to develop expressions for Φ, the active fraction, and Pd, the probability that the ...

[3] [2]

Practically speaking, we solve Eqs

Observables from the generating function Here, we summarize how to extract observables, such as the size fraction of the giant component g, suscep- tibility χ , and cluster distribution Pc(s) from the gen- erating function H0(x). Practically speaking, we solve Eqs. (B3) and ( B4) self-consistently for Hd(x) and Hp(x) via a Newton-Raphson scheme for a give...

[4] [3]

( B10), and inserting the solution into Eq

Phase-diagram for the k-regular network We can study the divergence of χ ∼⟨ s⟩n by solving Eq. ( B10), and inserting the solution into Eq. ( B9) to obtain ⟨s⟩n = 1− σ k (σ− σm) (1− σ )2− k−1 k σσ m (B27) where σm = A′ i(1) = ( k − 1)Pp1 = ( k −

[5] [4]

That is, σm describes the rate of merging of ini- tially independent clusters

Φ Pd ( 1− (1−Pd)2 1−Φ ) is the expected number of other ac- tive parents, to an active node with one already known parent. That is, σm describes the rate of merging of ini- tially independent clusters. Clearly, ⟨s⟩n diverges when k(1− σ )2− (k− 1)σσ m = 0 (Eq. (

[6] [5]

This result could also have been arrived at by setting the determinant of Eq

of the main text). This result could also have been arrived at by setting the determinant of Eq. ( B10) to zero. A reparameterization that will be convenient when considering the correlation length is to replace σm with β = (k−1)σ m kσ , meaning that the critical line diverges when (1− σ )2 = σ 2β . (B28) The set of ( pc, q c) that cause this divergence d...

[7] [6]

At the critical point, Hp(1) = Hd(1) = 1

The giant component The giant component fraction g is given by g = Φ(1 − H0(1)) = Φ ( 1− Hd(1) [ Pd + PdHp(1) ]) . At the critical point, Hp(1) = Hd(1) = 1. So for δ = q−qc≪ 1, we have that g ≈ ( Φ Pd ∂H p ∂q + Φ ∂H d ∂q ) δ. Since both Hd(1) and Hp(1) are strictly decreasing functions of q, g∼ (q− qc) identifying the critical exponent β = 1. Appendix C: ...

[8] [7]

The excluded nodes on the perimeter occur with probability Pd, which is the probability of not activating, despite having an ac- tive parent

Nodes are included in the tree with probability Pd1 = Φ k−1 (1− p q) , (C1) denoting the probability that a given daughter node is activated while having exactly one parent. The excluded nodes on the perimeter occur with probability Pd, which is the probability of not activating, despite having an ac- tive parent. Hence, the probability of observing a mer...