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REVIEW 2 major objections 5 minor 38 references

Critical and Nonpercolating Phases in Bond Percolation on the Song-Havlin-Makse Network

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Bond percolation on the small-world Song–Havlin–Makse tree is critical for every bond probability, while on the finite-dimensional tree it never percolates; dimensionality, not degree heterogeneity, is the controlling factor.

desk verdict Exact fractal exponents for the two SHM tree limits are clean and well-checked; the p-dependent cluster-size exponent is imported from prior work, a support gap that is fixable and not fatal. read the letter →

arxiv 2505.22166 v3 pith:M6NEWORC submitted 2025-05-28 cond-mat.stat-mech cond-mat.dis-nn

Kazuki Wataya , Takehisa Hasegawa This is my paper
classification cond-mat.stat-mechcond-mat.dis-nn MSC 82B4382B27
keywords bondpercolationcriticalphaseSong-Havlin-Maksenetworkscale-freetreefractalexponentgeneratingfunctionsmall-worldfinite-sizescaling
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 asks what structural feature of an inhomogeneous tree decides whether bond percolation has a critical phase, a range of bond probabilities in which clusters follow scale-free power laws rather than being finite or giant. On the Song–Havlin–Makse (SHM) tree, a recursively built scale-free tree whose dimensionality can be tuned, the authors derive the average root-cluster size and its fractal exponent using generating functions. They find that when the SHM tree is small-world (infinite-dimensional), the fractal exponent $\psi(p)$ is continuous in $p$, placing the system in a critical phase for every $0

What carries the argument

The central objects are two generating functions $W_n(x)$ and $V_n(x)$ that encode the size distribution of the root cluster in generation $n$, split according to whether the two original roots are connected by retained bonds ($W_n$) or not ($V_n$); recursive equations (16)-(17) for $p_G=0$ and (32)-(33) for $p_G=1$ express these functions in terms of the previous generation and carry the whole calculation. The derived quantity that does the phase classification is the fractal exponent $\psi(p)$, defined by the scaling $R_n(p)\sim N_n^{\psi(p)}$ of the average root-cluster size, with $\psi=0$, $0<\psi<1$, and $\psi=1$ marking nonpercolating, critical, and percolating phases. The relation $\tau(p)=1+1/\psi(p)$ (Eq. 46), taken from the enhanced binary tree model, converts $\psi$ into the exponent of the cluster size distribution $n(s)\sim s^{-\tau(p)}$ used to characterize the critical phase.

What would settle it

On the small-world SHM tree, fit the cluster-size distribution $n(s)$ directly from Monte Carlo data at, say, $p=0.2$ and $p=0.8$ across generations $n=8,10,12$, without using the theoretical $\psi(p)$ collapse, and compare the fitted slopes with $1+1/\psi(p)$. If the slopes do not vary with $p$, or do not approach the predicted values as $n$ grows, the central claim of a p-dependent critical phase is falsified.

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Extended reading notes

Core claim

For the deterministic small-world SHM tree (rewiring probability $p_G=0$), the generating-function recurrences give the average root-cluster size $R_n(p)=1+p(1+m+mp)^{n-1}$ (Eq. 23), so the fractal exponent is $\psi(p)=\ln(1+m+mp)/\ln(1+2m)$ (Eq. 25). This lies strictly between $0$ and $1$ for all $0<p<1$ and varies continuously with $p$, which the authors identify as a critical phase throughout the whole range, with $p_{c1}=0$ and $p_{c2}=1$. For the fractal SHM tree ($p_G=1$), the same method yields $\psi(p)=\ln m/\ln(2m+1)$ (Eq. 42), independent of $p$ and equal to the exponent $\alpha$ governing the growth of the maximum degree. The authors interpret this constant nonzero exponent as coming from the large number of nodes directly attached to the root rather than from a cluster that extends through the network, and they classify the fractal tree as nonpercolating for all $p<1$. Monte Carlo simulations for the intermediate case $p_G=0.5$ behave like the fractal case, supporting the paper's conclusion that finite versus infinite dimensionality, not the degree exponent, is the decisive factor.

Load-bearing premise

The p-dependent cluster-size exponent that defines the small-world critical phase is carried by the finite-size scaling ansatz together with the relation $\tau(p)=1+1/\psi(p)$ imported from the enhanced binary tree model; if that relation does not transfer to the SHM tree, the paper's own derivations do not establish that the cluster-size exponent actually varies with $p$.

Editorial extensions

If this is right

  • On the small-world SHM tree the giant component never forms for $p<1$; the system is critical for every $0<p<1$, with the cluster-size exponent $\tau(p)=1+1/\psi(p)$ varying continuously with $p$.
  • On the fractal SHM tree both critical points coincide at $p=1$: for every $p<1$ the system is nonpercolating, and the constant fractal exponent reflects the root's huge degree rather than a cluster that reaches far into the tree.
  • Changing the branching parameter $m$, which changes the degree exponent $\gamma$ and the fractal dimension, does not alter this dichotomy.
  • For intermediate rewiring probability $0<p_G<1$, simulations show the same nonpercolating behavior, with the fractal exponent equal to the maximum-degree exponent $\alpha$.
  • The correlation-volume argument gives a mechanism: in a tree the two-point correlation decays as $p^l$, so a critical phase can appear only when the number of nodes reachable at distance $l$ grows exponentially fast, as it does in the small-world case.

Reading between the lines

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

  • The correlation-volume heuristic implies a sharper, testable prediction for other recursive trees: any tree whose number of reachable nodes grows subexponentially with distance should have no critical phase, while exponential growth of reachable nodes should allow $p_{c1}<1$.
  • Because the relation $\tau(p)=1+1/\psi(p)$ is imported from another tree model, the most direct confirmation of the small-world result would be an independent power-law fit of the cluster-size distribution in simulations, without using the theoretical $\psi(p)$ collapse.
  • If the geometric distinction is the real mechanism, the same dichotomy should appear in other models placed on the SHM tree, such as an Ising model, with qualitatively different relaxation behavior in the small-world and fractal cases.
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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 / 5 minor

Summary. The paper analyzes bond percolation on the Song–Havlin–Makse (SHM) tree, a scale-free recursive tree with rewiring probability pG, using a generating function approach. For the small-world case pG=0, it derives the average root cluster size R_n(p)=1+p(1+m+mp)^{n-1} and the fractal exponent ψ(p)=ln(1+m+mp)/ln(1+2m), which lies in (0,1) for 0<p<1. For the fractal case pG=1, it derives R_n(p)=1+p∏_{n'=1}^{n-1}[m+p^{3^{n'-1}}+p^{2·3^{n'-1}}+(m-1)p^{3^{n'}}] and ψ(p)=ln m/ln(2m+1), independent of p. Monte Carlo simulations confirm the formulas. The authors conclude that the small-world tree is in a critical phase for all 0<p<1 (pc1=0, pc2=1), the fractal tree in a nonpercolating phase for all p<1, and that dimensionality, not degree heterogeneity, controls the appearance of the critical phase.

Significance. The exact recurrence solutions are elegant and the Monte Carlo agreement is excellent, providing a rare analytic handle on percolation in scale-free trees. The identification of the root-degree artifact (nonzero but constant ψ) and its quantification via the exponent α in Eq. (47) is careful and useful. If the classification is correct, the paper would establish a clean principle: infinite-dimensionality is necessary for a critical phase in tree networks. The main caveat is that the p-dependent cluster-size exponent τ(p), a headline feature, is not independently derived or measured but imported from Ref. [11] via Eq. (46).

major comments (2)
  1. [Section IV, Eq. (46) and Fig. 6] The abstract's claim that the cluster size distribution is a power law with a p-dependent exponent is not established by the paper's own derivations. Equation (46), τ(p)=1+ψ^{-1}(p), is imported from Ref. [11] rather than derived for the SHM tree, and the collapse in Fig. 6 rescales the axes using the theoretically computed ψ(p) from Eq. (25), so it presupposes Eq. (46). Because the generating functions in Eqs. (27)-(29) give n(s) exactly, τ(p) can be measured directly from the slopes of n(s) or from a scaling collapse that treats τ(p) as a free parameter; without such a check, the p-dependence of τ(p) remains an assumption rather than a result.
  2. [Section IV, Eqs. (42) and Fig. 8] The conclusion that the fractal SHM tree is in a nonpercolating phase for all p<1 conflicts with the definitions used earlier in the paper. Section I defines the critical phase via the existence of infinitely many infinite clusters, and Section III defines the nonpercolating phase via ψ=0; for pG=1, Eq. (42) gives ψ(p)=α>0, and Eq. (39) implies the root cluster size diverges as N^α, so in the n→∞ limit the root cluster is infinite for every p>0. The observation in Fig. 8 that the root cluster's density decays with distance l does not show that the cluster is finite; the cluster may still contain nodes at arbitrarily large distances. The authors should either prove that no infinite clusters exist away from the hub, or base the nonpercolating-phase claim on a consistently defined order parameter (e.g., the largest finite cluster or the typical cluster size), rather than the special root cluster.
minor comments (5)
  1. [Abstract and Section IV] Please specify the range 0<p<1 for the small-world critical phase; at exactly p=0 the root cluster is a single node and ψ is undefined, so the phrase 'entire range of p' is imprecise.
  2. [Section III, Eq. (29)] The definition of \tilde{U}_n(x) as \sum_s n(s)x^s for per-node n(s) and Eq. (29) with the factor 1/N_n are inconsistent; either \tilde{U}_n(x) counts total clusters (in which case the text should not call it the generating function of n(s)) or the 1/N_n factor should be removed.
  3. [Section III, Eq. (24)] The cancellation of p in the finite-size estimate ψ_n(p) requires p>0; please state this restriction explicitly.
  4. [Section IV] The statement that similar results hold for varying m '(data not shown)' is not verifiable; consider including a supporting figure or moving this claim to a supplementary file.
  5. [Figure 5] The generating-function curves for pG=0.5 in panel (c) are numerical evaluations, not closed-form analytic results; the text should distinguish these from the analytic curves for pG=0 and pG=1.

Circularity Check

1 steps flagged · score 4.0 of 10

The exact ψ(p) derivations are self-contained and Monte-Carlo-validated, but the abstract's p-dependent cluster-size exponent τ(p) is not established here: it comes from the assumed relation τ = 1 + 1/ψ (Eq. 46) cited to Ref. [11], co-authored by the present author, and the Fig. 6 collapse that 'confirms' it uses that relation to fix the rescaling rather than measuring τ(p).

  1. self citation load bearing [Sec. IV, Eqs. (44)-(46) and the discussion of Fig. 6; abstract.]
    "We further assume that the exponent τ(p) is related to the fractal exponent ψ(p) via τ(p) = 1 + ψ^{-1}(p), (46) as reported in Ref. [11]. ... The data collapse onto a single curve for each value of p, confirming the validity of the scaling form. This implies that, in the large size limit, the cluster size distribution n(s) follows a power law of the form n(s) ∝ s^{-τ(p)}."

    The abstract's headline claim—that the small-world SHM tree exhibits a critical phase 'where the cluster size distribution follows a power-law with a p-dependent exponent'—is quantified only by inserting the derived ψ(p) (Eq. 25) into the assumed relation (46), whose cited basis is Ref. [11], a paper co-authored by the present author (T. Hasegawa); relation (46) is not derived here for the SHM tree. The Fig. 6 collapse rescales the ordinate as n≥(s)·N_n, i.e. N^{ψ(τ−1)} with ψ(τ−1)=1 by (46), so it presupposes the relation and cannot independently determine τ(p); and although n(s) is exactly computable from the generating functions (27)-(29), no direct slope fit of τ(p) is reported.

full rationale

The core analytic content is self-contained and externally validated: the recurrences (16)-(17) and (32)-(33) are derived from the recursive SHM construction, yield exact root-cluster sizes (22)-(23) and (39)-(40) and fractal exponents (25) and (42), and agree with independent Monte Carlo results (Figs. 4, 5, 7). The small-world critical-phase claim (pc1=0, pc2=1) follows from the derived ψ(p) ∈ (0,1) with ψ(p) p-dependent, and the fractal-tree nonpercolating conclusion is supported by the distance-resolved root-cluster data (Fig. 8); both retain content beyond the cited framework of Refs. [11,14,15]. The one load-bearing self-citation is Eq. (46), τ(p) = 1 + 1/ψ(p), 'as reported in Ref. [11]' (Nogawa and Hasegawa 2009, the latter a present author): the abstract's 'power-law with a p-dependent exponent' becomes quantitative only through this assumed relation, and the Fig. 6 collapse uses (46) to set its y-rescaling, so it is a consistency test rather than an independent measurement of τ(p). The exact n(s) computation (Fig. 5a) does show slopes varying with p, which independently supports the qualitative claim; the specific exponent value is what remains assumed. Two correctness risks, distinct from circularity and flagged for completeness: the stated classification (0<ψ<1 ⇒ critical) is contradicted by the fractal tree, which has ψ = ln m/ln(2m+1) ∈ (0,1) yet is declared nonpercolating; and the claim that the correlation volume 'does not diverge' in finite-dimensional trees is in tension with the paper's own Eq. (40), which gives R_n(p) → ∞ as N_n^α. Verdict: partial circularity on the quantitative exponent claim, score 4.

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

No free parameters are fitted in this paper: m, pG, and N1 are structural model inputs and p is the control variable; the ψ(p) formulas are derived, not fit. The load-bearing premises are the adopted phase-classification framework and τ-ψ scaling relation from the authors' prior work (Refs. [11, 14, 15]), plus the correlation-volume heuristic used to extend the results beyond the two exactly solved deterministic cases. No new entities are postulated.

assumptions (5)
  • domain assumption SHM network construction defines the model: recursion with branching m and rewiring probability pG yields a scale-free tree with degree exponent γ = 1 + log(2m+1)/log(m+1-pG), finite-dimensional for pG > 0 and small-world for pG = 0.
    Model definition in Sec. II; the dimensionality and degree-exponent relations are taken from Refs. [26, 27, 29].
  • domain assumption Phase classification by the fractal exponent ψ: ψ = 0 nonpercolating, 0 < ψ < 1 critical, ψ = 1 percolating.
    Adopted in Sec. III from Refs. [11, 14, 15], two of which share an author with this paper; used to convert root-cluster scaling into phase labels.
  • domain assumption Cluster size distribution obeys the finite-size scaling form of Eqs. (44)-(45) with τ(p) = 1 + 1/ψ(p).
    Imported from Ref. [11], a self-authored reference; the Fig. 6 data collapse is consistent with the scaling variable s/N^ψ but does not independently fix τ.
  • domain assumption Correlation-volume principle for trees: pair correlations decay as p^l, so the correlation length does not diverge for p < 1 (pc2 = 1), and a critical phase requires the number of reachable nodes to grow exponentially with distance l.
    Stated in Sec. IV with attribution to Refs. [15, 11]; this is the load-bearing heuristic for generalizing the nonpercolating classification to all finite-dimensional trees.
  • standard math Standard generating function calculus and uniqueness of paths in trees (two roots connected iff the single path between them is fully retained).
    Basis for recurrences (16)-(21) and the exact result P_n = p for the small-world tree.

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Pith. "Pith review of Critical and Nonpercolating Phases in Bond Percolation on the Song-Havlin-Makse Network." pith.science (2026). https://pith.science/paper/M6NEWORC

@misc{pith2026250522166,
  author       = {Pith},
  title        = {Pith review of: Critical and Nonpercolating Phases in Bond Percolation on the Song-Havlin-Makse Network},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M6NEWORC}},
  note         = {Machine review of arXiv:2505.22166}
}
abstract

We investigate bond percolation on the Song-Havlin-Makse (SHM) network, a scale-free tree with a tunable degree exponent and dimensionality. Using a generating function approach, we analytically derive the average size and the fractal exponent of the root cluster for deterministic cases. Our analysis reveals that bond percolation on the SHM network remains in a nonpercolating phase for all $p < 1$ when the network is fractal (i.e., finite-dimensional), whereas it exhibits a critical phase, where the cluster size distribution follows a power-law with a $p$-dependent exponent, throughout the entire range of $p$ when the network is small-world (i.e., infinite-dimensional), regardless of the specific dimensionality or degree exponent. The analytical results are in excellent agreement with Monte Carlo simulations.

Figures

Figures reproduced from arXiv: 2505.22166 by the authors.

Figure 1
Figure 1. FIG. 1: Illustration of the construction of the SHM network fr [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Examples of the SHM network [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Recursive construction of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4: (a) ¯s [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Cumulative cluster size distribution [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Finite-size scaling of the cumulative cluster size d [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: (b), both the theoretical curve and the Monte Carlo results indicate that ψn(p) converges to a constant value in the large size limit, specifically ψ = log 2/ log 5 for all 0 < p < 1. Although the fractal exponent ψ(p) is nonzero, we conclude that bond percolation on t…
Figure 8
Figure 8. Figure 8: FIG. 8: Distribution of [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

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Reviewed August 7, 2026 · model on record in the stance chip above.