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REVIEW 1 major objections 2 minor 33 references

Sandpile Models on complex networks

T0 review · 1 major / 2 minor · reviewed 2026-07-03 · grok-4.3

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Pith's one-line read Dissipation in sandpile models on networks produces exponential cutoffs in avalanche sizes while network topology continues to set the scaling.

desk verdict Dissipation adds exponential cutoffs to sandpile avalanches on networks and clustering shifts the exponent, but the branching approximation still needs checking on cycles. read the letter →

arxiv 2607.02023 v1 pith:3X6QVSVL submitted 2026-07-02 cond-mat.stat-mech nlin.AO

Komlan Fiagbe , Jean-François de Kemmeter , Timoteo Carletti This is my paper · ORCID
classification cond-mat.stat-mechnlin.AO
keywords sandpilemodelcomplexnetworksavalanchesizedistributiondissipationbranchingprocessclusteringscale-freecriticalphenomena
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

The paper develops a branching-process description that inserts grain loss directly into the offspring distribution to handle dissipative sandpile dynamics on complex networks. This yields generalized generating functions whose predictions match simulations on sparse random graphs but show systematic shifts when short cycles or low edge density are present. Increasing clustering is shown to reduce the avalanche-size exponent and raise the probability of large events. A reader would care because the results set quantitative limits on when independent-branch approximations can be trusted for real networked systems that always contain both dissipation and cycles.

What carries the argument

Generalized generating functions formed by inserting grain-loss directly into the offspring distribution of a branching process.

What would settle it

A direct comparison of the predicted avalanche-size exponent versus measured exponent on Holme-Kim networks as the clustering coefficient is varied from zero to its maximum value.

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Extended reading notes

Core claim

In the dissipative regime, avalanche-size distributions acquire exponential cutoffs while preserving topology-dependent scaling behavior; increasing clustering continuously lowers the avalanche exponent and enhances the probability of large cascades. The generalized generating functions obtained by inserting grain-loss directly into the offspring distribution remain the central theoretical tool, yet numerical checks on Holme-Kim clustered scale-free networks and on trees reveal that short cycles and leaf nodes produce strong correlations that invalidate the classical independent-branch approximation.

Load-bearing premise

The generalized generating functions obtained by inserting grain-loss directly into the offspring distribution remain accurate on networks containing short cycles.

Editorial extensions

If this is right

  • Avalanche distributions on sparse random networks follow the predicted exponential cutoffs.
  • Higher clustering in scale-free networks lowers the power-law exponent and increases the weight of the largest events.
  • Even loop-free trees deviate from pure power-law behavior because leaves and low edge density limit propagation.
  • Short cycles create correlations that break the independent-branch assumption used in earlier models.

Reading between the lines

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

  • Models of real infrastructure or epidemic spread that ignore both dissipation and clustering will mis-estimate the frequency of extreme events.
  • The same offspring-distribution modification could be applied to other threshold-driven processes such as bootstrap percolation or neural avalanches.
  • Cycle-aware corrections beyond the present generating-function approach may be needed for dense or modular networks.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 2 minor

Summary. The paper develops a branching-process framework for sandpile models on complex networks by inserting grain-loss effects directly into the offspring distribution, producing generalized generating functions for dissipative dynamics. It claims that dissipation induces exponential cutoffs in avalanche-size distributions while preserving topology-dependent scaling on sparse random networks; simulations confirm this but reveal systematic deviations on clustered (Holme-Kim) scale-free networks, where increasing clustering lowers the avalanche exponent and raises the probability of large cascades. The work also reports substantial deviations on trees due to low edge density and leaves, and concludes that short cycles invalidate the classical independent-branch approximation.

Significance. If the generalized generating functions remain accurate once dissipation is included, the results would usefully quantify how dissipation, clustering, and sparsity reshape avalanche statistics beyond conservative branching-process models, while explicitly mapping the breakdown of the tree-like approximation. The demonstration that clustering enhances large cascades on networks is potentially relevant for applications to real networked systems.

major comments (1)
  1. [Abstract / central construction] Abstract / central construction: the claim that inserting grain-loss into the offspring distribution yields generalized generating functions that 'preserve topology-dependent scaling behavior' in the dissipative regime rests on an unverified extension of the independent-branching assumption. The manuscript itself states that short cycles invalidate this assumption in the classical case, yet provides no derivation or isolated test showing that the same assumption holds (or is mitigated) after dissipation is added. This is load-bearing for the central assertion that exponential cutoffs coexist with topology-dependent scaling.
minor comments (2)
  1. [Abstract] Abstract: the statement that 'numerical simulations confirm the theoretical predictions' is given without any reference to the specific networks, system sizes, number of realizations, fitting procedures, or error bars used, making the strength of the confirmation impossible to assess from the provided text.
  2. [Abstract] Abstract: 'approx imation' contains a typographical space and should read 'approximation'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive critique of the central construction. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract / central construction] Abstract / central construction: the claim that inserting grain-loss into the offspring distribution yields generalized generating functions that 'preserve topology-dependent scaling behavior' in the dissipative regime rests on an unverified extension of the independent-branching assumption. The manuscript itself states that short cycles invalidate this assumption in the classical case, yet provides no derivation or isolated test showing that the same assumption holds (or is mitigated) after dissipation is added. This is load-bearing for the central assertion that exponential cutoffs coexist with topology-dependent scaling.

    Authors: The generalized generating functions are obtained by substituting a dissipative offspring distribution (incorporating a per-step grain-loss probability) into the standard branching-process formalism; the independence of branches remains an input assumption that follows solely from the locally tree-like character of the underlying network and is not altered by the loss term. Consequently the exponential cutoff is produced by the modified offspring distribution while the scaling exponents continue to be controlled by the network degree distribution, exactly as in the conservative case. This structure is verified numerically on sparse random networks (Erdős–Rényi and configuration-model scale-free) where the tree-like approximation is known to hold, and the same networks exhibit the predicted topology-dependent exponents together with the cutoff. On clustered or low-density topologies the approximation breaks down for both conservative and dissipative dynamics, as already reported. We acknowledge that the manuscript does not isolate an analytic test of the assumption’s validity specifically after dissipation is introduced. We will therefore add (i) an explicit derivation of the dissipative generating functions in the theory section and (ii) a concise paragraph stating that the independence assumption is inherited unchanged from the network topology. These additions will make the load-bearing claim fully transparent. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

Derivation self-contained with no reduction to inputs by construction

full rationale

The paper modifies the offspring distribution to incorporate explicit grain loss and derives generalized generating functions from that modified distribution; the resulting predictions for exponential cutoffs and topology-dependent scaling are direct mathematical consequences of the branching-process construction rather than fitted quantities or self-referential definitions. These predictions are then tested against separate numerical simulations on random and clustered networks, with deviations from the tree-like approximation reported as simulation observations rather than asserted by the theory. No load-bearing self-citations, imported uniqueness theorems, or renamings of known results appear in the derivation chain, and the central claims remain independent of the simulation data used for confirmation.

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

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the framework implicitly relies on a modified branching-process offspring distribution whose validity on loopy graphs is asserted without derivation details.

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Cite this review

Pith. "Pith review of Sandpile Models on complex networks." pith.science (2026). https://pith.science/paper/3X6QVSVL

@misc{pith2026260702023,
  author       = {Pith},
  title        = {Pith review of: Sandpile Models on complex networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3X6QVSVL}},
  note         = {Machine review of arXiv:2607.02023}
}
read the original abstract

We investigate the sandpile model on complex networks by developing a branching-process framework that explicitly incorporates dissipation during avalanche propagation. Unlike classical branching descriptions, which assume conservative transport and locally tree-like independence, the present approach introduces grain-loss effects directly into the offspring distribution, yielding generalized generating functions for dissipative avalanche dynamics. In the dissipative regime, avalanche-size distributions acquire exponential cutoffs while preserving topology-dependent scaling behavior. Numerical simulations confirm the theoretical predictions on sparse random networks and reveal systematic deviations in highly structured topologies. In particular, by using Holme-Kim clustered scale-free networks, we show that increasing clustering continuously lowers the avalanche exponent and enhances the probability of large cascades, demonstrating that short cycles generate strong correlations that invalidate the classical independent-branch approx imation. Surprisingly, trees also exhibit substantial deviations from power-law because low edge density and the abundance of leaves constrain avalanche propagation. These results show that dissipation, clustering, and sparse connectivity fundamentally reshape avalanche size distribution of the sandpile model on networks and establish quantitative limits for branching-process descriptions of avalanche dynamics.

Figures

Figures reproduced from arXiv: 2607.02023 by the authors.

Figure 1
Figure 1. Main panel shows the distributions of avalanche [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Variation of exponents τ we obtained by fitting the distribution by power-law with respect to γ obtained from data shown in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Main panel shows the distributions of avalanche [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Variation of exponents τ we obtained by fitting the distribution by power-law with respect to γ obtained from data shown in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: In Fig. 16 we report similar results for different [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Distributions of avalanche sizes obtained with [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: Avalanche size distributions for different values [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: Avalanche size distributions for different [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 10
Figure 10. Figure 10: Main panel shows the distributions of avalanche sizes obtained with the sandpile model on different Holme-Kim scale-free graphs with γ = 3. The number of nodes in each graph is 10 000. The dissipation probability is 5 × 10−3 in each case. The data are grouped into log…
Figure 13
Figure 13. Figure 13: Distributions of avalanche sizes obtained with [PITH_FULL_IMAGE:figures/full_fig_p012_13.png]
Figure 14
Figure 14. Figure 14: Distributions of avalanche sizes obtained with [PITH_FULL_IMAGE:figures/full_fig_p013_14.png]
Figure 16
Figure 16. Figure 16: Distributions of avalanche sizes obtained with [PITH_FULL_IMAGE:figures/full_fig_p017_16.png]
Figure 15
Figure 15. Figure 15: We show the root w ∗ as a function of the dissi￾pation f in the Erdős–Rényi case with λ = 10. In [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]
Figure 17
Figure 17. Figure 17: We show the root w ∗ as a function of the dissipa￾tion probability f in the random regular case with k0 = 10. Appendix C: Determination of the asymptotic expan￾sion of p(s) for large s. To analyze the singularity of Q(w) at w = 1, we use the Bose-Einstein type expansi…
Figure 18
Figure 18. Figure 18: Comparison between the exact distribution [PITH_FULL_IMAGE:figures/full_fig_p022_18.png]
Figure 19
Figure 19. Figure 19: Evolution of the fitted exponent τ as a function of the dissipation parameter f for scale-free networks of different sizes with fixed degree exponent γ = 3. For small f, τ remains close to the value γ/(γ − 1) = 1.5, while for larger f it approaches γ = 3. The vertical…

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Works this paper leans on

33 extracted references · 33 canonical work pages

  1. [1]

    2.ψ(z)admits the following square-root singular expan- sion atz=ρ: ψ(z) =τ−γ r 1− z ρ +O 1− z ρ , γ= s 2ϕ(τ) ϕ′′(τ)

    The radius of convergence ofψ(z)isρ >0. 2.ψ(z)admits the following square-root singular expan- sion atz=ρ: ψ(z) =τ−γ r 1− z ρ +O 1− z ρ , γ= s 2ϕ(τ) ϕ′′(τ) . Moreover, a full locally convergent expansion exists in powers of p 1−z/ρ

    work page

  2. [2]

    branches

    The coefficients ofψ(z)satisfy the asymptotic esti- mate [zn]ψ(z) = s ϕ(τ) 2ϕ′′(τ) ρ−n √ πn3 1 +O 1 n .(14) Theorem 2(Lagrange Inversion Theorem).Let ϕ(u) = X k≥0 ϕkuk be a power series inC[[u]]withϕ 0 ̸= 0. Then the equation ψ(z) =zϕ(ψ(z)) admits a unique solutionψ(z)∈C[[z]], whose coefficients are given by (Lagrange form) ψ(z) = ∞X n=1 ψnzn, ψ n = 1 n[u...

    work page

  3. [3]

    P. Bak, C. Tang, and K. Wiesenfeld, Physical review. A, General physics38 1, 364 (1988)

    work page 1988

  4. [4]

    P. Bak, C. Tang, and K. Wiesenfeld, Physical review letters 59 4, 381 (1987)

    work page 1987

  5. [5]

    H. J. Jensen,Self-Organized Criticality, Cambridge lecture notes in physics ; 10. (Cambridge University Press, 1998)

    work page 1998

  6. [6]

    Statistical mechanics of complex networks

    R. Albert and A.-L. Barabàsi, Reviews of Modern Physics 74, 47 (2002), arXiv:cond-mat/0106096v1

    work page Pith review arXiv 2002

  7. [7]

    S. H. Strogatz, Nature410, 268 (2001)

    work page 2001

  8. [8]

    Barrat, M

    A. Barrat, M. Barthlemy, and A. Vespignani,Dynami- cal Processes on Complex Networks(Cambridge University Press, New York, 2001)

    work page 2001

Show all 33 references
  1. [9]

    Newman,Networks: An Introduction(Oxford Univer- sity Press, Oxford, 2010)

    M. Newman,Networks: An Introduction(Oxford Univer- sity Press, Oxford, 2010)

    2010

  2. [10]

    J.Lahtinen, J.Kertész,andK.Kaski,PhysicaA:Statistical Mechanics and its Applications349, 535 (2005)

    2005

  3. [11]

    K. B. Lauritsen, S. Zapperi, and H. E. Stanley, arXiv preprint (1996), arXiv:cond-mat/9603154 [cond-mat]

    1996 arXiv

  4. [12]

    Bonabeau, Journal of the Physical Society of Japan64, 327 (1995)

    E. Bonabeau, Journal of the Physical Society of Japan64, 327 (1995)

    1995

  5. [13]

    Dhar and S

    D. Dhar and S. N. Majumdar, Journal of Physics A: Math- ematical and General23, 4333 (1990)

    1990

  6. [14]

    Goh, D.-S

    K.-I. Goh, D.-S. Lee, B. Kahng, and D. Kim, Physical Re- view Letters91, 148701 (2003)

    2003

  7. [15]

    Lee, K.-I

    D.-S. Lee, K.-I. Goh, B. Kahng, and D. Kim, Journal of the Korean Physical Society44, 633 (2004)

    2004

  8. [16]

    Lee, K.-I

    D.-S. Lee, K.-I. Goh, B. Kahng, and D. Kim, arXiv preprint (2004), arXiv:cond-mat/0401531 [cond-mat.stat-mech]

    2004 arXiv

  9. [17]

    Flajolet and R

    P. Flajolet and R. Sedgewick,Analytic Combinatorics (Cambridge University Press, 2009)

    2009

  10. [18]

    Alstott, powerlaw: A python package for analysis of heavy-tailed distributions (2014)

    J. Alstott, powerlaw: A python package for analysis of heavy-tailed distributions (2014)

    2014

  11. [19]

    Clauset, C

    A. Clauset, C. R. Shalizi, and M. E. J. Newman, SIAM Review51, 661 (2009)

    2009

  12. [20]

    Virkar and A

    Y. Virkar and A. Clauset, The Annals of Applied Statistics 8, 89 (2014)

    2014

  13. [21]

    N. H. Kuiper, Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Series A63, 38 (1960)

    1960

  14. [22]

    M. A. Stephens, Journal of the Royal Statistical Society, Series B32, 115 (1970)

    1970

  15. [23]

    T.W.AndersonandD.A.Darling,AnnalsofMathematical Statistics23, 193 (1952)

    1952

  16. [24]

    M. A. Stephens, Journal of the American Statistical Asso- ciation69, 730 (1974)

    1974

  17. [25]

    Barabàsi and R

    A.-L. Barabàsi and R. Albert, Science286, 509 (1999)

    1999

  18. [26]

    J. E. Robinson, Physical Review83, 678 (1951)

    1951

  19. [27]

    D. C. Wood,The Computation of Polylogarithms, Technical Report (University of Kent, Canterbury, United Kingdom, 1992)

    1992

  20. [28]

    Flajolet and A

    P. Flajolet and A. Odlyzko,Singularity Analysis of Gen- erating Functions, Rapport de recherche 826 (INRIA, Roc- quencourt, France, 1988) rapport de recherche INRIA

    1988

  21. [29]

    R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, Advances in Computational Mathematics 5, 329 (1996), received 18 January 1993; revised 27 Febru- ary 1996

    1996

  22. [30]

    Erdõs and A

    P. Erdõs and A. Rényi, Publicationes Mathematicae , 290 (1959), dedicated to O. Varga on the occasion of his 50th birthday

    1959

  23. [31]

    Mckay and N

    B. Mckay and N. C. Wormald, Journal of Algorithms11, 52 (1990), dOI unavailable

    1990

  24. [32]

    Holme and B

    P. Holme and B. J. Kim, Physical Review E65, 026107 (2002)

    2002

  25. [33]

    kY i=1 p(αi) # δα1+···+αk, s−1, whereδ a,b is the Kronecker symbol. Plugging this expres- sion into (A1) gives p(s) = ∞X k=0 qk X α1,...,αk≥1

    G. Mikaberidze, A. Plaud, and R. M. D’Souza, Dragon kings in self-organized criticality systems (2023), arXiv:2308.02658 [nlin.AO]. 14 Appendix A: Determination of the functional equa- tionP(y) =y Q P(y) The aim of this Appendix is to prove the main functional equation (6) rel...

    2023

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