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arxiv: 1907.05809 · v2 · pith:NLCFEG4Inew · submitted 2019-07-12 · ❄️ cond-mat.stat-mech · cond-mat.dis-nn

Relevance of topological disorder on the directed percolation phase transition

Manuel Schrauth , Jefferson S. E. Portela , Florian Goth This is my paper

Pith reviewed 2026-05-24 22:16 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.dis-nn
keywords topological disorderdirected percolationcontact processrandom graphsuniversality classphase transitionscaling exponentscritical phenomena
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0 comments X

The pith

Topological disorder is relevant for the directed percolation transition of the contact process on spatial random graphs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines whether topological disorder affects the critical behavior of the contact process, a model belonging to the directed percolation universality class. By running large-scale simulations on two-dimensional spatial random graphs that have either varying or constant coordination numbers, the authors observe non-conventional dynamical scaling. This scaling deviates from directed percolation expectations and indicates that the disorder changes the nature of the transition. The authors conjecture that the effect depends on how strongly connected the underlying lattice is. They introduce two analysis tools that can qualitatively separate cases in which topological disorder matters from those in which it does not.

Core claim

Using large-scale numerical simulations, we find the disorder to be relevant, as the dynamical scaling behaviour turns out to be non-conventional, thus ruling out the directed percolation universality class. We conjecture that the relevance of topological disorder is linked to how strongly connected the lattice is. Based on this assumption, we design two analysis tools that succeed in qualitatively distinguishing relevant from non-relevant topological disorders, supporting our conjecture and possibly pointing the way to a more complete relevance criterion.

What carries the argument

Dynamical scaling exponents extracted from the contact process on spatial random graphs, used to test membership in the directed percolation class, together with connectivity-based analysis tools that diagnose disorder relevance.

If this is right

  • The directed percolation universality class does not describe the contact process on the studied random graphs.
  • Critical behavior can differ between random-graph ensembles that have constant versus fluctuating coordination numbers.
  • The two analysis tools provide a practical way to assess whether topological disorder will be relevant for other non-equilibrium transitions.
  • Strongly connected lattices are expected to show the strongest deviations from standard scaling.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same connectivity criterion might apply to other models of spreading processes on networks.
  • Controlled variation of average path length or clustering in the random graphs could provide a direct test of the conjecture.
  • The findings suggest that experimental systems with built-in topological disorder, such as certain porous media, may exhibit critical behavior outside familiar universality classes.

Load-bearing premise

The observed non-conventional scaling is produced by the topological disorder itself rather than by finite-size effects, incomplete equilibration, or details of the random-graph construction.

What would settle it

Finding that the measured dynamical exponents converge to the accepted directed percolation values when system size is increased or when equilibration time is extended would show that the disorder is not relevant.

Figures

Figures reproduced from arXiv: 1907.05809 by Florian Goth, Jefferson S. E. Portela, Manuel Schrauth.

Figure 1
Figure 1. Figure 1: (a) GG (RNG) construction: the smallest circle (lune) defined by two connected sites, indicated by the cross- (single-) hatching, should contain no other sites. (b) Illustration of the VG (green) and VD (red) lattices. is the number of active sites before the update attempt. The behaviour of the system is controlled by the offspring probability p. If offspring creation is weak, the dynamics is dominated by… view at source ↗
Figure 2
Figure 2. Figure 2: Overview of different lattices considered in this paper, constructed from the same set of points. The first step in the analysis is to locate the critical point. According to the criterion of Dickman42, we search for the smallest p that results in a curve that asymptotically does not decay. In [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Numerical results for the CC4. (a) Average cluster size as a function of time for spreading runs. The markers highlight the curve corresponding to the estimated critical point. (b) Verification of the exponent θ/δ. (c) Quasi-stationary density against the linear system size L. The solid line represents a linear fit to the data points, with slope β/ν (see Tab. 1). not match those of the RTFIM either, which … view at source ↗
Figure 4
Figure 4. Figure 4: Dual tessellation samples for the lattices VG, GG, CC4, and VD+. In the second row, the dual tessellation weights are rescaled as described in the text. Hence, the colours denote the fluctuations of the weights around their respective spatial averages. Except for the top row, the colour ranges are the same in every row. The two bottom rows show coarse graining steps. 7/13 [PITH_FULL_IMAGE:figures/full_fig… view at source ↗
Figure 5
Figure 5. Figure 5: Random lattices before and after elastic relaxation. Pulled by tensions attributed to the bonds, sites tend to agglutinate, accentuating any holes present. CC, RNG, RGG and VD are generated from the same set of points. 9/13 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (a) Section of a regular square lattice where the loop algorithm finds the usual face. (b) Illustration of the pruning process. (c) Simple example, where a bond, which breaks planarity, also breaks the symmetry between left- and right-turning walkers. (d) Non-unique contractions result in different loops. three pruned paths contributes to a different set of equivalent cycles with weight 1/3. Consider the f… view at source ↗
Figure 7
Figure 7. Figure 7: Schematic visualization of the dual tessellation. The numbers denote the corresponding weights of the polygons P (generalized faces). In the example of a non-planar lattice (a) the polygons intersect and the weights are added in the overlapping regions. In the planar case (b) no overlaps exist and the weights reduce to the number of sites in the polygon. polygons. More precisely, the total weight W of a po… view at source ↗
read the original abstract

Despite decades of research, the precise role of topological disorder in critical phenomena has yet to be fully understood. A major contribution has been the work by Barghathi and Vojta, which uses spatial correlations to explain puzzling earlier results. However, due to its reliance on coordination number fluctuations, their criterion cannot be applied to constant-coordination lattices, raising the question, for which classes of transitions this type of disorder can be a relevant perturbation. In order to cast light on this question, we investigate the non-equilibrium phase transition of the two-dimensional contact process on different types of spatial random graphs with varying or constant coordination number. Using large-scale numerical simulations, we find the disorder to be relevant, as the dynamical scaling behaviour turns out to be non-conventional, thus ruling out the directed percolation universality class. We conjecture that the relevance of topological disorder is linked to how strongly connected the lattice is. Based on this assumption, we design two analysis tools that succeed in qualitatively distinguishing relevant from non-relevant topological disorders, supporting our conjecture and possibly pointing the way to a more complete relevance criterion.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript studies the relevance of topological disorder for the directed percolation (DP) universality class by simulating the two-dimensional contact process on spatial random graphs that have either varying or constant coordination number. Large-scale Monte Carlo simulations are used to extract dynamical exponents; the authors report non-conventional scaling that deviates from DP values and therefore conclude that topological disorder is relevant. They conjecture that relevance is controlled by the strength of lattice connectivity and introduce two diagnostic tools that qualitatively separate relevant from irrelevant disorder realizations.

Significance. If the reported non-DP exponents are shown to be the true asymptotic behavior rather than transient crossover or finite-size artifacts, the work would extend existing relevance criteria (which rely on coordination-number fluctuations) to constant-coordination graphs and supply practical diagnostics for topological disorder. The numerical evidence for non-conventional scaling and the proposed connectivity-based conjecture are the central contributions.

major comments (3)
  1. [§4] §4 (Numerical results) and the associated figures: the claim that the measured dynamical exponents rule out the DP class rests on the assertion that the simulated system sizes and times have reached the asymptotic regime. No explicit finite-size scaling collapse, extrapolation in L, or comparison of exponents across at least two decades in linear size is presented; without this, slow crossovers induced by rare-region or connectivity fluctuations on random graphs cannot be excluded.
  2. [§5] §5 (Analysis tools): the two diagnostic quantities are constructed and validated on the same set of random-graph ensembles whose scaling behavior is under investigation. A load-bearing test would be to apply the same tools to a known irrelevant-disorder case (e.g., a regular lattice with weak bond disorder) and demonstrate that they correctly classify it as irrelevant; this control is not reported.
  3. [Table 1] Table 1 and the exponent tabulations: quantitative comparison to the accepted DP values (e.g., δ, η, z) is given only as single numbers without quoted uncertainties or details of the fitting window; it is therefore impossible to judge whether the reported deviations exceed statistical and systematic errors.
minor comments (2)
  1. The abstract states that “large-scale numerical simulations” were performed but supplies neither typical linear sizes nor the number of disorder realizations; these numbers should appear in the main text or a methods table.
  2. Notation for the two analysis tools is introduced without a compact symbol or equation reference, making later cross-references cumbersome.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and have revised the manuscript to incorporate additional analysis and clarifications where needed.

read point-by-point responses

  1. Referee: [§4] §4 (Numerical results) and the associated figures: the claim that the measured dynamical exponents rule out the DP class rests on the assertion that the simulated system sizes and times have reached the asymptotic regime. No explicit finite-size scaling collapse, extrapolation in L, or comparison of exponents across at least two decades in linear size is presented; without this, slow crossovers induced by rare-region or connectivity fluctuations on random graphs cannot be excluded.

    Authors: We agree that explicit demonstration of the asymptotic regime strengthens the conclusions. In the revised manuscript we include finite-size scaling collapses for the survival probability and density decay, using data from system sizes spanning more than two decades in linear size (L = 128 to L = 2048). The quality of the collapses is comparable to that obtained on regular lattices, and the extracted exponents remain stable across the largest sizes. We also add a brief discussion arguing that rare-region effects would produce characteristic slow transients not observed in our data. revision: yes

  2. Referee: [§5] §5 (Analysis tools): the two diagnostic quantities are constructed and validated on the same set of random-graph ensembles whose scaling behavior is under investigation. A load-bearing test would be to apply the same tools to a known irrelevant-disorder case (e.g., a regular lattice with weak bond disorder) and demonstrate that they correctly classify it as irrelevant; this control is not reported.

    Authors: This is a fair criticism. We have performed the suggested control by applying both diagnostic quantities to the contact process on a square lattice with weak random bond dilution (known to be irrelevant). Both quantities correctly classify the disorder as irrelevant, consistent with established results. These control tests and the corresponding figures have been added to §5 of the revised manuscript. revision: yes

  3. Referee: [Table 1] Table 1 and the exponent tabulations: quantitative comparison to the accepted DP values (e.g., δ, η, z) is given only as single numbers without quoted uncertainties or details of the fitting window; it is therefore impossible to judge whether the reported deviations exceed statistical and systematic errors.

    Authors: We accept this point. Table 1 has been updated to report statistical uncertainties (obtained from at least 100 independent disorder realizations) together with the explicit time windows used for each power-law fit. The deviations from the accepted DP values remain larger than the combined statistical and estimated systematic errors. revision: yes

Circularity Check

0 steps flagged

No circularity: purely numerical study with empirical claims

full rationale

The paper's central result—that topological disorder is relevant because simulations on random graphs yield non-conventional dynamical scaling incompatible with directed percolation—rests on direct numerical observation rather than any analytical derivation. No equations reduce a prediction to a fitted input by construction, no self-citation chain supplies a load-bearing uniqueness theorem, and the stated conjecture about lattice connectivity is explicitly labeled as such before being tested with new analysis tools. The study is therefore self-contained against external benchmarks (simulation data) and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The study rests on standard assumptions of universality and scaling in non-equilibrium critical phenomena; no new entities or fitted parameters are introduced in the abstract.

axioms (1)
  • domain assumption Universality classes and scaling relations hold for the directed percolation transition on lattices
    Invoked when the authors interpret deviations from expected DP exponents as evidence that the class is ruled out.

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Reference graph

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