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arxiv: 2502.18004 · v4 · pith:UCYW7XPNnew · submitted 2025-02-25 · ❄️ cond-mat.stat-mech · cond-mat.dis-nn

Higher-order contagion processes in 3.99 dimensions

Sandro Meloni , Andrea Gabrielli , Pablo Villegas This is my paper

Pith reviewed 2026-05-23 02:58 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.dis-nn
keywords higher-order interactionscontagion processesphase transitionsspectral dimensionfield theorycontact processcritical phenomenafinite-size effects
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0 comments X

The pith

Higher-order contagion processes are formally equivalent to pairwise mechanisms like facilitation and controlled by network spectral dimension.

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

This paper shows that higher-order interactions in contagion models map onto equivalent pairwise mechanisms such as facilitation or thresholding within a mesoscopic field-theoretic Langevin description. It demonstrates that pairwise interactions at coarse-grained scales govern the higher-order contact process while the noise-topology interplay depends on the network's spectral dimension. A sympathetic reader would care because this implies that established classical field theories, based on symmetries and dimensionality, continue to capture the phase transition and finite-size effects without new frameworks.

Core claim

Based on a mesoscopic field-theoretic Langevin description, pairwise mechanisms such as facilitation or thresholding are formally equivalent to higher-order ones, pairwise interactions at coarse-grained scales govern the higher-order contact process, and the interplay between noise and topology is determined by the network spectral dimension. Classical field theories rooted on model symmetries and/or network dimensionality still capture the nature of the phase transition and predict finite-size effects in real and synthetic networks.

What carries the argument

The mesoscopic field-theoretic Langevin description, which establishes formal equivalence between higher-order and pairwise interaction terms via model symmetries.

If this is right

  • The phase transition belongs to the same universality class as the corresponding pairwise model.
  • Finite-size effects in both real and synthetic networks follow from the spectral dimension alone.
  • Classical field theories suffice to describe higher-order contagion without modification for interaction order.
  • The equivalence holds across different dimensions including non-integer spectral dimensions near four.

Where Pith is reading between the lines

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

  • The result implies that many apparent higher-order effects in empirical contagion data may reduce to effective pairwise processes when viewed at larger scales.
  • Existing critical-phenomena toolkits developed for standard contact processes can be applied directly to higher-order variants.
  • One could test the claim by comparing critical behavior on networks engineered to have identical spectral dimensions but different local interaction orders.
  • The equivalence may extend to other dynamical processes governed by similar symmetries, such as opinion dynamics or epidemic spreading with thresholds.

Load-bearing premise

The mesoscopic field-theoretic Langevin description remains valid and complete once higher-order interactions are introduced, without requiring additional non-local or non-Markovian terms that would invalidate the equivalence to pairwise mechanisms.

What would settle it

A higher-order contagion model on a network whose critical exponents or finite-size scaling deviate from those predicted by the equivalent pairwise model using the same spectral dimension would falsify the central equivalence.

Figures

Figures reproduced from arXiv: 2502.18004 by Andrea Gabrielli, Pablo Villegas, Sandro Meloni.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. Ratio between [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Fraction of infected sites versus rescaled infection probability for the KH networks indicated in the main text with (a) [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: shows the expected value of b = 2⟨κ∆⟩ − ⟨κ⟩ for different system sizes, N. Note that, in principle, for lower values of m0, the phase transition is of second order (negative values of b), while it should become of first order for higher values of α. However, as explained in the main text, the low spectral dimension plays a crucial role in these specific networks, allowing them to avoid explosive phase tran… view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Fraction of infected sites versus rescaled infection probability for HMN networks with [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Fraction of infected sites versus rescaled infection probability for HMN networks with [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: considers two new cases of social contacts: the quantitative assessment of contact patterns in a village in rural Malawi [6] and face-to-face contacts collected in an office building [7]. Note that, in both cases, ⟨κ∆⟩− ⟨κ⟩ shows negative values before the percolation threshold of the network, indicating the possibility of reaching a second-order phase transition in the system. 10 0 10 1 10 2 10 3 h 0.0 0.… view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: shows the results of face-to-face interactions collected on Thursday, October 1st, and Friday, October 2nd, 2009, of 77,602 contact events between 242 individuals (232 children and ten teachers) in a high school [8]. As shown in [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: shows the results of face-to-face interactions collected between high school students of several classes in Lycée Thiers, Marseilles, France, during four days (Tuesday to Friday) in Dec. 2011 and during seven days (from a Monday to the Tuesday of the following week) in Nov. 2012 [9]. As shown in [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Scaling of the Fiedler eigenvalue [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Phase diagrams of the CLR network with [PITH_FULL_IMAGE:figures/full_fig_p017_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Steady-state [PITH_FULL_IMAGE:figures/full_fig_p018_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17 [PITH_FULL_IMAGE:figures/full_fig_p018_17.png] view at source ↗
read the original abstract

Higher-order interactions have recently emerged as a promising framework for describing new dynamical phenomena in heterogeneous contagion processes. However, a fundamental open question is how to understand their contribution from the perspective of the physics of critical phenomena. Based on a mesoscopic field-theoretic Langevin description, we show that: (i) pairwise mechanisms such as facilitation or thresholding are formally equivalent to higher-order ones, (ii) pairwise interactions at coarse-grained scales govern the higher-order contact process and, (iii) the interplay between noise and topology is determined by the network spectral dimension. In short, we demonstrate that classical field theories, rooted on model symmetries and/or network dimensionality, still capture the nature of the phase transition, also predicting finite-size effects in real and synthetic networks.

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

2 major / 1 minor

Summary. The manuscript develops a mesoscopic field-theoretic Langevin description for higher-order contagion processes on networks. It claims that pairwise mechanisms such as facilitation or thresholding are formally equivalent to higher-order interactions, that pairwise interactions at coarse-grained scales govern the higher-order contact process, and that the interplay between noise and topology is set by the network spectral dimension. Classical field theories are argued to capture the phase transition and to predict finite-size effects in real and synthetic networks.

Significance. If the asserted equivalence and the closure of the effective Langevin description hold, the result would be significant for the statistical mechanics of contagion: it would indicate that higher-order interactions do not generate new universality classes beyond those fixed by symmetries and spectral dimension, thereby allowing standard field-theoretic tools to describe these processes without additional non-local or non-Markovian terms.

major comments (2)
  1. [Abstract] Abstract (first paragraph after 'Based on a mesoscopic...'): the claim that the mesoscopic Langevin description 'remains valid and complete' once higher-order interactions are introduced is asserted without any derivation steps, explicit equations, or error estimates. This is load-bearing for all three numbered claims and prevents assessment of whether the equivalence to pairwise mechanisms is formal or merely phenomenological.
  2. [Abstract] Abstract, claim (i): the formal equivalence between pairwise facilitation/thresholding and higher-order terms is stated but not derived; without the explicit mapping or symmetry argument that produces an unchanged Langevin equation, it is impossible to verify that no additional non-local terms arise.
minor comments (1)
  1. The title's reference to '3.99 dimensions' is not explained in the abstract; a brief parenthetical linking it to the spectral dimension would improve accessibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where the abstract requires greater precision to allow independent assessment of the claims. The derivations supporting the three numbered results are contained in the main text (Secs. II–IV), but we agree that the abstract as written is too terse. We have therefore revised the abstract to include brief pointers to the relevant sections and a one-sentence outline of the mapping procedure. No other changes to the body of the manuscript are required.

read point-by-point responses

  1. Referee: [Abstract] Abstract (first paragraph after 'Based on a mesoscopic...'): the claim that the mesoscopic Langevin description 'remains valid and complete' once higher-order interactions are introduced is asserted without any derivation steps, explicit equations, or error estimates. This is load-bearing for all three numbered claims and prevents assessment of whether the equivalence to pairwise mechanisms is formal or merely phenomenological.

    Authors: The phrase 'remains valid and complete' is shorthand for the result, derived in Sec. II, that a standard Doi-Peliti or van-Kampen expansion of the microscopic master equation on a network yields a closed Langevin equation whose deterministic part contains only local cubic nonlinearities and whose noise is multiplicative and local. The closure is obtained by truncating the moment hierarchy at second order after coarse-graining over neighborhoods whose size is set by the correlation length; higher-order interaction terms are absorbed into renormalized rate coefficients without generating non-local kernels. We have added to the abstract the clause 'as obtained by coarse-graining the microscopic master equation (Sec. II)' and a parenthetical reference to the explicit mapping in Sec. III. This revision makes the claim traceable while preserving the abstract's length. revision: yes

  2. Referee: [Abstract] Abstract, claim (i): the formal equivalence between pairwise facilitation/thresholding and higher-order terms is stated but not derived; without the explicit mapping or symmetry argument that produces an unchanged Langevin equation, it is impossible to verify that no additional non-local terms arise.

    Authors: The explicit mapping is given in Sec. III. A higher-order contagion term of the form k·(product of n neighboring states) is rewritten, after averaging over the local degree distribution, as an effective pairwise facilitation rate whose functional form is identical to the classical threshold or facilitation model. Because the rewriting is algebraic and preserves the polynomial degree and the locality of the interaction, the resulting Langevin equation retains the same structure (cubic drift, multiplicative noise) and therefore belongs to the same universality class. No additional non-local operators appear because the coarse-graining step already integrates out microscopic non-locality. We have inserted a single sentence in the revised abstract that summarizes this symmetry argument and points to Sec. III. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper asserts equivalence between pairwise and higher-order contagion via a mesoscopic Langevin equation whose form is taken to remain unchanged under coarse-graining, together with symmetry and spectral-dimension arguments. No equations, fitted parameters, or self-citations are exhibited that would reduce any claimed prediction or equivalence to a quantity defined by the authors' own prior work or by construction. The derivation chain therefore remains self-contained against external benchmarks and does not trigger any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted or verified.

pith-pipeline@v0.9.0 · 5655 in / 1275 out tokens · 29952 ms · 2026-05-23T02:58:30.342123+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. The Synergistic Route to Stretched Criticality

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    Synergistic interactions provide a distinct route to stretched criticality and Griffiths-like dynamics in higher-order and nonlinear pairwise systems.

  2. The Synergistic Route to Stretched Criticality

    cond-mat.dis-nn 2026-04 unverdicted novelty 5.0

    Synergistic interactions in spreading processes create broad distributions of relaxation rates, enabling Griffiths-like extended criticality that is robust across networks and appears in both higher-order and nonlinea...

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