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arxiv: 2605.17004 · v1 · pith:FCAG2NVQnew · submitted 2026-05-16 · ❄️ cond-mat.stat-mech

Model of Simplicial Complexes with dimension-wise preferential attachment

Diego Febbe , Duccio Fanelli , Timoteo Carletti This is my paper

Pith reviewed 2026-05-19 19:05 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords simplicial complexespreferential attachmenthigher-order networkspower-law distributionsgenerative modelsgeneralized degreegrowing networks
0
0 comments X

The pith

A simplicial complex grows when new vertices attach to existing simplexes through independent preferential attachment rules at each dimension.

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

The paper develops a generative model for growing simplicial complexes that applies preferential attachment separately for simplexes of each dimension. Attachment probabilities depend on the generalized degree within that dimension alone, without extra coupling between dimensions. This produces power-law distributions in the generalized degrees for every dimension. A reader would care because the model offers a straightforward way to create higher-order structures that match the heterogeneous connectivity seen in real systems such as social or biological networks. If the claim holds, dimension-independent rules are enough to generate scale-free properties across multiple interaction orders.

Core claim

The authors define a growing simplicial complex by beginning with a seed structure and iteratively adding new vertices that form simplexes of chosen dimensions; the probability that a new k-simplex attaches to an existing one is proportional to the current generalized degree of that existing k-simplex, and the rule operates independently for each k.

What carries the argument

Dimension-wise preferential attachment: the attachment probability for simplexes of dimension k depends solely on their own generalized degree at that dimension.

If this is right

  • Generalized degree distributions for simplexes of every dimension follow power-law tails.
  • The model generates higher-order networks whose connectivity is heterogeneous at multiple interaction orders.
  • Scale-free properties emerge across dimensions from rules that do not mix information between dimensions.
  • The construction can be used to study dynamical processes on simplicial complexes with tunable higher-order structure.

Where Pith is reading between the lines

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

  • Real-world many-body systems might exhibit independent scaling exponents for interactions of different orders.
  • The framework could be tested against temporal datasets that record the order of added simplexes.
  • Extensions might add rules that occasionally couple dimensions while preserving the core power-law behavior.

Load-bearing premise

Preferential attachment can be applied independently in each dimension without needing coupling terms that link different dimensions.

What would settle it

Simulations of the model produce generalized degree distributions that are not power laws, or empirical higher-order networks require measurable cross-dimensional dependencies to fit observed data.

Figures

Figures reproduced from arXiv: 2605.17004 by Diego Febbe, Duccio Fanelli, Timoteo Carletti.

Figure 1
Figure 1. Figure 1: FIG. 1: Representation of a simplex (panel a) and a simplicial complex (panel b) together [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Growing process for the simplicial complex. With probability [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Examples of various simplex generations, by varying the parameters [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Panel (a): Degree evolution of the nodes over building time for various values of [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Spectral dimensions [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Tetrahedron, basis of the new 4-dimensional structure that will be created. Here, [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: The objective is to understand how to express the sum of the degrees of the links [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Here we want to express the sum of the degrees of all triangles that include node [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Here we want to express how the degree [PITH_FULL_IMAGE:figures/full_fig_p029_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: When a new tetrahedron is formed, three new links (dashed in the figure) appear. [PITH_FULL_IMAGE:figures/full_fig_p030_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Numerical simulation results for the values of [PITH_FULL_IMAGE:figures/full_fig_p031_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Plot of [PITH_FULL_IMAGE:figures/full_fig_p037_12.png] view at source ↗
read the original abstract

Network science is a powerful framework allowing to model complex systems, it is capable to describe and take into account the intricate web of connections existing among the constituting basic element of the system. Recently scholars have brought to the fore the relevance of higher-order networks, namely structures allowing to encode for many-body interaction, differently from the pairwise case handled by networks. This novel research field opens new avenues of research with applications ranging from neurosciences to social sciences; there is thus a need for generative models of higher-order network capable to reproduce features present in empirical data. In this work we present a model for growing simplicial complex rooted on a preferential attachment process acting dimension-wise, i.e., returning a power law distribution for the generalized degree of simplexes of different dimension.

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 / 2 minor

Summary. The manuscript introduces a generative model for simplicial complexes that grows by successively adding simplices of varying dimensions according to a dimension-wise preferential attachment rule. The central claim is that this process produces power-law distributions for the generalized degrees of simplices in each dimension separately.

Significance. If the power-law result can be derived rigorously, the model would supply a minimal, parameter-light mechanism for generating higher-order networks with scale-free generalized-degree statistics across dimensions. Such a construction could serve as a useful benchmark for empirical simplicial data in statistical mechanics and complex-systems modeling, especially given the explicit dimension-wise separation.

major comments (2)
  1. [Abstract and model definition] Abstract and model section: the claim that dimension-wise preferential attachment yields power-law generalized-degree distributions is asserted without any rate equation, mean-field closure, or stationary-solution derivation. The central result therefore rests on an unshown calculation.
  2. [Model definition] Model construction: attaching a d-simplex necessarily augments the generalized degrees of all its (d-1)-faces. The manuscript treats each dimension’s attachment probability as independent, yet provides no argument showing that the resulting topological correlations factor out or average to zero in the mean-field limit. This coupling is load-bearing for the independence of the per-dimension power laws.
minor comments (2)
  1. [Results/figures] The manuscript should include at least one figure comparing simulated generalized-degree histograms against the claimed power-law form, with the theoretical exponent indicated.
  2. [Notation] Notation for generalized degree and simplex incidence should be defined once at first use and used consistently thereafter.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which help clarify the presentation of the analytic results and the treatment of inter-dimensional effects. We address each major comment below and have revised the manuscript to incorporate the requested derivations and arguments.

read point-by-point responses

  1. Referee: [Abstract and model definition] Abstract and model section: the claim that dimension-wise preferential attachment yields power-law generalized-degree distributions is asserted without any rate equation, mean-field closure, or stationary-solution derivation. The central result therefore rests on an unshown calculation.

    Authors: We agree that the submitted version did not include an explicit derivation of the power-law distributions. The manuscript relied on numerical simulations to illustrate the emergence of power laws across dimensions. To address this, we have added a new subsection deriving the rate equations for the generalized degree of simplices in each dimension d, applying a mean-field approximation to close the equations, and solving the resulting stationary master equation to obtain the power-law form with exponent 1 + 1/α_d, where α_d is the attachment parameter for dimension d. revision: yes

  2. Referee: [Model definition] Model construction: attaching a d-simplex necessarily augments the generalized degrees of all its (d-1)-faces. The manuscript treats each dimension’s attachment probability as independent, yet provides no argument showing that the resulting topological correlations factor out or average to zero in the mean-field limit. This coupling is load-bearing for the independence of the per-dimension power laws.

    Authors: This observation correctly identifies a potential source of coupling. In the revised manuscript we have inserted a paragraph explaining that, within the mean-field limit, the expected increment to the generalized degree of a (d-1)-face arising from the attachment of a new d-simplex is proportional to the current generalized degree of that face in dimension d-1. Because the attachment probability for dimension d is itself linear in the generalized degrees of d-simplices, this contribution factors into the existing preferential-attachment term for dimension d-1 and does not introduce additional dimension-dependent correlations beyond those already captured by the per-dimension rate equations. The argument is supported by an explicit averaging step over the uniform random choice of the new simplex’s faces. revision: yes

Circularity Check

0 steps flagged

No significant circularity: power-law result derived from attachment rule via rate equations

full rationale

The paper defines a generative growth process with dimension-wise preferential attachment and derives the stationary power-law form for generalized degrees from the resulting mean-field rate equations. This is a forward derivation from the model rules rather than a re-statement of fitted inputs or self-referential definitions. No load-bearing step reduces to a self-citation chain, an ansatz smuggled via prior work, or a uniqueness theorem imported from the same authors. The topological coupling concern raised in the skeptic note pertains to the validity of the mean-field closure, not to circularity in the derivation chain itself. The model is self-contained against external benchmarks for this class of preferential-attachment constructions.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The model rests on a single core attachment rule and at least one tunable parameter controlling attachment strength per dimension; no new particles or forces are postulated.

free parameters (1)
  • dimension-specific attachment probability
    Controls the strength of preferential attachment within each dimension and must be chosen to match desired exponents.
axioms (1)
  • domain assumption Preferential attachment acts independently on each dimension of the simplicial complex.
    Invoked as the defining mechanism that produces the claimed power-law distributions.

pith-pipeline@v0.9.0 · 5658 in / 1083 out tokens · 35032 ms · 2026-05-19T19:05:40.251706+00:00 · methodology

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Reference graph

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    General case We now consider the general case in which simplices of different dimensions can enter the network. The evolution of the node degree can be written as dk(0) i dt = DX d=1 pd P σ(d−1)∋i k(d−1) σ P σ(d−1) k(d−1) σ .(C21) where we used the notationσ (d−1) ∋ito denote the fact that the sum is restricted to (d−1)-simplexes containing nodei. As in t...

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