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arxiv: 2511.05869 · v2 · submitted 2025-11-08 · 🧮 math.CO

Iterative Generation and Generalized Degree Distribution of Higher-Order Fractal Scale-Free Networks

Lin Qi , Jiaxin Zhang This is my paper

Pith reviewed 2026-05-17 23:45 UTC · model grok-4.3

classification 🧮 math.CO
keywords higher-order networksfractal networkssimplicial complexesscale-free networksiterative generationgeneralized degree distributionfractal dimension
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The pith

Iterative construction with parameters K, m, and t produces higher-order networks as pure simplicial complexes that are fractal and scale-free in generalized degree distribution for large m.

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

The paper introduces an iterative generation model for higher-order fractal networks controlled by the simplicial complex dimension K, multiplier m, and iteration count t. This process builds a pure simplicial complex whose fractal character is established theoretically by the similarity dimension and confirmed experimentally by the box-counting dimension. When the multiplier m becomes large, the generalized degree distribution of the resulting networks follows the power-law form that defines scale-free behavior. A sympathetic reader would care because the model supplies a controllable way to generate networks that capture both fractal self-similarity and higher-order interactions simultaneously.

Core claim

The authors construct higher-order networks as pure simplicial complexes through an iterative process governed by parameters K, m, and t; they prove these networks are fractal by matching similarity-dimension calculations to box-counting measurements and show that the generalized degree distribution is scale-free once the multiplier m is taken sufficiently large.

What carries the argument

The iterative generation process that builds a pure simplicial complex from an initial structure using the three parameters K (dimension), m (multiplier), and t (steps), which simultaneously enforces fractal scaling and allows analysis of the generalized degree sequence.

If this is right

  • The generated networks possess self-similar fractal structure that can be quantified by both similarity and box-counting dimensions.
  • Generalized degree distributions become scale-free once the multiplier m exceeds a sufficient threshold.
  • Higher-order interactions among multiple nodes can be represented inside a single pure simplicial complex generated by the same iteration.
  • The three-parameter construction supplies a tunable family of networks that simultaneously exhibit fractal and scale-free traits.

Where Pith is reading between the lines

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

  • The same iterative scheme could be used to embed additional higher-order motifs such as triangles or tetrahedra while preserving the fractal scaling.
  • Real-world datasets of multi-way interactions could be compared against the model's degree distribution to test whether the large-m scale-free regime appears in practice.
  • Extending the construction to weighted or directed simplicial complexes might reveal whether the fractal and scale-free properties survive when edge directions or strengths are introduced.

Load-bearing premise

The iterative process with chosen K, m, and t produces a pure simplicial complex whose fractal and scale-free properties hold independently of specific data-fitting choices.

What would settle it

Generate networks for fixed K and increasing t, then check whether the measured box-counting dimension converges to the theoretical similarity dimension; separately, increase m and test whether the generalized degree distribution approaches a stable power law.

Figures

Figures reproduced from arXiv: 2511.05869 by Jiaxin Zhang, Lin Qi.

Figure 1
Figure 1. Figure 1: Simplicial Complex, 1-Skeleton, and Clique Complex. (a) A two-dimensional simplicial complex, with blue triangles representing the 2-simplices it contains, edges denoting the 1-simplices it encompasses, and points signifying the 0-simplices it holds; (b) The 1-skeleton corresponding to the two-dimensional simplicial complex in (a) is also the 1-skeleton of the network in (c); (c) The clique complex generat… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic Diagram of the Iterative Construction Process. The grey nodes in the diagram represent nodes present at the outset, the blue nodes denote midpoints inserted along edges during iteration, and the green nodes signify multiplier nodes. (a) Schematic of one iteration when K = 1 and m = 2, where the blue node form bottom; (b) Schematic of one iteration when K = 2 and m = 1, and the blue edges form bot… view at source ↗
Figure 3
Figure 3. Figure 3: Networks Obtained After Multiple Iterations. (a)-(c) Networks after four iterations with K = 1 for m = 1, 2, 3, respectively; (d)-(f) Networks after four iterations with K = 2 for m = 1, 2, 3, respectively; (g)-(i) Networks after three iterations with m = 1 for K = 2, 3, 4, respectively. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Plot of the fractal dimension as a function of dimension K and multiplier m. (a) The fractal dimension of the graph generated with m = 3 after two iterations, for K ranging from 1 to 10; (b) The fractal dimension of the graph generated with K = 2 after two iterations, for m ranging from 0 to 10. 3.3 Generalized Degree Distribution In network science, degree distributions are extensively studied due to thei… view at source ↗
Figure 5
Figure 5. Figure 5: Plot of the 1-dimensional generalized degree distribution PK,1(k) for networks under different combinations of parameters K and m in a double-logarithmic coordinate system. 4 Conclusion This study proposes an iterative generative model for higher-order fractal scale-free networks. First, a fractal network is generated through an iterative process. Subsequently, the clique complex method is employed to deri… view at source ↗
Figure 6
Figure 6. Figure 6: Plot of the 2-dimensional generalized degree distribution PK,2(k) for networks under different combinations of parameters K and m in a double-logarithmic coordinate system. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
read the original abstract

Fractals represent one of the fundamental manifestations of complexity, and fractal networks serve as tools for characterizing and investigating the fractal structures and properties of large-scale systems. Higher-order networks have emerged as a research hotspot due to their ability to express interactions among multiple nodes. This study proposes an iterative generation model for higher-order fractal networks. The iteration is controlled by three parameters: the dimension K of the simplicial complex, the multiplier m, and the iteration count t. The constructed network is a pure simplicial complex. Theoretical analysis using the similarity dimension and experimental verification using the box-counting dimension demonstrate that the generated networks exhibit fractal characteristics. When the multiplier m is large, the generalized degree distribution of the generated networks is characterized by its scale-free nature.

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

1 major / 1 minor

Summary. The manuscript proposes an iterative construction for higher-order networks realized as pure simplicial complexes, parameterized by the simplicial dimension K, multiplier m, and iteration depth t. It establishes fractal properties via a theoretical similarity-dimension calculation and experimental box-counting verification, and asserts that the generalized degree distribution becomes scale-free for large m, supported by numerical log-log plots.

Significance. A rigorously justified generative model that simultaneously produces fractal and scale-free higher-order networks would be a useful addition to the literature on complex systems and simplicial complexes. The combination of a closed-form similarity-dimension argument with box-counting experiments is a constructive feature; strengthening the scale-free claim would increase the model's utility for theoretical and simulation studies.

major comments (1)
  1. [Abstract] Abstract (and the corresponding experimental section on generalized degree distribution): the statement that 'when the multiplier m is large, the generalized degree distribution ... is characterized by its scale-free nature' is supported only by log-log plots at selected finite m values. No closed-form expression for the degree sequence, its generating function, or an asymptotic analysis (e.g., m → ∞ with t fixed) is provided, so the power-law regime may depend on the particular m-range, binning, or least-squares window chosen.
minor comments (1)
  1. [Experimental verification section] The manuscript should specify the precise fitting procedure, the range of m values examined, the number of independent realizations, and any data-exclusion rules used for the degree-distribution plots to permit independent verification.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The single major comment raises a valid point about the evidential basis for the scale-free claim, which we address below. We will revise the manuscript to incorporate additional analysis while preserving the existing numerical results.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the corresponding experimental section on generalized degree distribution): the statement that 'when the multiplier m is large, the generalized degree distribution ... is characterized by its scale-free nature' is supported only by log-log plots at selected finite m values. No closed-form expression for the degree sequence, its generating function, or an asymptotic analysis (e.g., m → ∞ with t fixed) is provided, so the power-law regime may depend on the particular m-range, binning, or least-squares window chosen.

    Authors: We agree that the current support for the scale-free character rests on numerical log-log plots for finite m and that a closed-form or asymptotic treatment would make the claim more robust. The iterative construction multiplies the number of higher-order simplices by m at each step, which produces a multiplicative broadening of the generalized degree sequence; this mechanism is expected to yield a power-law tail whose exponent stabilizes for large m. In the revision we will add a recurrence relation for the number of nodes of each generalized degree after t iterations, derive its generating function, and analyze the asymptotic tail as m → ∞ with t held fixed. We will also specify the binning method, fitting window, and regression procedure used for the plots so that the numerical evidence can be assessed independently of arbitrary choices. These additions will appear in the abstract, the theoretical analysis section, and the experimental results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are independent of target claims

full rationale

The paper defines an iterative generative process controlled by explicit parameters K, m, and t that produces a pure simplicial complex. Fractal properties are obtained via a theoretical similarity-dimension calculation on the construction and separately verified by box-counting on the resulting graphs; neither step presupposes the other or the scale-free claim. The scale-free characterization for large m is presented as an empirical observation from degree-distribution plots rather than an analytic derivation or a fitted parameter renamed as a prediction. No self-citations, uniqueness theorems, or ansatzes imported from prior author work are invoked to close the argument. The central claims therefore remain logically independent of the inputs they are asserted to explain.

Axiom & Free-Parameter Ledger

3 free parameters · 1 axioms · 0 invented entities

The model rests on choosing three free parameters to define the iteration and on the domain assumption that the output is always a pure simplicial complex.

free parameters (3)
  • K
    Dimension of the simplicial complex, selected as input to control the order of interactions.
  • m
    Multiplier that determines how many copies are added at each iteration step.
  • t
    Number of iterations that determines the final network size and depth.
axioms (1)
  • domain assumption The constructed network is a pure simplicial complex.
    Stated directly in the abstract as a property of the generated object.

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Reference graph

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