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arxiv: 2509.08350 · v2 · submitted 2025-09-10 · ⚛️ physics.soc-ph · cs.LG· math.AT

Chordless cycle filtrations for dimensionality detection in complex networks via topological data analysis

Aina Ferr\`a Marc\'us , Robert Jankowski , Meritxell Vila Mi\~nana , Carles Casacuberta , M. \'Angeles Serrano This is my paper

Pith reviewed 2026-05-18 18:19 UTC · model grok-4.3

classification ⚛️ physics.soc-ph cs.LGmath.AT
keywords complex networkstopological data analysishyperbolic geometrydimensionality estimationchordless cyclespersistent homologymachine learningnetwork embedding
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The pith

Chordless cycle filtrations in topological data analysis yield descriptors that a neural network uses to estimate network dimensionality.

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

The paper introduces a weighting scheme for graphs that incorporates chordless cycles into topological data analysis filtrations. These filtrations generate descriptors that capture structural features tied to the latent geometry of the network. A neural network trained exclusively on synthetic graphs with known dimensionalities then uses those descriptors to predict the dimension. The same trained model applies without retraining to real-world networks from social and biological domains. If correct, this supplies a data-driven route to uncovering the hidden hyperbolic space that many complex networks are thought to inhabit, with direct consequences for modeling communities, navigation, and connectivity patterns.

Core claim

We introduce a topological data analysis weighting scheme for graphs based on chordless cycles to estimate network dimensionality in a data-driven way. We further show that the resulting descriptors can effectively estimate network dimensionality using a neural network architecture trained on a synthetic graph database constructed for this purpose, which requires no retraining to transfer effectively to real-world networks.

What carries the argument

Chordless cycle filtrations, a weighting scheme in topological data analysis that assigns importance to edges and higher-order structures according to the lengths and frequencies of chordless cycles to encode dimensionality information.

Load-bearing premise

The distribution of chordless cycles in a network reliably encodes the dimensionality of its underlying hyperbolic geometry.

What would settle it

Generate synthetic graphs with independently controlled hyperbolic dimensions, apply the chordless-cycle descriptors and trained neural network, and check whether the predicted dimensions match the controlled inputs within a small error margin; mismatch on this controlled test would falsify the central claim.

Figures

Figures reproduced from arXiv: 2509.08350 by Aina Ferr\`a Marc\'us, Carles Casacuberta, M. \'Angeles Serrano, Meritxell Vila Mi\~nana, Robert Jankowski.

Figure 1
Figure 1. Figure 1: FIG. 1: 3D view of a point cloud representing an ensemble of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: (a) Pipeline of our first method and (b) confusion matrices. Given a target network, we generate an ensemble of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: (a) Pipeline of dimensionality estimation using [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Validation accuracies after five repetitions of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: (a) Number of networks for each network domain. (b) Inferred dimension of real networks using [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Degree-splitting subdivision of a graph [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
read the original abstract

Many complex networks, ranging from social to biological systems, exhibit structural patterns consistent with an underlying hyperbolic geometry. Revealing the dimensionality of this latent space can disentangle the structural complexity of communities, impact efficient network navigation, and fundamentally shape connectivity and system behavior. We introduce a topological data analysis weighting scheme for graphs based on chordless cycles to estimate network dimensionality in a data-driven way. We further show that the resulting descriptors can effectively estimate network dimensionality using a neural network architecture trained on a synthetic graph database constructed for this purpose, which requires no retraining to transfer effectively to real-world networks. Thus, by combining cycle-aware filtrations, algebraic topology, and machine learning, our approach provides a robust and effective method for uncovering the hidden geometry of complex networks and guiding accurate modeling and low-dimensional embedding.

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 introduces a topological data analysis weighting scheme based on chordless cycles to estimate the dimensionality of latent hyperbolic geometry in complex networks. A neural network is trained on descriptors derived from a synthetic database of hyperbolic random graphs with controlled dimensions and is shown to transfer effectively to real-world networks without retraining, combining cycle-aware filtrations, algebraic topology, and machine learning to uncover hidden network geometry.

Significance. If the central claims hold, the work offers a data-driven alternative to existing methods for detecting latent dimensionality, with potential applications in network navigation, community detection, and embedding. The construction of a dedicated synthetic graph database and the zero-shot transfer demonstration are explicit strengths that support reproducibility and broader applicability. The integration of chordless-cycle filtrations with neural networks is a novel contribution to the intersection of TDA and network geometry.

major comments (2)
  1. [Abstract] Abstract: The claim that the resulting descriptors 'can effectively estimate network dimensionality' and enable transfer 'effectively to real-world networks' with no retraining is load-bearing for the paper's contribution. However, the abstract provides no quantitative metrics (e.g., prediction error, accuracy on held-out real networks, or comparison to baselines), making it impossible to evaluate whether the chordless-cycle descriptors isolate hyperbolic dimension from other topological features.
  2. [Methods and Results] Methods and Results sections: The transfer success assumes chordless cycle filtrations produce descriptors insensitive to non-geometric confounders such as degree correlations or modular overlays. No ablation is described that adds controlled non-hyperbolic structure to the synthetic graphs while holding dimension fixed and checks whether the NN output remains stable; without this, the zero-shot claim rests on an untested assumption.
minor comments (1)
  1. The abstract would be strengthened by including at least one key quantitative result (e.g., mean absolute error on real networks) to support the effectiveness claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment, as well as for the constructive major comments. We address each point below and agree to revisions that strengthen the manuscript without misrepresenting our results.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that the resulting descriptors 'can effectively estimate network dimensionality' and enable transfer 'effectively to real-world networks' with no retraining is load-bearing for the paper's contribution. However, the abstract provides no quantitative metrics (e.g., prediction error, accuracy on held-out real networks, or comparison to baselines), making it impossible to evaluate whether the chordless-cycle descriptors isolate hyperbolic dimension from other topological features.

    Authors: We agree that the abstract would be strengthened by the inclusion of quantitative metrics to support the central claims. In the revised manuscript we will update the abstract to report key performance figures, including mean absolute error on held-out synthetic graphs and observed accuracy when applied to real-world networks, together with a brief comparison to baseline dimensionality estimators. revision: yes

  2. Referee: [Methods and Results] Methods and Results sections: The transfer success assumes chordless cycle filtrations produce descriptors insensitive to non-geometric confounders such as degree correlations or modular overlays. No ablation is described that adds controlled non-hyperbolic structure to the synthetic graphs while holding dimension fixed and checks whether the NN output remains stable; without this, the zero-shot claim rests on an untested assumption.

    Authors: This observation correctly identifies a gap in the current presentation. Although the zero-shot transfer to real networks (which contain such confounders) offers indirect support, an explicit controlled ablation would provide stronger evidence. We will add a dedicated ablation subsection in the revised Methods and Results, in which we overlay modular structure or degree correlations onto synthetic hyperbolic graphs at fixed dimension and report the resulting stability of the neural-network predictions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses independent synthetic training and external transfer

full rationale

The paper constructs a synthetic graph database from hyperbolic random graphs with controlled dimensions, computes chordless-cycle filtration descriptors, trains a neural network on those descriptors to predict dimension, and applies the fixed model to real networks. This workflow does not reduce any claimed prediction to a fitted parameter by construction, nor does it define dimensionality in terms of the descriptors themselves. No self-citation is invoked as a load-bearing uniqueness theorem, no ansatz is smuggled via prior work, and the central result is not a renaming of a known empirical pattern. The approach remains self-contained against external benchmarks because the synthetic data generation and real-network application are distinct steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central approach rests on the domain assumption that complex networks exhibit hyperbolic geometry; no free parameters or invented entities are described in the abstract.

axioms (1)
  • domain assumption Many complex networks exhibit structural patterns consistent with an underlying hyperbolic geometry.
    Stated in the opening sentence of the abstract as the motivation for dimensionality detection.

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Reference graph

Works this paper leans on

55 extracted references · 55 canonical work pages

  1. [1]

    Ballester and B

    R. Ballester and B. Rieck, inThe Third Learning on Graphs Conference(2024), URLhttps://openreview. net/forum?id=phs5A7bUPA

    work page 2024

  2. [2]

    Carlsson and M

    G. Carlsson and M. Vejdemo-Johansson,Topological data analysis with applications(Cambridge University Press, 2021)

    work page 2021

  3. [3]

    Edelsbrunner and J

    H. Edelsbrunner and J. L. Harer,Computational Topol- ogy: An Introduction(American Mathematical Soceity, 2022)

    work page 2022

  4. [4]

    Wasserman, Annual Review of Statistics and its Ap- plication5, 501 (2018)

    L. Wasserman, Annual Review of Statistics and its Ap- plication5, 501 (2018)

    work page 2018

  5. [5]

    Forman, Discrete & Computational Geometry29, 323 (2003)

    R. Forman, Discrete & Computational Geometry29, 323 (2003)

    work page 2003

  6. [6]

    L. C. Freeman, Sociometry40, 35 (1977)

    work page 1977

  7. [7]

    G.Bianconi,Higher-Order Networks(CambridgeUniver- sity Press, 2021)

    work page 2021

  8. [8]

    Bobrowski and D

    O. Bobrowski and D. Krioukov, inHigher-Order Systems (Springer, 2022), pp. 59–96

    work page 2022

  9. [9]

    Hernández Serrano, J

    D. Hernández Serrano, J. Hernández Serrano, and D. Sánchez Gómez, Chaos, Solitons and Fractals137, 109839 (2020)

    work page 2020

  10. [10]

    Salnikov, D

    V. Salnikov, D. Cassese, and R. Lambiotte, European Journal of Physics40, 014001 (2018)

    work page 2018

  11. [11]

    J. J. Torres and G. Bianconi, Journal of Physics: Com- plexity1, 015002 (2020)

    work page 2020

  12. [12]

    Horak, S

    D. Horak, S. Maletić, and M. Rajković, J. Stat. Mech. 2009, P03034 (2009)

    work page 2009

  13. [13]

    Taylor, Klimm, et al., Nature Communications6, 7723 (2015)

    D. Taylor, Klimm, et al., Nature Communications6, 7723 (2015)

    work page 2015

  14. [14]

    Kannan, E

    H. Kannan, E. Saucan, I. Roy, and A. Samal, Scientific Reports9(2019)

    work page 2019

  15. [15]

    Myers, E

    A. Myers, E. Munch, and F. A. Khasawneh, Phys. Rev. E100(2019)

    work page 2019

  16. [16]

    Guerra, A

    M. Guerra, A. D. Gregorio, U. Fugacci, G. Petri, and F. Vaccarino, Scientific Reports11(2021)

    work page 2021

  17. [17]

    Jhun, Chaos32(2022)

    B. Jhun, Chaos32(2022)

    work page 2022

  18. [18]

    Petri, P

    G. Petri, P. Expert, F. Turkheimer, R. Carhart-Harris, D. Nutt, P. J. Hellyer, and F. Vaccarino, Journal of the Royal Society Interface11(2014)

    work page 2014

  19. [19]

    Giusti, E

    C. Giusti, E. Pastalkova, C. Curto, and V. Itsko, PNAS 112, 13455 (2015)

    work page 2015

  20. [20]

    A. E. Sizemore, J. E. Phillips-Cremins, R. Ghrist, and D. S. Bassett, Network Neuroscience3, 656 (2019)

    work page 2019

  21. [21]

    S. Das, D. V. Anand, and M. K. Chung, PLoS One18, e0276419 (2023)

    work page 2023

  22. [22]

    Boguñá, I

    M. Boguñá, I. Bonamassa, M. De Domenico, S. Havlin, D. Krioukov, and M. Á. Serrano, Nature Reviews Physics 3, 114 (2021)

    work page 2021

  23. [23]

    Almagro, M

    P. Almagro, M. Boguñá, and M. A. Serrano, Nature Communications13, 6096 (2022)

    work page 2022

  24. [24]

    M. Á. Serrano, D. Krioukov, and M. Boguñá, Physical Review Letters100, 078701 (2008)

    work page 2008

  25. [25]

    Budel, M

    G. Budel, M. Kitsak, R. Aldecoa, K. Zuev, and D. Kri- oukov, Physical Review E109, 054131 (2024)

    work page 2024

  26. [26]

    Jankowski, A

    R. Jankowski, A. Allard, M. Boguñá, and M. Á. Serrano, Nature Communications14, 7585 (2023)

    work page 2023

  27. [27]

    M. A. Serrano and M. Boguñá,The Shortest Path to Network Geometry(Cambridge University Press, Cam- bridge, UK, 2021)

    work page 2021

  28. [28]

    Krioukov, F

    D. Krioukov, F. Papadopoulos, M. Kitsak, A. Vahdat, 12 and M. Boguñá, Physical Review E – Statistical, Nonlin- ear, and Soft Matter Physics82(2010)

    work page 2010

  29. [29]

    Starnini, E

    M. Starnini, E. Ortiz, and M. Á. Serrano, New Journal of Physics21, 053039 (2019)

    work page 2019

  30. [30]

    Ghasemian, H

    A. Ghasemian, H. Hosseinmardi, and A. Clauset, IEEE Transactions on Knowledge and Data Engineering32, 1722 (2019)

    work page 2019

  31. [31]

    Vila Miñana, R

    M. Vila Miñana, R. Jankowski, A. Ferrà Marcús, R. Ballester, M. A. Serrano, and C. Casacuberta, In preparation (2025)

    work page 2025

  32. [32]

    M. E. Aktas, E. Akbas, and A. El Fatmaoui, Applied Network Science4(2019)

    work page 2019

  33. [33]

    Boguñá, R

    M. Boguñá, R. Pastor-Satorras, and A. Vespignani, The European Physical Journal B38, 205 (2004)

    work page 2004

  34. [34]

    Marsaglia, The Annals of Mathematical Statistics43, 645 (1972)

    G. Marsaglia, The Annals of Mathematical Statistics43, 645 (1972)

    work page 1972

  35. [35]

    M. Á. Serrano and M. Boguñá, Physical Review E – Sta- tistical, Nonlinear, and Soft Matter Physics74, 056114 (2006)

    work page 2006

  36. [36]

    J.Nickolls, I.Buck, M.Garland, andK.Skadron, inACM SIGGRAPH 2008 Classes(Association for Computing Machinery, New York, NY, USA, 2008), SIGGRAPH ’08, ISBN 9781450378451, URLhttps://doi.org/10.1145/ 1401132.1401152

    work page arXiv 2008

  37. [37]

    Hatcher,Algebraic Topology(Cambridge University Press, 2002)

    A. Hatcher,Algebraic Topology(Cambridge University Press, 2002)

    work page 2002

  38. [38]

    Cohen-Steiner, H

    D. Cohen-Steiner, H. Edelsbrunner, and J. Harer, Foun- dations of Computational Mathematics9, 79 (2009)

    work page 2009

  39. [39]

    Carrière, F

    M. Carrière, F. Chazal, Y. Ike, T. Lacombe, M. Royer, and Y. Umeda, inProceedings of the 23rd International Conference on Artificial Intelligence and Statistics (AIS- TATS)(2020), vol. 108 ofPMLR

    work page 2020

  40. [40]

    The GUDHI Project,GUDHI User and Reference Man- ual(GUDHI Editorial Board, 2015), URLhttp:// gudhi.gforge.inria.fr/doc/latest/

    work page 2015

  41. [41]

    Albert and A.-L

    R. Albert and A.-L. Barabási, Rev. Mod. Phys74(2002)

    work page 2002

  42. [42]

    Krizhevsky, I

    A. Krizhevsky, I. Sutskever, and G. E. Hinton, inAd- vances in Neural Information Processing Systems(2012), pp. 1097–1105

    work page 2012

  43. [43]

    LeCun, Y

    Y. LeCun, Y. Bengio, and G. Hinton, Nature521, 436 (2015)

    work page 2015

  44. [44]

    Touvron et al., IEEE Transactions on Pattern Analy- sis and Machine Intelligence45, 5314 (2023)

    H. Touvron et al., IEEE Transactions on Pattern Analy- sis and Machine Intelligence45, 5314 (2023)

    work page 2023

  45. [45]

    Chordless cycle filtrations for dimensionality detection in complex networks via topological data analysis

    K. He, X. Zhang, S. Ren, and J. Sun, in2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)(2016), pp. 770–778. Supplementary Information for “Chordless cycle filtrations for dimensionality detection in complex networks via topological data analysis” Aina Ferrà Marcús,1 Robert Jankowski,2, 3 Meritxell Vila Miñana,4 Carles Casacuberta,1 a...

    work page 2016

  46. [46]

    Dimensionality estimation of real networks with persistence descriptors 2

    work page

  47. [47]

    Neural network architecture and predictions 4

    work page

  48. [48]

    Topological properties of real networks and their inferred dimensions 7

    work page

  49. [49]

    Network generation algorithms 11 Supplementary References 11 ∗ marian.serrano@ub.edu 2

    work page

  50. [50]

    DIMENSIONALITY ESTIMA TION OF REAL NETWORKS WITH PERSISTENCE DESCRIPTORS FIG. S1: 2D projections representing an ensemble of1330 surrogates for each real network in the phase space of total persistence computed from three chordless cycle densities (triangles, squares, and pentagons). Points are colored by dimension, and the target network is marked with a...

    work page

  51. [51]

    S3: Schematic representation of theDimNN neural network architecture

    NEURAL NETWORK ARCHITECTURE AND PREDICTIONS FIG. S3: Schematic representation of theDimNN neural network architecture. Boxes indicate layers within the MLP, with the number above each box specifying the number of neurons in that layer. Blue arrows are used to denote skip (residual) connections. Dashed gray boxes denote repeated layer blocks, with the repe...

    work page

  52. [52]

    For more details about each network, we refer to [1] and [2]

    TOPOLOGICAL PROPER TIES OF REAL NETWORKS AND THEIR INFERRED DIMENSIONS We compile a dataset of 53 real-world networks from various domains. For more details about each network, we refer to [1] and [2]. 1000 2000 3000 N 0.0 2.5 5.0 7.5 10.0 12.5Count 10 20 30 k 0 5 10 15 20 5 10 k2 / k 2 0 2 4 6 8 5 10 15 kmin 0 10 20 30 40 50Count 0 500 1000 1500 kmax 0 2...

    work page 2000

  53. [53]

    NETWORK GENERA TION ALGORITHMS Algorithm 1 SD network generation algorithm 1: Input: Dimension D, number of nodesN, power-law exponentγ, average degree⟨κ⟩, inverse temperatureβ 2: Output: Graph G = (V,E ) with nodes expressed in angular coordinates 3: V←{} 4: E←{} 5: κ0←⟨k⟩(γ−2)(1−N − 1)/[(γ−1)(1−N (2−γ)/(γ− 1))] 6: κc←κ0N 1/(γ− 1) 7: for i = 1 to N do 8:...

    work page

  54. [54]

    Almagro, M

    P. Almagro, M. Boguñá, and M. Á. Serrano, Nature Communications13, 6096 (2022)

    work page 2022

  55. [55]

    Ghasemian, H

    A. Ghasemian, H. Hosseinmardi, and A. Clauset, IEEE Transactions on Knowledge and Data Engineering32, 1722 (2019) 24

    work page 2019