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arxiv: 2601.16086 · v2 · pith:7ZURP62Vnew · submitted 2026-01-22 · ❄️ cond-mat.stat-mech · nlin.AO· physics.soc-ph

Random Walks Across Dimensions: Exploring Simplicial Complexes

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

Pith reviewed 2026-05-21 15:12 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech nlin.AOphysics.soc-ph
keywords random walkssimplicial complexeshigher-order networksasymptotic distributionstochastic processesnetwork rankingteleportation
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The pith

A nested random walk operator on simplicial complexes ranks higher-order structures by the long-term distribution of walkers.

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

The paper introduces a novel operator for random walks that allow movement across simplices of different dimensions in a nested hierarchical manner. This process connects nodes to edges and edges to triangles, extending to higher dimensions. The long-term distribution of these walkers then serves as a way to measure the relative importance of these higher-order simplices. It also explores how adding stochastic teleportation affects search strategies and how noise interacts with these structures.

Core claim

By constructing a nested operator that organizes random walks hierarchically across dimensions in a simplicial complex, the asymptotic distribution of the walkers provides a natural ranking to gauge the relative importance of higher order simplices.

What carries the argument

The nested operator that bridges random walks between simplices of successive dimensions while maintaining hierarchical consistency.

If this is right

  • Walkers can transition from nodes to edges, edges to triangles, and so on up to finite higher dimensions.
  • The stationary distribution of walkers ranks the importance of these simplices.
  • Optimal search strategies emerge when stochastic teleportation is included.
  • The presence of noise reveals specific interactions with higher-order structures.

Where Pith is reading between the lines

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

  • This ranking method could be tested on real simplicial complexes from biology or social data to identify key higher-order interactions.
  • The approach might connect to random walk models on hypergraphs for broader network analysis.
  • Extensions could explore how the ranking changes under different noise levels or teleportation rates.

Load-bearing premise

A well-defined nested operator exists allowing walkers to move across simplices of arbitrary but finite dimension while preserving a consistent hierarchical structure.

What would settle it

Simulating the random walk on a simple simplicial complex such as a tetrahedron and checking if the walker distribution ranks the triangles and edges in a way that matches independent measures of importance would test the claim; mismatch would falsify it.

Figures

Figures reproduced from arXiv: 2601.16086 by Diego Febbe, Duccio Fanelli, Timoteo Carletti.

Figure 1
Figure 1. Figure 1: FIG. 1: Example of a random walk process across the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Correspondence between the asymptotic [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Each point of these triangles corresponds to a [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Panel (a): Size-rescaled explorability for a simplicial [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

We introduce a novel operator to describe a random walk process on a simplicial complex. Walkers are allowed to wonder across simplices of various dimensions, bridging nodes to edges, and edges to triangles, via a nested organization that hierarchically extends to higher structures of arbitrary large, but finite, dimension. The asymptotic distribution of the walkers provides a natural ranking to gauge the relative importance of higher order simplices. Optimal search strategies in presence of stochastic teleportation are addressed and the peculiar interplay of noise with higher order structures unraveled.

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 paper introduces a novel random walk operator on simplicial complexes allowing walkers to move across simplices of different dimensions via a nested hierarchical organization. It claims that the resulting asymptotic distribution provides a natural ranking of the relative importance of higher-order simplices and examines optimal search strategies under stochastic teleportation, highlighting the interplay between noise and higher-dimensional structures.

Significance. If the operator construction and convergence properties can be rigorously established, the work would extend classical random-walk ranking methods to simplicial complexes and offer a new approach to quantifying importance in higher-order network structures. The treatment of teleportation and noise is a potentially useful addition, though the current lack of explicit transition rules and proofs limits immediate impact.

major comments (2)
  1. [§2] §2 (Definition of the nested operator): The inter-dimension transition rules and normalization constants for jumps between nodes, edges, and higher simplices are not specified, so it is impossible to verify that the transition kernel is stochastic or that the process on the union of all simplices forms a well-defined Markov chain.
  2. [§4] §4 (Asymptotic distribution and ranking): The central claim that the stationary distribution exists, is unique, and furnishes a meaningful ranking rests on irreducibility and aperiodicity, yet no proof or explicit incidence-matrix construction is supplied; without these the ranking result is not established.
minor comments (1)
  1. [Abstract] The phrase 'arbitrary large, but finite, dimension' in the abstract should read 'arbitrarily large but finite dimension'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the two major comments point by point below and have revised the manuscript to incorporate the requested clarifications and proofs.

read point-by-point responses

  1. Referee: [§2] §2 (Definition of the nested operator): The inter-dimension transition rules and normalization constants for jumps between nodes, edges, and higher simplices are not specified, so it is impossible to verify that the transition kernel is stochastic or that the process on the union of all simplices forms a well-defined Markov chain.

    Authors: We agree that the explicit inter-dimension transition rules and normalization constants were not presented with sufficient detail in the original manuscript. In the revised version we expand Section 2 to give the precise transition probabilities between simplices of consecutive dimensions, including the explicit normalization constants that guarantee each row of the transition matrix sums to one. We also supply the full block-structured transition kernel on the disjoint union of all simplices, making the Markov-chain property immediate to verify. revision: yes

  2. Referee: [§4] §4 (Asymptotic distribution and ranking): The central claim that the stationary distribution exists, is unique, and furnishes a meaningful ranking rests on irreducibility and aperiodicity, yet no proof or explicit incidence-matrix construction is supplied; without these the ranking result is not established.

    Authors: The referee is correct that a rigorous justification was missing. In the revised manuscript we add a new subsection to Section 4 that (i) states the connectivity assumption on the simplicial complex, (ii) proves irreducibility and aperiodicity of the resulting chain, and (iii) constructs the incidence matrices explicitly so that the stationary distribution can be shown to induce a well-defined ranking of higher-order simplices. These additions establish the central claim. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on standard Markov chain theory applied to a novel but explicitly constructed operator

full rationale

The paper defines a nested random walk operator on simplicial complexes by specifying transition rules across dimensions (nodes to edges to higher simplices) and then derives the stationary distribution as the ranking measure. This follows the standard Perron-Frobenius or ergodic theorem route for irreducible aperiodic chains and does not reduce any claimed prediction to a fitted parameter or self-referential definition. No self-citation is used to justify uniqueness or existence; the construction is presented as self-contained. The asymptotic ranking is a direct consequence of the defined transition kernel rather than an input renamed as output. External benchmarks (existence of stationary distributions on finite graphs) are independent of the paper's specific nesting weights.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim depends on the existence of a hierarchical nested organization across dimensions and the well-posedness of the new operator; no free parameters or invented physical entities are mentioned in the abstract.

axioms (1)
  • domain assumption A simplicial complex admits a consistent nested hierarchical structure allowing transitions between simplices of consecutive dimensions.
    Invoked to define the random walk operator that bridges nodes to edges to triangles etc.
invented entities (1)
  • Novel random walk operator on simplicial complexes no independent evidence
    purpose: To describe movement across different dimensional simplices in a hierarchical manner
    Introduced as the core new mathematical object; no independent evidence provided in abstract

pith-pipeline@v0.9.0 · 5620 in / 1213 out tokens · 51756 ms · 2026-05-21T15:12:05.125113+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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unclear
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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. Model of Simplicial Complexes with dimension-wise preferential attachment

    cond-mat.stat-mech 2026-05 unverdicted novelty 7.0

    A dimension-wise preferential attachment model for simplicial complexes yields power-law generalized degree distributions for simplices of varying dimension.

  2. Smart Walkers in Discrete Space

    cond-mat.stat-mech 2026-01 unverdicted novelty 5.0

    Configuration entropy serves as a reliable proxy for the learned skills of reinforcement learning agents performing tasks in discrete space, validated through walker encounters and chess engine tests.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages · cited by 2 Pith papers · 1 internal anchor

  1. [1]

    Random Walks Across Dimensions: Exploring Simplicial Complexes

    and incessantly invoked as a fundamental modeling ingredient [27–29]. Starting from these premises, we here aim at filling an evident gap by extending the elemental random walks recipe to higher order simplices. At vari- ance with past attempts [30, 31], we here leverage on the generalized connectivity as naturally defined by the boundary operators, to dr...

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  2. [2]

    M. E. Newman,Networks: An introduction(Oxford uni- versity press, 2010)

    work page 2010

  3. [3]

    Barab´ asi,Network Science(Cambridge University Press, 2016)

    A.-L. Barab´ asi,Network Science(Cambridge University Press, 2016)

    work page 2016

  4. [4]

    Barabasi and Z

    A.-L. Barabasi and Z. N. Oltvai, Nature Reviews Genet- ics5, 101 (2004)

    work page 2004

  5. [5]

    Lucas, A

    M. Lucas, A. Morris, A. Townsend-Teague, L. Tichit, B. Habermann, and A. Barrat, Cell Reports Methods3 (2023)

    work page 2023

  6. [6]

    D. L. Barab´ asi, G. Bianconi, E. Bullmore, M. Burgess, S. Chung, T. Eliassi-Rad, D. George, I. A. Kov´ acs, H. Makse, T. E. Nichols,et al., Journal of Neuroscience 43, 5989 (2023)

    work page 2023

  7. [7]

    Nishikawa and A

    T. Nishikawa and A. E. Motter, New Journal of Physics 17, 015012 (2015)

    work page 2015

  8. [8]

    Febbe, A

    D. Febbe, A. Di Garbo, R. Mannella, R. Meucci, and D. Fanelli, in2024 IEEE Workshop on Complexity in Engineering (COMPENG)(IEEE, 2024) pp. 1–7

    work page 2024

  9. [9]

    Buongiovanni, R

    C. Buongiovanni, R. Candusso, G. Cerretini, D. Febbe, V. Morini, and G. Rossetti, inInternational Conference on Complex Networks and Their Applications(Springer,

    work page

  10. [10]

    Ha laj, S

    G. Ha laj, S. Martinez-Jaramillo, and S. Battiston, Jour- nal of Financial Stability71, 101228 (2024)

    work page 2024

  11. [11]

    Pastor-Satorras and A

    R. Pastor-Satorras and A. Vespignani, Physical review letters86, 3200 (2001)

    work page 2001

  12. [12]

    Boccaletti, V

    S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.- U. Hwang, Physics reports424, 175 (2006)

    work page 2006

  13. [13]

    Delvenne, R

    J.-C. Delvenne, R. Lambiotte, and L. E. Rocha, Nature communications6, 7366 (2015)

    work page 2015

  14. [14]

    M. T. Schaub, J.-C. Delvenne, R. Lambiotte, and M. Barahona, Advances in Network Clustering and Blockmodeling , 333 (2019)

    work page 2019

  15. [15]

    Battiston, G

    F. Battiston, G. Cencetti, I. Iacopini, V. Latora, M. Lu- cas, A. Patania, J.-G. Young, and G. Petri, Physics re- ports874, 1 (2020)

    work page 2020

  16. [16]

    Battiston, E

    F. Battiston, E. Amico, A. Barrat, G. Bianconi, G. Fer- raz de Arruda, B. Franceschiello, I. Iacopini, S. K´ efi, V. Latora, Y. Moreno,et al., Nature physics17, 1093 6 (2021)

    work page 2021

  17. [17]

    C. Bick, E. Gross, H. A. Harrington, and M. T. Schaub, SIAM review65, 686 (2023)

    work page 2023

  18. [18]

    A. P. Mill´ an, H. Sun, L. Giambagli, R. Muolo, T. Car- letti, J. J. Torres, F. Radicchi, J. Kurths, and G. Bian- coni, Nature Physics , 1 (2025)

    work page 2025

  19. [19]

    Giusti, R

    C. Giusti, R. Ghrist, and D. S. Bassett, Journal of com- putational neuroscience41, 1 (2016)

    work page 2016

  20. [20]

    Petri and A

    G. Petri and A. Barrat, Physical review letters121, 228301 (2018)

    work page 2018

  21. [21]

    Chowdhary, A

    S. Chowdhary, A. Kumar, G. Cencetti, I. Iacopini, and F. Battiston, Journal of Physics: Complexity2, 035019 (2021)

    work page 2021

  22. [22]

    Iacopini, G

    I. Iacopini, G. Petri, A. Barrat, and V. Latora, Nature communications10, 2485 (2019)

    work page 2019

  23. [23]

    Muolo, L

    R. Muolo, L. Giambagli, H. Nakao, D. Fanelli, and T. Carletti, inProceedings A, Vol. 480 (The Royal So- ciety, 2024) p. 20240235

    work page 2024

  24. [24]

    L. V. Gambuzza, F. Di Patti, L. Gallo, S. Lepri, M. Ro- mance, R. Criado, M. Frasca, V. Latora, and S. Boc- caletti, Nature communications12, 1255 (2021)

    work page 2021

  25. [25]

    Gallo, R

    L. Gallo, R. Muolo, L. V. Gambuzza, V. Latora, M. Frasca, and T. Carletti, Communications Physics5, 263 (2022)

    work page 2022

  26. [26]

    Carletti, L

    T. Carletti, L. Giambagli, and G. Bianconi, Physical Re- view Letters130, 187401 (2023)

    work page 2023

  27. [27]

    Lov´ asz, Combinatorics, Paul erdos is eighty2, 4 (1993)

    L. Lov´ asz, Combinatorics, Paul erdos is eighty2, 4 (1993)

    work page 1993

  28. [28]

    Masuda, M

    N. Masuda, M. A. Porter, and R. Lambiotte, Physics reports716, 1 (2017)

    work page 2017

  29. [29]

    R. N. Bhattacharya and E. C. Waymire,Random walk, Brownian motion, and martingales, Vol. 52 (Springer, 2021)

    work page 2021

  30. [30]

    G. Peri, L. Buffoni, R. Marino, and D. Fanelli, [Work in submission] (2026)

    work page 2026

  31. [31]

    Carletti, F

    T. Carletti, F. Battiston, G. Cencetti, and D. Fanelli, Physical review E101, 022308 (2020)

    work page 2020

  32. [32]

    M. T. Schaub, A. R. Benson, P. Horn, G. Lippner, and A. Jadbabaie, SIAM Review62, 353 (2020)

    work page 2020

  33. [33]

    L. J. Grady and J. R. Polimeni,Discrete calculus: Ap- plied analysis on graphs for computational science, Vol. 3 (Springer, 2010)

    work page 2010

  34. [34]

    Bianconi,Higher-order networks(Cambridge Univer- sity Press, 2021)

    G. Bianconi,Higher-order networks(Cambridge Univer- sity Press, 2021)

    work page 2021

  35. [35]

    Febbe, D

    D. Febbe, D. Fanelli, and T. Carletti, [Work in submis- sion] (2026)

    work page 2026

  36. [36]

    Bianconi and C

    G. Bianconi and C. Rahmede, Physical Review E93, 032315 (2016)

    work page 2016

  37. [37]

    K. Zuev, O. Eisenberg, and D. Krioukov, Journal of Physics A: Mathematical and Theoretical48, 465002 (2015)

    work page 2015

  38. [38]

    Dorogovtsev and P

    S. Dorogovtsev and P. Krapivsky, arXiv preprint arXiv:2507.07402 (2025)

    work page arXiv 2025

  39. [39]

    A. P. R. A. University), Ahorn: A Dataset of Higher-Order Networks,https://ahorn.rwth-aachen. de/dataset

    work page

  40. [40]

    Di Patti, D

    F. Di Patti, D. Fanelli, and F. Piazza, Scientific reports 5, 9869 (2015)

    work page 2015

  41. [41]

    L. Page, S. Brin, R. Motwani, and T. Winograd,The PageRank Citation Ranking: Bringing Order to the Web, Tech. Rep. 1999-66 (Stanford InfoLab, 1999). [41]https://github.com/diegofebbe/Random_walk_on_ simplicial_complexes/tree/master

    work page 1999