pith. sign in

arxiv: 2605.16987 · v1 · pith:RBDBGBKRnew · submitted 2026-05-16 · ❄️ cond-mat.stat-mech

Percolation transition of strongly connected clusters in finite dimensions and on complete graphs

Qi Wang , Ming Li This is my paper

Pith reviewed 2026-05-19 19:25 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords percolationstrongly connected clustersdirected bondsfractal dimensionhyperscalingmean-field limitcomplete graphsfinite dimensions
0
0 comments X

The pith

Critical strongly connected clusters in directed percolation remain fractal across all dimensions with a change at six.

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

The paper examines percolation of strongly connected clusters, groups of sites mutually reachable by directed paths, on hypercubic lattices in dimensions 2 through 7 and on complete graphs with randomly oriented bonds. It establishes that below the upper critical dimension of 6 these clusters possess a nontrivial fractal dimension and their size distribution obeys a hyperscaling relation linking the exponent to that dimension. Above six dimensions mean-field scaling is recovered, with the fractal dimension over spatial dimension equal to one third, and this is consistent with complete-graph results. Complete graphs additionally display a double-scaling form in the size distribution, with large clusters following the mean-field exponent of 4 and small clusters following a separate exponent of 1. Giant in-clusters and out-clusters at criticality share the fractal dimension of ordinary undirected percolation clusters.

Core claim

Below the upper critical dimension d_u=6, the critical SCCs exhibit nontrivial fractal dimension d_SCC, and the size distribution scales as ∼s^{-τ_SCC} with the hyperscaling relation τ_SCC=1+d/d_SCC. For d≥d_u, mean-field behavior is recovered with d_SCC/d=1/3, consistent with complete-graph results. However, in contrast to hypercubic lattices, complete graphs exhibit a double-scaling structure in the SCC size distribution: large SCCs are governed by mean-field value τ_SCC=4, while small SCCs follow a distinct power law with exponent τ'=1. At criticality, the giant in- and out-clusters are also fractal, sharing the same dimension as standard percolation clusters.

What carries the argument

Strongly connected cluster (SCC), a set of sites mutually reachable through directed paths, which serves as the object whose fractal dimension and size-distribution exponents are measured at the percolation transition.

If this is right

  • Critical SCCs possess a nontrivial fractal dimension d_SCC below six dimensions.
  • The size-distribution exponent of SCCs satisfies the hyperscaling relation τ_SCC equals 1 plus d over d_SCC below the upper critical dimension.
  • Above six dimensions and on complete graphs the ratio d_SCC over d equals one third.
  • On complete graphs the SCC size distribution splits into two power-law regimes with τ_SCC=4 for large clusters and τ'=1 for small clusters.
  • Giant in- and out-clusters at criticality share the fractal dimension of standard undirected percolation clusters.

Where Pith is reading between the lines

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

  • The double-scaling observed only on complete graphs suggests that mean-field descriptions of directed percolation may conceal a separation between local and global cluster regimes.
  • The persistence of distinct fractal geometry for SCCs below six dimensions implies that directed connectivity can produce geometric criticality that survives longer into the finite-dimensional regime than undirected percolation alone would suggest.
  • These scaling relations provide a concrete benchmark for testing whether other directed network models, such as those with heterogeneous degrees, recover the same hyperscaling or double-scaling structure.

Load-bearing premise

The simulations correctly locate the percolation transition and extract fractal dimensions and size-distribution exponents without dominant finite-size or sampling artifacts that would alter the reported scaling relations.

What would settle it

A simulation in dimension 5 on much larger lattices that finds the measured τ_SCC deviating from 1 plus 5 divided by the measured d_SCC would falsify the hyperscaling claim.

Figures

Figures reproduced from arXiv: 2605.16987 by Ming Li, Qi Wang.

Figure 1
Figure 1. Figure 1: FIG. 1. Sketch of the bow-tie structure of a directed graph. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Illustration of SCC percolation on a directed square [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Finite-size scaling of the Binder cumulant [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Finite-size scaling of the largest SCC on finite [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The rescaled largest SCC, [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: , data for different system sizes collapse onto a sin￾gle universal curve, confirming the validity of the finite￾size scaling form. These results demonstrate that SCCs exhibit conven￾tional critical scaling behavior analogous to standard per￾colation clusters, despite their higher-order connectivity constraints. 5. Fractal dimensions of ICs and OCs We further find that both ICs and OCs exhibit fractal geom… view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Finite-size scaling of SCCs on complete graphs. (a) [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Mean number of SCCs, [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
read the original abstract

We study the percolation of strongly connected clusters (SCCs), in which sites are mutually reachable through directed paths, in systems with randomly oriented bonds by extensive simulations on hypercubic lattices from dimension $d=2$ to $7$ and complete graphs. Below the upper critical dimension $d_u=6$, the critical SCCs exhibit nontrivial fractal dimension $d_{\rm SCC}$, and the size distribution scales as $\sim s^{-\tau_{\rm SCC}}$ with the hyperscaling relation $\tau_{\rm SCC}=1+d/d_{\rm SCC}$. For $d \ge d_u$, mean-field behavior is recovered with $d_{\rm SCC}/d=1/3$, consistent with complete-graph results. However, in contrast to hypercubic lattices, complete graphs exhibit a double-scaling structure in the SCC size distribution: large SCCs are governed by mean-field value $\tau_{\rm SCC}=4$, while small SCCs follow a distinct power law with exponent $\tau'=1$. At criticality, the giant in- and out-clusters are also fractal, sharing the same dimension as standard percolation clusters. These results show that critical SCCs remain well-defined fractal objects across dimensions, while their approach to the mean-field limit involves nontrivial changes in cluster statistics.

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 studies the percolation transition of strongly connected clusters (SCCs) in systems with randomly oriented bonds. Using extensive simulations on hypercubic lattices (d=2 to 7) and complete graphs, it reports that below the upper critical dimension du=6 the critical SCCs are fractal with dimension d_SCC satisfying the hyperscaling relation τ_SCC=1+d/d_SCC. For d≥du mean-field behavior is recovered with d_SCC/d=1/3. Complete graphs exhibit a double-scaling structure in the SCC size distribution (τ_SCC=4 for large SCCs, τ'=1 for small SCCs), while lattices are stated to recover mean-field scaling. Giant in- and out-clusters at criticality are fractal with the same dimension as ordinary percolation clusters.

Significance. If substantiated, the results establish that critical SCCs remain well-defined fractal objects in all dimensions and that the crossover to mean-field involves nontrivial changes in cluster statistics. The direct comparison between finite-d lattices and complete graphs is a strength, as is the numerical support for the scaling relations and the exploration of both lattice and complete-graph limits.

major comments (2)
  1. [Results for d≥6] Results for d≥6 and comparison with complete graphs: the claim that mean-field behavior (d_SCC/d=1/3) is recovered on hypercubic lattices for d≥6 is presented as consistent with complete-graph results, yet the manuscript explicitly contrasts the SCC size-distribution structure (double scaling with τ_SCC=4 tail on complete graphs versus the lattice data). If d=7 lattices were in the true asymptotic mean-field regime they should exhibit the same double-scaling or at least the τ=4 tail for large SCCs; the reported contrast together with the known slow crossover near du=6 indicates possible residual finite-size or dimensional corrections that need to be ruled out before the mean-field recovery claim can be considered fully supported.
  2. [Numerical Methods and Data Analysis] Simulation methodology and data analysis: the abstract states that extensive simulations support the scaling claims, but the manuscript provides insufficient detail on how the percolation transition is located, the finite-size scaling procedures employed, error-bar estimation, and any data-exclusion rules. Without these, it is difficult to assess whether the reported fractal dimensions and exponents are free of dominant finite-size or sampling artifacts that could alter the scaling relations.
minor comments (2)
  1. [Abstract and Introduction] The notation τ' for the small-SCC exponent is introduced only in the abstract and should be defined explicitly in the main text when the double-scaling structure is first discussed.
  2. [Figures] Figure captions should explicitly state the system sizes and number of disorder realizations used for each dimension so that the reader can judge the statistical quality of the data.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which will help improve its clarity and rigor. We address the major comments point by point below.

read point-by-point responses

  1. Referee: Results for d≥6 and comparison with complete graphs: the claim that mean-field behavior (d_SCC/d=1/3) is recovered on hypercubic lattices for d≥6 is presented as consistent with complete-graph results, yet the manuscript explicitly contrasts the SCC size-distribution structure (double scaling with τ_SCC=4 tail on complete graphs versus the lattice data). If d=7 lattices were in the true asymptotic mean-field regime they should exhibit the same double-scaling or at least the τ=4 tail for large SCCs; the reported contrast together with the known slow crossover near du=6 indicates possible residual finite-size or dimensional corrections that need to be ruled out before the mean-field recovery claim can be considered fully supported.

    Authors: We agree that the approach to mean-field behavior is slow near d_u=6 and that d=7 simulations may retain some corrections to scaling. Our primary evidence for mean-field recovery on lattices is the measured d_SCC/d approaching 1/3, which is consistent with the complete-graph value. The lack of double power-law scaling in the lattice data (as opposed to complete graphs) is expected because hypercubic lattices at finite d retain local connectivity and spatial structure absent from the complete-graph limit. We will revise the manuscript to expand the discussion of crossover effects, include additional finite-size scaling data for d=7, and clarify that the fractal dimension provides the main indicator of mean-field behavior while the full distribution structure may require even higher dimensions or larger sizes. revision: partial

  2. Referee: Simulation methodology and data analysis: the abstract states that extensive simulations support the scaling claims, but the manuscript provides insufficient detail on how the percolation transition is located, the finite-size scaling procedures employed, error-bar estimation, and any data-exclusion rules. Without these, it is difficult to assess whether the reported fractal dimensions and exponents are free of dominant finite-size or sampling artifacts that could alter the scaling relations.

    Authors: We acknowledge that the original manuscript lacks sufficient detail on the numerical procedures. In the revised version we will add an expanded Methods section (or subsection) that explicitly describes: the procedure for locating the percolation threshold (including use of Binder cumulants and peak positions of the susceptibility), the finite-size scaling protocols and system sizes employed, the methods for error-bar estimation (bootstrap resampling), and any criteria used to exclude poorly sampled or strongly finite-size-affected data points. This will enable readers to fully assess the robustness of the reported exponents and scaling relations. revision: yes

Circularity Check

0 steps flagged

No circularity: results obtained from direct numerical simulations on lattices and graphs

full rationale

The paper reports fractal dimensions, size-distribution exponents, and scaling relations extracted from extensive Monte Carlo simulations on hypercubic lattices (d=2 to 7) and complete graphs. The hyperscaling relation τ_SCC = 1 + d/d_SCC is the standard percolation hyperscaling relation applied to the measured d_SCC; it is not redefined or fitted from the same data. Mean-field recovery for d ≥ 6 is asserted by direct comparison of the simulated d_SCC/d = 1/3 on lattices to the independently simulated complete-graph limit, without any parameter fitting that renames an input as a prediction. No self-citations, ansatzes, or uniqueness theorems are invoked to close the argument. The derivation chain is therefore self-contained against external numerical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

With only the abstract available, the work rests on standard assumptions of bond percolation with random orientations and the conventional definition of strongly connected clusters; no explicit free parameters, ad-hoc axioms, or new invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5754 in / 1097 out tokens · 47347 ms · 2026-05-19T19:25:25.827828+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

69 extracted references · 69 canonical work pages

  1. [1]

    The terms (p−p c)kLk/ν describe the scaling behavior approaching criticality, whileL −ωk accounts for finite-size corrections atp c

    Percolation threshold Near criticality, the Binder cumulantQfollows the finite-size scaling form [49, 50] Q(p, L) =Q c + X k=1 ak(p−p c)kLk/ν + X k=1 bkL−ωk ,(8) whereQ c is the universal critical value in the thermody- namic limit. The terms (p−p c)kLk/ν describe the scaling behavior approaching criticality, whileL −ωk accounts for finite-size correction...

    work page

  2. [2]

    In all finite dimensions,C 1 exhibits a clear power-law scaling, C1 ∼L dSCC ∼V dSCC/d,(9) indicating a continuous percolation transition of SCCs, with fractal dimensiond SCC

    Fractal dimensions of SCCs Figure 4(a) shows the size of the largest SCC,C 1, at criticality as a function of the system volumeV=L d for dimensionsd= 2-7. In all finite dimensions,C 1 exhibits a clear power-law scaling, C1 ∼L dSCC ∼V dSCC/d,(9) indicating a continuous percolation transition of SCCs, with fractal dimensiond SCC. 103 104 105 106 107 108 V 1...

    work page 2048

  3. [3]

    The mean- field valued SCC/d= 1/3 [40, 41] givesd SCC = 2 ind= 6

    Logarithmic corrections atd u = 6 At the upper critical dimensiond u = 6, multiplicative logarithmic corrections are expected [59–61]. The mean- field valued SCC/d= 1/3 [40, 41] givesd SCC = 2 ind= 6. To present multiplicative logarithmic corrections visually, we show a log-log plot of the ratioC 1/L2 versus lnL in Fig. 6. The observed trend for largeLind...

    work page

  4. [4]

    The size distribution of SCCs Besides the largest SCC, we now examine the size dis- tribution of SCCs at criticality. In standard percolation 7 101 102 103 104 105 106 100 10−6 10−12 n(s, L) » s−2.1086 (a) L = 1024 L = 2048 L = 4096 L = 8192 10−2 100 10−1 10−3 s¿SCCn(s, L) vs s/LdSCC L = 1024 L = 2048 L = 4096 L = 8192 101 102 103 104 105 s 10−2 10−8 10−1...

    work page 2048

  5. [5]

    Fractal dimensions of ICs and OCs We further find that both ICs and OCs exhibit fractal geometry. Applying the same finite-size scaling analysis, their fractal dimensions are consistent with that of ordi- nary percolation clusters, i.e.,d IC =d OC =d f, as sum- marized in Table I. Since the giant IC and OC are sta- tistically symmetric, and their intersec...

    work page

  6. [6]

    Ma,Modern theory of critical phenomena(Rout- ledge, New York, 2018)

    S.-K. Ma,Modern theory of critical phenomena(Rout- ledge, New York, 2018)

    work page 2018

  7. [7]

    Stauffer and A

    D. Stauffer and A. Aharony,Introduction to Percolation Theory, 2nd ed. (Taylor & Francis, London, 1994)

    work page 1994

  8. [8]

    P. W. Kasteleyn and C. M. Fortuin, Phase transitions in lattice systems with random local properties, J. Phys. Soc. Jpn. Suppl.26, 11 (1969)

    work page 1969

  9. [9]

    C. M. Fortuin and P. W. Kasteleyn, On the random- cluster model: I. Introduction and relation to other mod- els, Physica57, 536 (1972)

    work page 1972

  10. [10]

    Nienhuis, Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas, J

    B. Nienhuis, Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas, J. Stat. Phys.34, 731 (1984)

    work page 1984

  11. [11]

    Nienhuis, Coulomb gas formulations of two- dimensional phase transitions, inPhase Transitions and Critical Phenomena, Vol

    B. Nienhuis, Coulomb gas formulations of two- dimensional phase transitions, inPhase Transitions and Critical Phenomena, Vol. 11, edited by C. Domb and J. L. Lebowitz (Academic Press, London, UK, 1987) pp. 1–53

    work page 1987

  12. [12]

    Di Francesco, P

    P. Di Francesco, P. Mathieu, and M. S´ en´ echal,Conformal Field Theory(Springer–Verlag, New York, NY, 1997)

    work page 1997

  13. [13]

    Kagery and B

    W. Kagery and B. Nienhuis, A guide to stochastic L¨ owner 10 evolution and its applications, J. Stat. Phys.115, 1149 (2004), 0312056v3

    work page 2004

  14. [14]

    Cardy, SLE for theoretical physicists, Ann

    J. Cardy, SLE for theoretical physicists, Ann. Phys.318, 81 (2005)

    work page 2005

  15. [15]

    Coniglio, Fractal structure of ising and potts clusters: Exact results, Phys

    A. Coniglio, Fractal structure of ising and potts clusters: Exact results, Phys. Rev. Lett.62, 3054 (1989)

    work page 1989

  16. [16]

    Smirnov and W

    S. Smirnov and W. Werner, Critical exponents for two- dimensional percolation, Math. Res. Lett.8, 729 (2001)

    work page 2001

  17. [17]

    Nolin, W

    P. Nolin, W. Qian, X. Sun, and Z. Zhuang, Backbone exponent for two-dimensional percolation, arXiv preprint , 2309.05050 (2023)

    work page arXiv 2023

  18. [18]

    Nolin, W

    P. Nolin, W. Qian, X. Sun, and Z. Zhuang, Backbone ex- ponent and annulus crossing probability for planar per- colation, Phys. Rev. Lett.134, 117101 (2025)

    work page 2025

  19. [19]

    S. Fang, D. Ke, W. Zhong, and Y. Deng, Backbone and shortest-path exponents of the two-dimensionalq- state potts model, Phys. Rev. E105, 044122 (2022), 2112.10162

    work page arXiv 2022

  20. [20]

    Hara and G

    T. Hara and G. Slade, Mean-field critical behaviour for percolation in high dimensions, Commun. Math. Phys. 128, 333 (1990)

    work page 1990

  21. [21]

    Grimmett,Percolation(Springer-Verlag, Heidelberg, 1999)

    G. Grimmett,Percolation(Springer-Verlag, Heidelberg, 1999)

    work page 1999

  22. [22]

    Fitzner and R

    R. Fitzner and R. van der Hofstad, Mean-field behavior for nearest-neighbor percolation ind >10, Electron. J. Probab.22, 1 (2017)

    work page 2017

  23. [23]

    Domany and W

    E. Domany and W. Kinzel, Directed percolation in two dimensions: Numerical analysis and an exact solution, Phys. Rev. Lett.47, 5 (1981)

    work page 1981

  24. [24]

    J. W. Essam, A. J. Guttmann, and K. De’Bell, On two- dimensional directed percolation, J. Phys. A: Math. Gen. 21, 3815 (1988)

    work page 1988

  25. [25]

    Grassberger, Directed percolation in 2+1 dimensions, J

    P. Grassberger, Directed percolation in 2+1 dimensions, J. Phys. A: Math. Gen.22, 3673 (1989)

    work page 1989

  26. [26]

    J. Wang, Z. Zhou, Q. Liu, T. M. Garoni, and Y. Deng, High-precision monte carlo study of directed percolation in (d+1) dimensions, Phys. Rev. E88, 042102 (2013)

    work page 2013

  27. [27]

    Z. Zhou, J. Yang, R. M. Ziff, and Y. Deng, Crossover from isotropic to directed percolation, Phys. Rev. E86, 021102 (2012)

    work page 2012

  28. [28]

    Hof, Directed percolation and the transition to turbu- lence, Nat

    B. Hof, Directed percolation and the transition to turbu- lence, Nat. Rev. Phys.5, 62 (2022)

    work page 2022

  29. [29]

    G. P. Shrivastav, P. Chaudhuri, and J. Horbach, Yielding of glass under shear: A directed percolation transition precedes shear-band formation, Phys. Rev. E94, 042605 (2016)

    work page 2016

  30. [30]

    Chantelot and D

    P. Chantelot and D. Lohse, Leidenfrost effect as a di- rected percolation phase transition, Phys. Rev. Lett.127, 124502 (2021)

    work page 2021

  31. [31]

    Piroli, Y

    L. Piroli, Y. Li, R. Vasseur, and A. Nahum, Triviality of quantum trajectories close to a directed percolation transition, Phys. Rev. B107, 224303 (2023)

    work page 2023

  32. [32]

    H. Hu, R. M. Ziff, and Y. Deng, Universal critical behav- ior of percolation in orientationally ordered Janus parti- cles and other anisotropic systems, Phys. Rev. Lett.129, 278002 (2022)

    work page 2022

  33. [33]

    R. E. Tarjan, Depth-first search and linear graph algo- rithms, SIAM J. Comput.1, 146 (1972)

    work page 1972

  34. [34]

    Broder, R

    A. Broder, R. Kumar, F. Maghoul, P. Raghavan, S. Ra- jagopalan, R. Stata, A. Tomkins, and J. Wiener, Graph structure in the Web, Comput. Networks33, 309 (2000)

    work page 2000

  35. [35]

    A. W. T. de Noronha, A. A. Moreira, A. P. Vieira, H. J. Herrmann, J. S. Andrade, and H. A. Carmona, Percola- tion on an isotropically directed lattice, Phys. Rev. E98, 062116 (2018)

    work page 2018

  36. [36]

    S. N. Dorogovtsev, J. F. F. Mendes, and A. N. Samukhin, Giant strongly connected component of directed net- works, Phys. Rev. E64, 025101 (2001)

    work page 2001

  37. [37]

    Schwartz, R

    N. Schwartz, R. Cohen, D. ben Avraham, A.-L. Barab´ asi, and S. Havlin, Percolation in directed scale-free networks, Phys. Rev. E66, 015104 (2002)

    work page 2002

  38. [38]

    Bogu˜ n´ a and M.´A

    M. Bogu˜ n´ a and M.´A. Serrano, Generalized percolation in random directed networks, Phys. Rev. E72, 016106 (2005)

    work page 2005

  39. [39]

    ´Angeles Serrano and P

    M. ´Angeles Serrano and P. De Los Rios, Interfaces and the edge percolation map of random directed networks, Phys. Rev. E76, 056121 (2007)

    work page 2007

  40. [40]

    Kryven, Emergence of the giant weak component in directed random graphs with arbitrary degree distribu- tions, Phys

    I. Kryven, Emergence of the giant weak component in directed random graphs with arbitrary degree distribu- tions, Phys. Rev. E94, 012315 (2016)

    work page 2016

  41. [41]

    van Ieperen and I

    F. van Ieperen and I. Kryven, Percolation in simple di- rected random graphs with a given degree distribution, Probab. Eng. Inf. Sci.38, 268 (2023)

    work page 2023

  42. [42]

    Li, R.-R

    M. Li, R.-R. Liu, L. L¨ u, M.-B. Hu, S. Xu, and Y.-C. Zhang, Percolation on complex networks: Theory and application, Phys. Rep.907, 1 (2021)

    work page 2021

  43. [43]

    S. d. S. Costa, A. W. T. de Noronha, A. P. Vieira, J. S. Andrade, and A. A. Moreira, Critical exponents for isotropically directed percolation on hierarchical lattices, Phys. Rev. E111, 054129 (2025)

    work page 2025

  44. [44]

    J. A. Garofalo, N. A. M. Ara´ ujo, L. de Arcangelis, A. Sar- racino, and E. Lippiello, Janus percolation in anisotropic limited-degree networks, arXiv preprint , 2512.10566 (2025)

    work page arXiv 2025

  45. [45]

    Luczak, The phase transition in the evolution of ran- dom digraphs, J

    T. Luczak, The phase transition in the evolution of ran- dom digraphs, J. Graph Theor.14, 217 (1990)

    work page 1990

  46. [46]

    Luczak and T

    T. Luczak and T. G. Seierstad, The critical behavior of random digraphs, Random Struct. Algor.35, 271 (2009)

    work page 2009

  47. [47]

    Huang, P

    W. Huang, P. Hou, J. Wang, R. M. Ziff, and Y. Deng, Critical percolation clusters in seven dimensions and on a complete graph, Phys. Rev. E97, 022107 (2018)

    work page 2018

  48. [48]

    Mertens and C

    S. Mertens and C. Moore, Percolation thresholds and fisher exponents in hypercubic lattices, Phys. Rev. E98, 022120 (2018)

    work page 2018

  49. [49]

    Bollob´ as,Random Graphs, 2nd ed

    B. Bollob´ as,Random Graphs, 2nd ed. (Cambridge Uni- versity Press, Cambridge, 2001)

    work page 2001

  50. [50]

    Sharir, A strong-connectivity algorithm and its appli- cations in data flow analysis, Comput

    M. Sharir, A strong-connectivity algorithm and its appli- cations in data flow analysis, Comput. Math. Appl.7, 67 (1981)

    work page 1981

  51. [51]

    Sedgewick,Algorithms in C++ Part 5: Graph Al- gorithms, 3rd ed

    R. Sedgewick,Algorithms in C++ Part 5: Graph Al- gorithms, 3rd ed. (Addison-Wesley Professional, Boston, 2002)

    work page 2002

  52. [52]

    Cheriyan and K

    J. Cheriyan and K. Mehlhorn, Algorithms for dense graphs and networks on the random access computer, Algorithmica15, 521 (1996)

    work page 1996

  53. [53]

    H. N. Gabow, Path-based depth-first search for strong and biconnected components, Inform. Process. Lett.74, 107 (2000)

    work page 2000

  54. [54]

    X. Qian, Y. Deng, and H. W. J. Bl¨ ote, Percolation in one ofqcolors near criticality, Phys. Rev. B71, 144303 (2005)

    work page 2005

  55. [55]

    Deng and H

    Y. Deng and H. W. J. Bl¨ ote, Surface critical phenom- ena in three-dimensional percolation, Phys. Rev. E71, 016117 (2005)

    work page 2005

  56. [56]

    J. Wang, Z. Zhou, W. Zhang, T. M. Garoni, and Y. Deng, Bond and site percolation in three dimensions, Phys. Rev. E87, 052107 (2013). 11

    work page 2013

  57. [57]

    X. Xu, J. Wang, J.-P. Lv, and Y. Deng, Simultaneous analysis of three-dimensional percolation models, Front. Phys.9, 113 (2013)

    work page 2013

  58. [58]

    Redner, Directed and diode percolation, Phys

    S. Redner, Directed and diode percolation, Phys. Rev. B 25, 3242 (1982)

    work page 1982

  59. [59]

    Kenna and B

    R. Kenna and B. Berche, Universal finite-size scaling for percolation theory in high dimensions, J. Phys. A: Math. Theor.50, 235001 (2017)

    work page 2017

  60. [60]

    M. Li, S. Fang, J. Fan, and Y. Deng, Crossover finite-size scaling theory and its applications in percolation, arXiv preprint , 2412.06228 (2024)

    work page arXiv 2024

  61. [61]

    M. Lu, S. Fang, Z. Zhou, and Y. Deng, Interplay of the complete-graph and gaussian fixed-point asymptotics in finite-size scaling of percolation above the upper critical dimension, Phys. Rev. E110, 044140 (2024)

    work page 2024

  62. [62]

    M. D. Rintoul and H. Nakanishi, A precise characteriza- tion of three-dimensional percolating backbones, J. Phys. A: Math. Gen.27, 5445 (1994)

    work page 1994

  63. [63]

    Zhang, P

    Z. Zhang, P. Hou, S. Fang, H. Hu, and Y. Deng, Critical exponents and universal excess cluster number of perco- lation in four and five dimensions, Physica A580, 126124 (2021)

    work page 2021

  64. [64]

    I. W. Essam, D. S. Gaunt, and A. J. Guttmann, Percola- tion theory at the critical dimension, J. Phys. A: Math. Gen.11, 1983 (1978)

    work page 1983

  65. [65]

    J. J. Ruiz-Lorenzo, Logarithmic corrections for spin glasses, percolation and Lee-Yang singularities in six di- mensions, J. Phys. A: Math. Gen.31, 8773 (1998)

    work page 1998

  66. [66]

    Kenna, D

    R. Kenna, D. A. Johnston, and W. Janke, Scaling re- lations for logarithmic corrections, Phys. Rev. Lett.96, 115701 (2006)

    work page 2006

  67. [67]

    Ben-Naim and P

    E. Ben-Naim and P. L. Krapivsky, Kinetic theory of ran- dom graphs: From paths to cycles, Phys. Rev. E71, 026129 (2005)

    work page 2005

  68. [68]

    H. Hu, R. M. Ziff, and Y. Deng, No-enclave percolation corresponds to holes in the cluster backbone, Phys. Rev. Lett.117, 185701 (2016)

    work page 2016

  69. [69]

    Yang and M

    L. Yang and M. Li, Emergence of biconnected clusters in explosive percolation, Phys. Rev. E110, 014122 (2024)

    work page 2024