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REVIEW 4 major objections 6 minor 1 cited by

Graphon as a Bridge between Graphs and Manifolds

T0 review · 4 major / 6 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Graphons, the limit objects of dense graph sequences, are shown to interpolate between Riemannian manifolds and weighted geometric graphs, with a single monotonicity inequality tying conductance, maxcut, capacity, and packing radius togethe

desk verdict The monotonicity inequality and the graphon-to-manifold convergence are valuable; the graphing leg is broken as stated. read the letter →

arxiv 2607.20213 v1 pith:PQHHFYCB submitted 2026-07-22 math.CO math.DGmath.MG

Dong Zhang This is my paper · ORCID
classification math.COmath.DGmath.MG MSC 05C8005C99
keywords graphonweightedgeometricgraphsRiemannianmanifoldGammaconvergenceSobolevconstantsconductancemaxcutpackingradius
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that graphons—the standard limit objects for dense graph sequences—sit naturally between weighted geometric graphs and Riemannian manifolds. For graphs built by uniform sampling on a manifold with a radial kernel, the usual graph-to-manifold approximation splits into a graph-to-graphon limit followed by a graphon-to-manifold limit. The paper also proves a monotonicity inequality for (p,q)-Sobolev constants on graphons, from which it derives relations among conductance, maxcut, capacity, and packing radius, and shows which quantities survive the passage to manifolds. The payoff would be a principled explanation of why some graph parameters have geometric counterparts (conductance) while others (maxcut) blow up.

What carries the argument

The central object is the graphon W_r(x,y)=K_r(dist_M(x,y)) on M×M, which acts as the hidden limit of the sampled graphs and as an r-ball-bundle approximation of M. The argument for the combinatorial-geometric bridge is carried by the (p,q)-Sobolev constants λ_k(W^{p,q}) defined through Krasnoselskii-genus minimax, together with the Mazur map f↦|f|^t sgn(f), whose two-sided estimate (Lemma 2.15) yields the monotonicity inequality (Theorem 2.4).

What would settle it

Take M to be the 2-sphere and V_n an i.i.d. uniform sample of size n. Let G_n be the graph whose edges are the vertex pairs at the global minimum distance (as in Theorem 1.7). With high probability, G_n is a tiny graph, and the Hausdorff distance between G_n (realised with geodesic edges) and the sphere does not converge to 0 as n→∞. This would violate Theorem 1.8 unless the sample happens to lie in the asserted 'good position'.

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Extended reading notes

Core claim

The central claim is that for a closed Riemannian manifold M, a radial kernel K, and uniform i.i.d. samples V_n, there is a graphon W_r(x,y)=K_r(dist(x,y)) such that the weighted graph G_{n,r} converges to W_r in cut distance and TL^p sense as n→∞ (Theorem 1.1), and W_r converges to M as r→0 in the sense that its diagonal-restricted measure weak-star converges to volume (Theorem 1.2) and its r-ball bundle measure weak-star converges to the volume measure (Theorem 1.3). Together these imply the factorization G_{n,r}→W_r→M. Separately, the paper defines (p,q)-Sobolev constants λ_k(W^{p,q}) and proves a two-sided monotonicity inequality (Theorem 2.4) relating them for different (p,q) via the Ma

Load-bearing premise

The graph-to-manifold leg (Theorem 1.8) depends on the unproved assertion that uniformly sampled vertices form an equal-edge-length polyhedralization of the manifold, so that the nearest-neighbor graph is its 1-skeleton; uniform sampling alone does not imply this.

Editorial extensions

If this is right

  • The graph-to-manifold approximation in manifold learning is decomposed into two independent limits, so quantitative error estimates can be split between graph-to-graphon and graphon-to-manifold steps.
  • Conductance, p-capacity, and packing radius have well-defined limits under graphon-to-manifold convergence, giving geometric counterparts on manifolds.
  • Maxcut, signed conductance, and graph-theoretic Sobolev constants blow up in the same limit, explaining why they have no geometric analog.
  • The monotonicity inequality yields explicit bounds between these parameters, some of which are new even for finite simple graphs and closed manifolds.
  • Nonlinear p-Laplacian-type eigenvalues are bounded by linear graphon eigenvalues (Remark 2.20).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The 'good position' step in the proof of Theorem 1.8 is not a consequence of uniform sampling; if nearest-neighbor graphs on typical random samples are not 1-skeleta of polyhedralizations, the claimed graphing leg G_n→M would require additional hypotheses.
  • If the blowup phenomenon persists in finite samples, it could serve as a practical diagnostic: parameters that scale differently with the bandwidth r are combinatorial, while those that stabilise are geometric.
  • The monotonicity inequality may yield quantitative stability estimates for spectral-clustering algorithms by comparing λ_k at different (p,q) pairs on the same graphon.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. This manuscript proposes a graphon W_r on a compact Riemannian manifold M, defined by W_r(x,y)=r^{-k}K(dist(x,y)/r), as an intermediate object between weighted geometric graphs and M. It claims that (i) uniformly sampled weighted graphs G_{n,r} converge to W_r as n→∞ in cut distance and TL^p, with Gamma-convergence of p-Rayleigh quotients (Theorem 1.1); (ii) W_r converges to M as r→0, both in a measure/diagonal sense and in a bundle sense, with renormalized Rayleigh quotients converging to manifold Dirichlet energies (Theorems 1.2, 1.3); (iii) for fixed n, r→0 yields a graphing G_n supported on closest-pair edges, and G_n→M in Hausdorff distance (Theorems 1.7, 1.8); and (iv) in Section 2, (p,q)-Sobolev constants on graphons satisfy a monotonicity inequality linking conductance, maxcut, capacity, and packing radius, with convergence and blowup phenomena under W_r→M. The paper also derives explicit constants and a dimension-asymptotic formula for the Rayleigh-quotient renormalization.

Significance. The W_r path is potentially significant: it provides a concrete continuum limit object for manifold learning and a transfer principle that could justify treating graphon inequalities as manifold inequalities. The monotonicity inequality in Theorem 2.4, if correct, unifies several graph and geometric parameters, and the explicit computation of C_0, C_p and the large-dimension asymptotics in Theorem 1.6 are concrete contributions. However, the manuscript currently contains a false stated theorem in the graphing leg, several central Gamma-convergence proofs are omitted, and key identifications depend on the author's own unpublished preprints. The significance is therefore conditional on substantial revision.

major comments (4)
  1. [§1.4, Theorem 1.8; §1.5 (text after Theorem 1.7)] Theorem 1.8 is false as stated. Under Theorem 1.7, E_n is the set of pairs attaining the global minimum distance. For an i.i.d. uniform sample from a non-atomic distribution on a compact manifold of dimension at least 1, the closest pair is unique almost surely, so #E_n=1 and G_n has O(1) edges. Realized as a metric graph, G_n is a single geodesic segment (or a finite set of segments), whose Hausdorff distance to M does not tend to 0 (on S^1 it tends to π). The 'good position' assertion in Section 1.5 is an extra structural hypothesis that is not implied by uniform sampling and is incompatible with the definition of E_n in Theorem 1.7. This invalidates the claimed commutative diagram's right leg. The W_r path (Theorems 1.1–1.3) may survive, but Theorem 1.8 and Section 1.4 must be rewritten or removed.
  2. [§1.5, Proposition 1.5 and Theorem 1.6] The liminf inequality in Proposition 1.5 is explicitly omitted ('we omit the detail'), and Theorem 1.6 relies on Gamma-convergence of the functionals Φ_{p,W_r} to C_p C_0^{-1} Φ_{p,M}, which is asserted as 'similar' without proof. These results are load-bearing: they justify the convergence of spectral constants used later in Theorem 2.5 and in the paper's geometric transfer claims. A complete proof or a precise, verifiable reference is required; an omitted-liminf statement is not sufficient for a central convergence theorem.
  3. [§2.5, Theorem 2.5] The proof of Theorem 2.5 is omitted ('The proof is similar to that of Theorem 1.3, and hence we omit the detail'). This theorem is central to Section 2: it is the basis for convergence of Cheeger constants, packing radii, and capacities under W_r→M. The passage from pointwise convergence of Rayleigh quotients to convergence of k-th min-max Sobolev constants requires Gamma-convergence and an equicoercivity/compactness argument, none of which is supplied. This needs a full proof or a precise citation to a result that covers exactly this setting.
  4. [§2, Definition 2 and Example 2.1] The identification λ_2(W^{p,q}) = inf_{f nonconstant} ∥f∥_{W,p}/inf_c∥f−c∥_q is cited to the author's preprint [29] (arXiv:2606.27004). This identification is used as a key input in Theorem 2.2 and in the interpretation of Theorem 2.4. Since [29] is not independently published or machine-checked, the manuscript should either state this as an explicitly assumed transfer principle or provide a self-contained proof. Reliance on an unpublished preprint for a load-bearing equality weakens verifiability.
minor comments (6)
  1. [General Overview] The displayed limit 'lim_{n→0} G_n = M' should read 'lim_{n→∞} G_n = M'.
  2. [§1.1, Assumption 2 and proof of Theorem 1.1] Assumption 2 states an exact equality #{B(x,r)∩V_n}/#V_n = vol(B(x,r))/vol(M) for every open ball, which cannot hold for finite n. It should be phrased as an asymptotic or limit condition. Also, the proof uses a partition M_i with vol(M_i)=vol(M)/n and M_i∩V_n={x_i} for arbitrary n; the existence of such a partition with max diameter o(1) for every admissible V_n is not justified and needs an argument (e.g., via quantization or Voronoi cells).
  3. [§1.5, proof of Theorem 1.2] There are duplicated phrases 'lim sup_{r→0+} lim_{r→0+}' and 'lim inf_{r→0+} lim_{r→0+}' in the Borel-measurable step; these should be corrected to single limits.
  4. [§1.5, text after Theorem 1.7] The term 'good position' is used without a definition. If it is intended as an additional assumption, it must be stated before Theorem 1.8 and checked for consistency with the edge set E_n defined in Theorem 1.7. As written, the paragraph is an unsupported assertion.
  5. [§2.1, Remark 2.10] The notation in 'MaxCut(W_r) ≥ ∥W_r∥_□/4 = ∥W_r∥_1/4 = ∥W∥_1/4 = ∥W∥_□/4' is confusing; use W_r consistently and justify each equality.
  6. [§2.7, proof of Theorem 2.19] The phrase 'decreasingly converges' is vague. Specify the mode of convergence (pointwise, monotone, etc.) and provide a brief justification for the interchange of limits.

Circularity Check

2 steps flagged · score 6.0 of 10

The W_r→M graphon leg is self-contained, but the G_n→M graphing leg of the central diagram is assumed via an unproved 'good position' polyhedralization hypothesis; a self-cited min-max equality is also load-bearing for the Section 2 parameter identifications.

  1. other [Section 1.5, paragraph following Proof of Theorem 1.7 (discussion of Theorem 1.8)]
    "Since an admissible sequence {x_i}_{i=1}^∞ is given, the set V_n={x_1,...,x_n} can be seen in “good position”, i.e., V_n is the vertex set of a “polyhedralization” of M with all equal edge-length, then V_n forms an ε-net of M when n is sufficiently large. Without loss of generality, we assume that G_n is geometrically realized as the 1-skeleton of polyhedralization of M with vertex set V_n and edge set E_n. The proof of Theorem 1.8 is then quite evident."

    Theorem 1.8 is not derived from Assumptions 1–2 or from admissibility; the proof imports 'good position', which asserts exactly that V_n is the vertex set of an equal-edge-length polyhedralization of M and that G_n is its 1-skeleton. Once this is assumed, d_H(G_n,M)→0 is immediate, so the theorem's conclusion is contained in the new hypothesis. The circularity is aggravated by Theorem 1.7, where E_n is defined as the set of pairs attaining the global minimum distance; such a graph generically has O(1) edges and cannot be the 1-skeleton of a polyhedralization. Thus the graphing-to-manifold leg of the paper's central commutative diagram is assumed rather than proved.

  2. self citation load bearing [Section 2, Definition 2 (Sobolev constants on graphon), after Eq. (14)]
    "For the case k=2, we simply call λ_2(W^{p,q}) the (p,q)-Sobolev constant on graphon W, which also equals inf_{f nonconstant} ∥f∥_{W,p} / inf_{c∈R} ∥f−c∥_q due to [29]."

    This k=2 equality is not proved in the paper; it is imported from the author's own preprint [29] (arXiv:2606.27004). The equality is then used in the proof of Theorem 2.2(i) and in Section 2.4 to identify λ_2 with p-capacity and conductance, replacing the genus min-max definition with a one-function infimum. Those identifications are therefore load-bearing on a self-citation that is not independently machine-checked or reproduced. It is not the central monotonicity inequality—Theorem 2.4 is proved directly—so this contributes partial, not total, circularity.

full rationale

The core graphon-to-manifold results (Theorems 1.1–1.3, 1.6) are not circular: W_r is defined from the manifold M and kernel K, the cut-distance/TL^p convergence is proved by a direct L^1 estimate, and the constants C_0, C_p are computed from K rather than fitted to the target limits. The monotonicity inequality Theorem 2.4 is also proved from the Mazur-map inequality (Lemma 2.15) without using the target identities. The main circularity is confined to the graphing leg: Theorem 1.8's proof silently strengthens the hypotheses through 'good position' and the assertion that G_n is the 1-skeleton of a polyhedralization, which is equivalent to the desired Hausdorff convergence. The manuscript itself flags this as an omitted proof ('quite evident') and also omits the proof of Theorem 2.5 ('we omit the detail'). Additionally, the k=2 characterization in Definition 2 rests on the author's own [29], which is load-bearing for the capacity/conductance identifications in Section 2. Because the principal graphon bridge and the monotonicity inequality have independent content, the overall circularity is partial rather than total.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

No free parameters are fitted: W_r, C_0, C_p are determined by M and K. The main assumptions are Assumptions 1 and 2, the admissibility/transport-map condition from [11], and an additional unproved 'good position' triangulation condition for the graphings-to-manifold limit. Section 2 also assumes bounded graphons with finite degree and uses the author's preprint [29] for the min-max identification of Sobolev constants.

assumptions (9)
  • domain assumption Smooth compact closed Riemannian manifold M of dimension k≥1 and threshold r>0; kernel K:[0,∞)→[0,∞) satisfies K(0)>0, non-increasing, finite moments; edge weights w_ij = r^{-k} K(dist(x_i,x_j)/r).
    Assumption 1; defines the sampling model and all W_r.
  • domain assumption Sampling V_n is uniformly distributed: #(V_n∩B(x,r))/n → vol(B(x,r))/vol(M).
    Assumption 2; needed for graph-to-graphon convergence and admissibility of point sequences.
  • standard math P-a.e. point sequence is admissible: there exist transport maps T_n with T_n#μ=μ_n and ‖Id−T_n‖∞→0.
    Cited from García Trillos–Slepčev [11]; used throughout to justify T_L^p and Riemann-sum limits.
  • ad hoc to paper For each n, V_n is in 'good position': vertex set of an equal-edge-length polyhedralization of M, so nearest-neighbor graph E_n is its 1-skeleton.
    Section 1.4; unproved geometric assumption needed for Theorem 1.8 (Hausdorff convergence of graphing G_n to M).
  • domain assumption For Theorem 1.7, K(t)=e^{-t^α} (α>0) with lim_{t→∞} K(Ct)/K(t)=0 for C>1.
    Needed so normalized edge weights concentrate on the nearest-neighbor set E_n; stated in §1.3.
  • domain assumption For Proposition 2.12, K has compact support (K(t)=1 for 0≤t≤1 and 0 otherwise).
    Needed for exact scaling r^{1-k} dist_Wr → dist_M and packing-radius limits.
  • standard math Standard machinery: Kolmogorov existence, Γ-convergence, T_L^p theory, Krasnoselskii genus, Mazur map, Laplace method, Stirling/Beta asymptotics.
    Used throughout Sections 1.5, 2.5, A.3; assumed without proof.
  • domain assumption For Section 2 p-Laplacian existence, W symmetric integrable with uniform positive lower bound near diagonal and Ω bounded.
    Assumed in §2.6 for Problems (19)–(21) and Proposition 2.17.
  • ad hoc to paper The identification λ_2(W^{p,q}) = inf_{f nonconstant} ‖f‖_{W,p}/inf_c‖f−c‖_q is taken from [29].
    Stated as 'due to [29]' (self-cited arXiv preprint) without proof; used to interpret Sobolev constants as mincut/conductance/maxcut.

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Pith. "Pith review of Graphon as a Bridge between Graphs and Manifolds." pith.science (2026). https://pith.science/paper/PQHHFYCB

@misc{pith2026260720213,
  author       = {Pith},
  title        = {Pith review of: Graphon as a Bridge between Graphs and Manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PQHHFYCB}},
  note         = {Machine review of arXiv:2607.20213}
}
read the original abstract

We show that there exist graphons that interpolate between Riemannian manifolds and weighted geometric graphs. Specifically, the graph-to-manifold approximation used in manifold learning can be regarded as the composition of a graph-to-graphon convergence and a graphon-to-manifold convergence in a certain sense. Furthermore, we establish a monotonicity inequality which reveals an implicit relationship between numerous combinatorial parameters and geometric quantities on graphons. Using this inequality, we find relations among conductance, maxcut problem, capacity, and packing radius, as well as their limiting behaviours under graph-to-graphon and graphon-to-manifold convergences; some of these relations are novel even for simple graphs and closed manifolds.

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Reference graph

Works this paper leans on

29 extracted references · 1 linked inside Pith · cited by 1 Pith paper

  1. [29]

    Hidden critical and morse equivalence behind duality: Theory and applications, 2026

    Dong Zhang. Hidden critical and morse equivalence behind duality: Theory and applications, 2026. arXiv:2606.27004. A Appendix A.1 Cut Distance for Graphons The cut norm was first introduced by Frieze and Kannan [10]. Following Lov´ asz’s notion, let𝑊:[0,1] 2→R be a bounded, measurable kernel. The cut norm of𝑊is defined as ∥𝑊∥ □=sup 𝑆,𝑇⊆[0,1] ∫ 𝑆×𝑇 𝑊(𝑥,𝑦)𝑑...

    arXiv 2026

  2. [1]

    Scalable implicit graphon learning

    Ali Azizpour, Nicolas Zilberstein, and Santiago Segarra. Scalable implicit graphon learning. In Yingzhen Li, Stephan Mandt, Shipra Agrawal, and Emtiyaz Khan, editors,Proceedings of The 28th International Con- ference on Artificial Intelligence and Statistics, volume 258 ofProceedings of Machine Learning Research, pages 3952–3960. PMLR, 03–05 May 2025

    2025

  3. [2]

    Towards a theoretical foundation for Laplacian-based manifold methods

    Mikhail Belkin and Partha Niyogi. Towards a theoretical foundation for Laplacian-based manifold methods. Journal of Computer and System Sciences, 74(8):1289–1308, 2008. Learning Theory 2005

    2008

  4. [3]

    Bhattacharya, Anirban Chatterjee, and Svante Janson

    Bhaswar B. Bhattacharya, Anirban Chatterjee, and Svante Janson. Fluctuations of subgraph counts in graphon based random graphs.Combinatorics, Probability and Computing, 32(3):428–464, 2023

    2023

  5. [4]

    Chayes, L´ aszl´ o Lov´ asz, Vera T

    Christian Borgs, Jennifer T. Chayes, L´ aszl´ o Lov´ asz, Vera T. S´ os, and Katalin Vesztergombi. Convergent sequences of dense graphs ii. multiway cuts and statistical physics.Annals of Mathematics, 176(1):151–219, 2012. 25

    2012

  6. [5]

    Oxford University Press, Oxford, 2002

    Andrea Braides.Γ-convergence for beginners, volume 22 ofOxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2002

    2002

  7. [6]

    Bhattacharya

    Anirban Chatterjee, Soham Dan, and Bhaswar B. Bhattacharya. Higher-order graphon theory: Fluctua- tions, degeneracies, and inference.Annals of Statistics, 2026

    2026

  8. [7]

    Birkh¨ auser Boston, Inc., Boston, MA, 1993

    Gianni Dal Maso.An introduction toΓ-convergence, volume 8 ofProgress in Nonlinear Differential Equa- tions and their Applications. Birkh¨ auser Boston, Inc., Boston, MA, 1993

    1993

Show all 29 references
  1. [8]

    On topological and metric critical point theory.Journal of Fixed Point Theory and Applications, 7(1):85–102, 2010

    Marco Degiovanni. On topological and metric critical point theory.Journal of Fixed Point Theory and Applications, 7(1):85–102, 2010

    2010

  2. [9]

    Limit of minimax values underΓ-convergence.Electron

    Marco Degiovanni and Marco Marzocchi. Limit of minimax values underΓ-convergence.Electron. J. Differential Equations, pages No. 266, 19, 2014

    2014

  3. [10]

    Quick approximation to matrices and applications.Combinatorica, 19(2):175–220, 1999

    Alan Frieze and Ravi Kannan. Quick approximation to matrices and applications.Combinatorica, 19(2):175–220, 1999

    1999

  4. [11]

    On the rate of convergence of empirical measures in∞- transportation distance.Canad

    Nicol´ as Garc ´ ıa Trillos and Dejan Slepˇ cev. On the rate of convergence of empirical measures in∞- transportation distance.Canad. J. Math., 67(6):1358–1383, 2015

    2015

  5. [12]

    Continuum limit of total variation on point clouds.Archive for Rational Mechanics and Analysis, 220(1):193–241, 2016

    Nicol´ as Garc ´ ıa Trillos and Dejan Slepˇ cev. Continuum limit of total variation on point clouds.Archive for Rational Mechanics and Analysis, 220(1):193–241, 2016

    2016

  6. [13]

    A variational approach to the consistency of spectral clustering

    Nicol´ as Garc ´ ıa Trillos and Dejan Slepˇ cev. A variational approach to the consistency of spectral clustering. Appl. Comput. Harmon. Anal., 45(2):239–281, 2018

    2018

  7. [14]

    Infinite-dimensional finitely forcible graphon.Pro- ceedings of the London Mathematical Society, 118(4):826–856, 2019

    Roman Glebov, Tereza Klimoˇ sov´ a, and Daniel Kr´ al’. Infinite-dimensional finitely forcible graphon.Pro- ceedings of the London Mathematical Society, 118(4):826–856, 2019

    2019

  8. [15]

    Compactness and finite forcibility of graphons.Journal of the European Mathematical Society, 21(10):3199–3223, 2019

    Roman Glebov, Daniel Kr´ al’, and Jan Volec. Compactness and finite forcibility of graphons.Journal of the European Mathematical Society, 21(10):3199–3223, 2019

    2019

  9. [16]

    Large deviation principles for graphon sampling.Electron

    Jan Greb ´ ık and Oleg Pikhurko. Large deviation principles for graphon sampling.Electron. J. Probab., 31(59):1–37, 2026

    2026

  10. [17]

    Metric critical point theory 1

    Alexander Ioffe and Efim Schwartzman. Metric critical point theory 1. morse regularity and homotopic stability of a minimum.Journal de mathematiques pures et appliqu´ ees, 75(2):125–154, 1996

    1996

  11. [18]

    Can smooth graphons in several dimensions be represented by smooth graphons on [0,1]?Examples and Counterexamples, 1:100011, 2021

    Svante Janson and Sofia Olhede. Can smooth graphons in several dimensions be represented by smooth graphons on [0,1]?Examples and Counterexamples, 1:100011, 2021

    2021

  12. [19]

    On nonlinear Rayleigh quotients.Potential Anal., 2(3):199–218, 1993

    Peter Lindqvist. On nonlinear Rayleigh quotients.Potential Anal., 2(3):199–218, 1993

    1993

  13. [20]

    Subgraph densities in signed graphons and the local Simonovits-Sidorenko conjecture

    L´ aszl´ o Lov´ asz. Subgraph densities in signed graphons and the local Simonovits-Sidorenko conjecture. Electron. J. Combin., 18(1):Paper 127, 21, 2011

    2011

  14. [21]

    Limits of dense graph sequences.J

    L´ aszl´ o Lov´ asz and Bal´ azs Szegedy. Limits of dense graph sequences.J. Combin. Theory Ser. B, 96(6):933– 957, 2006

    2006

  15. [22]

    Regularity partitions and the topology of graphons

    L´ aszl´ o Lov´ asz and Bal´ azs Szegedy. Regularity partitions and the topology of graphons. InAn irregular mind, volume 21 ofBolyai Soc. Math. Stud., pages 415–446. J´ anos Bolyai Math. Soc., Budapest, 2010

    2010

  16. [23]

    Maz´ on, Mayte P´ erez-Llanos, Julio D

    Jos´ e M. Maz´ on, Mayte P´ erez-Llanos, Julio D. Rossi, and Juli´ an Toledo. A nonlocal 1-Laplacian problem and median values.Publ. Mat., 60(1):27–53, 2016

    2016

  17. [24]

    Continuum limit of Lipschitz learning on graphs.Found

    Tim Roith and Leon Bungert. Continuum limit of Lipschitz learning on graphs.Found. Comput. Math., 23(2):393–431, 2023

    2023

  18. [25]

    Multi-class graph clustering via approximated effective𝑝-resistance

    Shota Saito and Mark Herbster. Multi-class graph clustering via approximated effective𝑝-resistance. In Andreas Krause, Emma Brunskill, Kyunghyun Cho, Barbara Engelhardt, Sivan Sabato, and Jonathan Scarlett, editors,Proceedings of the 40th International Conference on Machine Le...

    2023

  19. [26]

    Analysis of𝑝-Laplacian regularization in semisupervised learning

    Dejan Slepˇ cev and Matthew Thorpe. Analysis of𝑝-Laplacian regularization in semisupervised learning. SIAM J. Math. Anal., 51(3):2085–2120, 2019

  20. [27]

    Manifold learning in metric spaces.Appl

    Liane Xu and Amit Singer. Manifold learning in metric spaces.Appl. Comput. Harmon. Anal., 80:Paper No. 101813, 31, 2026

    2026

  21. [28]

    Homological eigenvalues of graph𝑝-Laplacians.J

    Dong Zhang. Homological eigenvalues of graph𝑝-Laplacians.J. Topol. Anal., 17(2):555–606, 2025

    2025

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