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The spectrum of dense kernel-based random graphs

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Pith's one-line read The paper proves that dense kernel-based random graphs on a torus have a universal limiting eigenvalue distribution, independent of the long-range parameter.

desk verdict Solid d=1 spectral limit for dense kernel-based random graphs with heavy-tailed weights, but the advertised d>1 extension is asserted, not proved. read the letter →

arxiv 2502.09415 v2 pith:OGWKXWSS submitted 2025-02-13 math.PR math.COmath.FA

Alessandra Cipriani , Rajat Subhra Hazra , Nandan Malhotra , Michele Salvi This is my paper · ORCID
classification math.PRmath.COmath.FA MSC 05C8046L5460B1060B20
keywords empiricalspectraldistributionfreeprobabilitykernel-basedrandomgraphsspectrumofmatricesscale-freepercolationoperator-valuedsemicirclelawheavy-tailedweightsnon-crossingpartitions
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

This paper studies the eigenvalues of dense spatial random graphs built on a discrete torus. Each vertex carries an independent Pareto weight, and two vertices at distance r are connected with probability κ_σ(W_i,W_j)/r^α ∧1. The paper proves that, after rescaling by the natural volume-dependent factor, the empirical eigenvalue distribution—the probability measure putting equal mass on each eigenvalue—converges to a deterministic limiting measure μ_{σ,τ}. The limit is symmetric and absolutely continuous, independent of the long-range parameter α, and it has a finite second moment even when the vertex weights have infinite variance. In the scale-free percolation case σ=1, the limiting measure is explicitly a free multiplicative convolution of the semicircle law with the weight distribution.

What carries the argument

The argument is carried by three pieces working together. First, Gaussianisation: an invariance theorem of Chatterjee (2005) lets the Bernoulli edges be replaced, for spectral purposes, by independent Gaussians with the same variances, provided the variance profile is smooth enough. Second, moment combinatorics: for the Gaussianised matrix, Wick's formula turns the 2k-th moment into sums over pair partitions; only non-crossing pair partitions survive, and each contributes the kernel product over the edge set of the tree G_{γπ}, yielding M_{2k}=Σ_{π∈NC2(2k)} E[∏_{(u,v)∈E(G_{γπ})} κ_σ(W_u^m,W_v^m)]. Third, a fixed-point characterisation: recursively decomposing the moment sums gives the quadratic equation a(z,x)(z+∫ a(z,y)κ_σ(x,y) μ_{W,m}(dy))=−1, whose limit as m→∞ is solved by a contraction on $L^{1}$([1,∞), $x^{{−β}}$dx) for τ>3 and σ<τ−2.

What would settle it

On the two-dimensional torus, compute the normalized index sum in Claim 4.10 for a small non-crossing partition (k=2) and compare it with the claimed limit 1; if it tends to a different constant, or if the second moment of the empirical spectrum fails to converge to the explicit value in Theorem 2.3, the d>1 statement is false.

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Extended reading notes

Core claim

The central claim, stated as Theorem 2.1, is that for connection probabilities p_ij = ((W_i∨W_j)(W_i∧W_j)^σ / ||i−j||^α) ∧ 1 on the d-dimensional torus, with τ>2, 0<α<d, and 0<σ<τ−1, the empirical spectral distribution of A_N = A_{G_N}/√c_N converges in probability to a deterministic measure μ_{σ,τ}. The normalisation c_N ∼ c0 $N^{{d−α}}$ is the only place the long-range parameter appears; the limiting measure depends on the Pareto tail exponent τ and the kernel exponent σ. The paper further shows μ_{σ,τ} is symmetric, absolutely continuous, has finite second moment, and for τ>3 and σ<τ−2 has Stieltjes transform S_{μ_{σ,τ}}(z)=∫_1^∞ a*(z,x) μ_W(dx), where a* is the unique fixed point of a contraction on a weighted $L^{1}$ space. For σ=1 it identifies μ_{1,τ}=μ_sc ⊠ μ_W, with power-law tails of exponent 2(τ−1).

Load-bearing premise

All proofs are carried out on a one-dimensional torus; the paper asserts without proof that the same estimates and combinatorial counting work unchanged in higher dimensions, and the central theorem as stated for any d relies on that assertion.

Editorial extensions

If this is right

  • For any 0<α<d, the bulk spectrum of the scaled adjacency matrix is asymptotically the same: the exponent α enters only through the scaling factor c_N, not through the limiting measure.
  • The limiting spectrum is absolutely continuous, so the empirical eigenvalue density has a well-defined bulk density for large N, with no atoms.
  • The second moment of μ_{σ,τ} is finite and explicitly computable even when the weights have infinite variance, so the spectrum is much lighter-tailed than the vertex weights.
  • When σ=1, the limiting spectrum is exactly the free multiplicative convolution μ_sc ⊠ μ_W, meaning eigenvalue histograms of the graph match those of a weighted Gaussian product P_N G_N P_N in the large-N limit.
  • For τ>3 and σ<τ−2, the Stieltjes transform is computable as the fixed point of an explicit contraction, giving a numerical route to the density for any parameter pair.

Reading between the lines

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

  • If the α-independence survives beyond the torus, the same μ_{σ,τ} should describe kernel-based random graphs on any vertex set with comparable volume growth and kernel regularity; a natural test is a square or cube with periodic boundary conditions at several dimensions.
  • The σ=1 identity suggests a cheap simulation scheme: instead of generating the graph, draw a GUE matrix and multiply it by a diagonal Pareto matrix; the eigenvalue histograms should coincide for large N.
  • The fixed-point equation for the Stieltjes transform is a candidate for an efficient numerical solver for μ_{σ,τ}; comparing its density to finite-N histograms would validate the contraction in the non-asymptotic regime.
  • The α=0 endpoint sits outside the proof because the error terms are O(N^{-α}) and fail at α=0; the paper's simulations indicate an atom there, so the α→0 limit of μ_{σ,τ} may be discontinuous.
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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

2 major / 4 minor

Summary. The paper studies the empirical spectral distribution of dense kernel-based random graphs on a discrete d-dimensional torus, with i.i.d. Pareto vertex weights and connection probability p_{ij} = ((W_i ∨ W_j)(W_i ∧ W_j)^σ / ||i-j||^α) ∧ 1. For τ>2, 0<α<d, and σ∈(0,τ-1), the authors claim that the scaled adjacency matrix has a deterministic limiting spectral measure μ_{σ,τ}; they further prove absolute continuity and symmetry of μ_{σ,τ}, non-degeneracy and finiteness of its second moment, an explicit identification μ_{1,τ} = μ_{sc} ⊠ μ_W with a power-law tail when σ=1, and a fixed-point equation for the Stieltjes transform. The proof strategy combines truncation of the heavy-tailed weights, centring, Gaussianisation via Chatterjee's invariance theorem, a moment method based on Wick's formula and non-crossing partitions, and free-probabilistic arguments. The paper's own Remark 2.6 states that the proofs are worked out in the d=1 setup and asserts without proof that the results remain unchanged for d>1.

Significance. If the stated results are correct, this is a substantial contribution to the spectral theory of inhomogeneous spatial random graphs. The paper is largely self-contained: the limiting moments are computed from the Gaussianized matrix via Wick's formula and non-crossing partitions, the free multiplicative convolution identification for σ=1 is proved rather than assumed, and the tail asymptotics and absolute continuity results go beyond what is currently available for such models. The proof of the second-moment finiteness under infinite-variance weights is a nice application of the two-parameter diagonal argument via Lemma 3.5. However, the claimed d-dimensional generality is the central issue: the entire technical core of Section 4 is written for d=1, with N vertices, one-dimensional torus estimates, and exponents 1-α rather than d-α. The unproved transfer in Remark 2.6 is load-bearing because Theorems 2.2-2.5 all rest on Theorem 2.1. There is also a gap in the analytic continuation argument for the Stieltjes transform in Section 8.

major comments (2)
  1. [Remark 2.6, §4 (esp. Lemma 3.4, eqs. (4.2), (4.9)-(4.10), Claim 4.10)] Theorem 2.1 is stated for the d-dimensional torus V_N of size N^d, but the proof of Section 4 is carried out only for d=1. The vertex set in the proof is {1,...,N}, Lemma 3.4 gives the one-dimensional estimate (1/N)∑_{i≠j} ||i-j||^{-β} ∼ c max{N^{1-β}, log N}, and the estimates in (4.2), the Gaussianisation bounds in (4.9)-(4.10), and the combinatorial Claim 4.10 all use the exponent 1-α rather than d-α. The statement in Remark 2.6 that 'the limiting spectral distribution and its properties remain unchanged for d>1' is an assertion, not a proof. For d>1 the normalisation is c_N ∼ c_0 N^{d-α}, Lemma 3.4 must be replaced by (1/N^d)∑_{i≠j} ||i-j||^{-β} ∼ c max{N^{d-β}, log N}, and the cancellation of constants in Claim 4.10 and the smallness of the error term R_1 need to be re-derived with these d-dependent exponents. Since Theorems 2.2-2.5 all build on Theorem 2.1, the central results of the paper are currently established only in dimension d=1.
  2. [§8, Lemmas 8.7-8.10 and Theorem 2.5] Theorem 2.5 asserts the existence of a unique analytic function a* on C+ × [1,∞) satisfying the fixed point equation (8.2). However, Lemma 8.7 proves that T_z is a contraction only when η² = (Im z)² > c̃ for a constant c̃ = c̃(τ,σ,β). Corollary 8.8 and Lemma 8.10 therefore produce the fixed point and the convergence a_z → a*_z only on the strip {z : Im z > √c̃}. The appeal to the identity theorem in Lemma 8.10 does not explain why the fixed point extends analytically to all of C+, nor why the extended function continues to satisfy (8.2) outside that strip. An additional argument, such as a z-dependent norm on the Banach space or an analytic-continuation argument for the fixed point equation, is needed before Theorem 2.5 is established as stated.
minor comments (4)
  1. [§3.1] The notation 'For an N × N matrix' at the start of Section 3.1 is inconsistent with the vertex set V_N of size N^d defined in Section 2; this notational mismatch contributes to the difficulty of tracking whether a claim is being proved for d=1 or for general d.
  2. [Theorem 2.2 statement] In the statement of Theorem 2.2, 'The the limiting spectral distribution' should read 'Then the limiting spectral distribution'.
  3. [Proof of Proposition 4.9] In the sentence beginning 'Let P (i) denote the expectation P (i) (4.19) := E[...]', the reference (4.19) appears to be incorrect, since (4.19) defines the matrix G rather than the expectation; please renumber the displayed equation.
  4. [General presentation] Because Remark 2.6 explicitly confines the proofs to d=1, the abstract and introduction should state this restriction prominently, or the main theorems should be restated for d=1 with the d>1 case presented as a conjecture or a remark requiring further proof.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the limiting spectrum is derived from the model via truncation, Gaussianization, Wick moments, and a proved Stieltjes fixed-point contraction; self-citations are not load-bearing.

full rationale

The derivation chain for Theorem 2.1 is self-contained: truncation (Lemma 4.2), centring (Lemma 4.3), Gaussianisation (Lemma 4.4 via Chatterjee), and the moment computation (Proposition 4.9) reduce the limit to the explicit sums in Claim 4.10, whose limit is proved by induction from the definition of cN in (2.4); no fitted parameter is renamed as a prediction. The measure μσ,τ,m is characterized by its moment sequence through Carleman's condition, and μσ,τ is obtained by the double-limit Lemma 3.5, not assumed. The σ=1 identification μ1,τ = μsc ⊠ μW follows from the moment formula using the Nica–Speicher combinatorial expression for free multiplicative convolution and external results of Arizmendi–Pérez-Abreu; the tail estimate uses Kołodziejek–Szpojankowski. The Stieltjes fixed-point equation (8.2) is derived from the moment recursion in Proposition 8.1, and the limit is taken through a contraction argument on the Banach space of Definition 8.6, so the fixed point is not an imported ansatz. Some technical inputs are cited from the authors' own prior work (Definition 4.7 from Avena et al.; Lemma 3.5 from Chakrabarty et al.), but these are elementary or standard lemmas and definitions rather than the target result, and the central combinatorial estimate is proved here. Remark 2.6's unproved assertion that the d>1 case is immediate is a completeness gap rather than a circular step; the same tree-sum factorization that proves Claim 4.10 would extend it, but the paper does not supply the calculation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper's central claims rest on a standard toolbox of random matrix and free-probability theorems from the literature (Chatterjee's invariance theorem, free multiplicative convolution continuity, Biane's absolute continuity, Nica-Speicher moment formulas, and a Breiman-type tail theorem). No parameter is fitted to data; the model parameters (α, τ, σ) are inputs, and the scaling constant c0 follows from the torus sum. The main unproved transfer is the d>1 extension flagged in Remark 2.6, plus the analyticity detail in Theorem 2.5.

assumptions (6)
  • standard math Chatterjee (2005, Theorem 1.1) invariance principle: replacing independent entries by Gaussians with the same mean and variance changes smooth functionals by a controlled amount.
    Used in Lemma 4.4 to Gaussianise the truncated adjacency matrix.
  • standard math Bercovici and Voiculescu (1993, Corollary 6.7): weak continuity of free multiplicative convolution for measures with unbounded support.
    Used in Theorem 2.2 to pass from truncated to untruncated weights in the σ=1 identification.
  • standard math Arizmendi and Pérez-Abreu (2009, Lemma 8 and Theorem 7): extension of free multiplicative convolution to symmetric measures with unbounded support, including the square identification.
    Used to justify µ_sc ⊠ µ_W for Pareto weights and the equality of squares in Theorem 2.2.
  • standard math Biane (1997, Corollary 2): the free additive convolution of any probability measure with a semicircle law is absolutely continuous.
    Used in Theorem 2.4 to conclude absolute continuity of µ_{σ,τ}.
  • standard math Nica and Speicher (2006, Theorem 14.4): combinatorial formula for moments of free multiplicative convolution.
    Used in the σ=1 identification to match moment formulas.
  • standard math Kolodziejek and Szpojankowski (2022, Lemma 7.2 and Theorem 1.3(iv)): Breiman-type tail theorem for free multiplicative convolution.
    Used for the tail asymptotics in Theorem 2.2.

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Pith. "Pith review of The spectrum of dense kernel-based random graphs." pith.science (2026). https://pith.science/paper/OGWKXWSS

@misc{pith2026250209415,
  author       = {Pith},
  title        = {Pith review of: The spectrum of dense kernel-based random graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OGWKXWSS}},
  note         = {Machine review of arXiv:2502.09415}
}
abstract

Kernel-based random graphs (KBRGs) are a broad class of random graph models that account for inhomogeneity among vertices. We consider KBRGs on a discrete $d-$dimensional torus $\mathbf{V}_N$ of size $N^d$. Conditionally on an i.i.d.~sequence of {Pareto} weights $(W_i)_{i\in \mathbf{V}_N}$ with tail exponent $\tau-1>0$, we connect any two points $i$ and $j$ on the torus with probability $$p_{ij}= \frac{\kappa_{\sigma}(W_i,W_j)}{\|i-j\|^{\alpha}} \wedge 1$$ for some parameter $\alpha>0$ and $\kappa_{\sigma}(u,v)= (u\vee v)(u \wedge v)^{\sigma}$ for some $\sigma\in(0,\tau-1)$. We focus on the adjacency operator of this random graph and study its empirical spectral distribution. For $\alpha<d$ and $\tau>2$, we show that a non-trivial limiting distribution exists as $N\to\infty$ and that the corresponding measure $\mu_{\sigma,\tau}$ is absolutely continuous with respect to the Lebesgue measure. $\mu_{\sigma,\tau}$ is given by an operator-valued semicircle law, whose Stieltjes transform is characterised by a fixed point equation in an appropriate Banach space. We analyse the moments of $\mu_{\sigma,\tau}$ and prove that the second moment is finite even when the weights have infinite variance. In the case $\sigma=1$, corresponding to the so-called scale-free percolation random graph, we can explicitly describe the limiting measure and study its tail.

Figures

Figures reproduced from arXiv: 2502.09415 by the authors.

Figure 1
Figure 1. [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. KBRG eigenvalue distribution and PN GN PN distribution. tions of scale-free percolation adjacency matrices of size 4000 × 4000 with σ = 1 and plot their eigenvalues (in green). We superpose to them the eigenvalues of the product PN GN PN of a GUE matrix GN with a diagonal matrix PN with i.i.d. entries distributed as p P areto(τ ) (in blue). Note that by Nica and Speicher (2006, Remark 14.2), Chakrabarty et al. (2021… view at source ↗
Figure 3
Figure 3. Negative of the log-empirical survival function and tails of Theo￾rem 2.2 for x ≥ 1.5 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: ESD for A˜ N,m,g. Remark 2.7 (Sparse case). We expect the case α > d to be very different due to the sparse nature of the graph. There has been a significant development in the area of spectral proper￾ties of sparse random graphs using the techniques of local weak conv…
Figure 5
Figure 5. Figure 5: Left: closed walk on [4]. Right: graph associated to γπ = {{1, 3}, {2}, {4}}. The root is in red. Finally, given π ∈ NC2(2k), we define the map T = Tπ : [2k] → [k + 1] as follows. By Lemma 4.8, we know that #γπ = k + 1 and let γπ = {V1, V2, . . . , Vk+1}. Define Tπ(i) …

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