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Central limit theorems for linear spectral statistics of inhomogeneous random graphs with graphon limits

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Pith's one-line read This paper proves Gaussian central limit theorems for the eigenvalue fluctuations of inhomogeneous random graphs in every sparsity regime, with covariances determined by the graphon limit of the variance profile.

desk verdict Strong, likely-correct CLTs for graphon-based inhomogeneous random graphs across all sparsity regimes, with a fixable gap in the stated graphon-continuity lemmas. read the letter →

arxiv 2412.19352 v2 pith:UCOFGFKW submitted 2024-12-26 math.PR math.CO

Xiangyi Zhu , Yizhe Zhu This is my paper · ORCID
classification math.PRmath.CO MSC 60B2005C8060F05
keywords centrallimittheoremlinearspectralstatisticsinhomogeneousrandomgraphsgraphonconvergenceWigner-typematriceseigenvaluefluctuationshomomorphismdensityphasetransition
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 proves central limit theorems for the linear spectral statistics of inhomogeneous random graphs in every sparsity regime. For both the adjacency matrix $A_n$ and its centered version $\overline{A}_n$, it identifies the correct rescaling and shows convergence to a centered Gaussian process whose covariances are explicit homomorphism densities of the graphon limit of the variance profile. The central structural finding is a phase transition at $np = n^{o(1)}$: when $np = n^{\Omega(1)}$ the two matrices need different scalings and have different limits, while when $np = n^{o(1)}$ they share the same scaling and the same limiting Gaussian process. For the non-centered adjacency matrix, infinitely many further phase transitions appear as $np$ ranges from $n^{1/m}$ to $n^{1/(m-1)}$ for $m \geq 2$. These results matter because they turn graphon limits into concrete predictions for spectrum fluctuations of sparse random networks.

What carries the argument

The central object is the variance-profile graphon $W_n$ built from the entrywise variance matrix $S_n$ of the random graph, together with its limit $W$ under one of three graphon topologies. The technical engine is the expansion of $\mathrm{tr}(M^k)$ as a sum over closed walks: a $k$-tuplet assigns a sequence of vertices, and the product of entries along the walk is a monomial whose graph is the union of the two walks in a covariance computation. The paper shows that only a short list of union-graph topologies survives the rescaling — trees glued along one edge, unicyclic graphs, and pairs of cycles with one shared edge — and each surviving topology contributes a homomorphism density times an explicit combinatorial multiplicity. Which topologies survive is determined by the size of $np$, which is why the covariance formulas and scalings change across regimes.

What would settle it

Simulate a homogeneous $G(n,p)$ random graph with $p = n^{-0.6}$, so that $n^{1/3} \ll np \ll n^{1/2}$. The theorem for the uncentered matrix predicts that $\mathrm{Var}(L_{n,4})$ diverges while $L_{n,5}$ satisfies a CLT with covariance $50$ (for constant graphon $W=1$). Monte Carlo data showing $\mathrm{Var}(L_{n,4})$ stabilising, or showing a non-Gaussian distribution for the standardized $L_{n,5}$, would refute the sparse phase-transition claim.

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Extended reading notes

Core claim

The paper's central claim is that global eigenvalue fluctuations of inhomogeneous random graphs are Gaussian at every sparsity scale, once the traces are centered and rescaled in the sparsity-dependent way spelled out in (1), (5), and (6). The limiting covariance between the $k$-th and $h$-th monomial statistics is a finite sum of homomorphism densities $t(F,W)$ (or $t(F,W')$ with $W' = W(1-pW)$ in the dense centered case), where $F$ ranges over glued-tree, unicyclic, and two-cycle graphs whose combinatorial structure is enumerated by the walk expansion. The theorem statement distinguishes dense $p \in (0,1)$, sparse $np\to\infty$ with $np=n^{\Omega(1)}$, critically sparse $np=n^{o(1)}$ with $np\to\infty$, and bounded-degree $np\to c\in(0,\infty)$ regimes. A direct corollary is the phase transition in the centering effect: for $np=n^{\Omega(1)}$ the adjacency matrix and its centered version obey different CLTs, while for $np=n^{o(1)}$ they coincide. The paper also asserts that for the non-centered matrix, in the window $n^{1/m}\ll np \ll n^{1/(m-1)}$ a CLT with finite variance holds exactly for monomial degrees $k=2$ and $k\ge 2m-1$, and that the required graphon-convergence assumption weakens as $p$ decreases.

Load-bearing premise

The entire description assumes that the variance-profile matrices converge to a single limiting graphon in the topology required by the sparsity regime; if that convergence fails, the stated Gaussian limits and covariances are not defined.

Editorial extensions

If this is right

  • A single framework now covers dense, sparse, and bounded-degree inhomogeneous random graphs, giving explicit Gaussian covariances in every case.
  • When $np = n^{\Omega(1)}$, the centered and non-centered adjacency matrices have genuinely different fluctuation laws; when $np = n^{o(1)}$, centering no longer changes the CLT.
  • For the non-centered matrix there are infinitely many sparsity windows labeled by $m \geq 2$, and inside each window the CLT applies to monomial degrees $2$ and $k \geq 2m-1$.
  • The bounded-degree result $np \to c$ recovers the earlier $G(n,c/n)$ CLT for the centered adjacency matrix and extends it to inhomogeneous variance profiles.
  • The three graphon-convergence assumptions form a hierarchy: dense regimes need $\delta_1$-convergence, sparse regimes need cut-metric or tree-homomorphism convergence.

Reading between the lines

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

  • A practical consequence not drawn by the paper: the covariance formulas supply a goodness-of-fit test for graphon models, comparing observed fluctuations of trace statistics with the variance predicted by a fitted graphon.
  • The infinitely many phase transitions imply that low-degree trace statistics are not stable across sparsities; empirical work using, say, $k=4$ must choose normalization for the specific $np$ window.
  • Because the proof is built on union-graph counting rather than special structure of adjacency matrices, the same closure could plausibly handle Laplacian or non-backtracking matrices, though the paper does not claim this.
  • The dense covariance's use of $W' = W(1-pW)$ suggests the natural effective limit is the variance graphon of the entries; finite-$n$ simulations could test whether the $-p t(C_2,W)$ correction is visible.
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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 / 5 minor

Summary. The paper establishes central limit theorems for linear spectral statistics of the adjacency matrix and the centered adjacency matrix of inhomogeneous random graphs, under the assumption that the variance profile converges to a graphon. The results cover the dense regime, the sparse regime with np tending to infinity, and the bounded-expected-degree regime. The limiting covariances are expressed as homomorphism densities to the graphon limit of the variance profile. A phase transition is identified: for np = n^{Ω(1)} the two matrices require different scalings, while for np = n^{o(1)} they share the same scaling and the same limiting Gaussian process. For the non-centered adjacency matrix, additional phase transitions are described when n^{1/m} ≪ np ≪ n^{1/(m-1)}. The proofs are based on a careful enumeration of union graphs associated with products of walk indicators and a moment-method verification of joint Gaussianity.

Significance. If the technical gap noted below is repaired, this is a substantial contribution. It provides the first unified graphon-based treatment of global eigenvalue fluctuations for inhomogeneous random graphs across all sparsity regimes, with explicit covariance formulas and a new phase transition for the centering effect. The paper also weakens the required graphon convergence assumptions as the graph becomes sparser, connecting graphon theory with random matrix theory in a novel way. The proof strategy is systematic and the counting of contributing graph classes is carried out in detail, including an appendix with enumerations of the relevant tree and unicyclic families.

major comments (2)
  1. [§2, Lemmas 2.4 and 2.6; used in §4.1 and §5.2] Lemmas 2.4 and 2.6 are stated for graphons in W0 = {0 ≤ W ≤ 1}, but Assumption 3.1 only bounds the variance-profile entries by an absolute constant C, and C > 1 is allowed. Consequently the associated graphons W_n and the modified profiles W'_n = W_n(1 - pW_n) need not lie in W0; for instance, if s_ij = C and p ≤ 1/C, then W'_n is close to C(1-pC), which exceeds 1 whenever C > 1 and p < 1 - 1/C. The proofs of Theorem 3.8, Theorem 3.16, and Theorem 3.17 pass to the limit using these lemmas (e.g., §4.1 around Eq. (13) uses Lemma 2.6 for the multigraphs in T^{k,h}_2; §5.2.1 uses Lemma 2.4 for simple graphs under Assumption 3.3). As stated, the lemmas do not apply. The gap is repairable: the telescoping proof of Lemma A.1 yields |t(F,W) - t(F,W')| ≤ |E| C^{|E|-1} δ1(W,W') for graphons uniformly bounded by C, and the analogous Lipschitz bound for simple graphs under δ□ holds with the same constant, so the lemmas should be stated and proved for uniformly bounded graphons. Because every covariance formula relies on this limiting step, the gap is load-bearing for rigor.
  2. [§3.2, Theorem 3.17 and Remark 3.18] Remark 3.18 asserts that for 3 ≤ k ≤ 2m−2 in the regime n^{1/m} ≪ np ≪ n^{1/(m-1)}, Var(L_{n,k}) diverges at different scales, and therefore no CLT holds for those fixed k. This is a necessary-condition claim that is used to justify the restriction 'k = 2 and k ≥ 2m−1' in parts (3) and (4) of Theorem 3.17, but it is not proved in the manuscript. The proof of sufficiency is given, but the claimed necessary part is left as a remark. Since the theorem statements explicitly restrict the range of k, this omission does not affect the validity of the stated CLTs, but it should be either proved or clearly labeled as a conjecture.
minor comments (5)
  1. [§3.1, Definition 3.7] The phrase 'where and all other edges are visited exactly twice' contains an extra 'and'; it should read 'where all other edges are visited exactly twice'.
  2. [§5.2.1, Case 2(ii)] The sentence 'In this case, h must be even' is imprecise: it is the closed walk on the tree G_{J_h} that forces h to be even, not the cycle case. Please clarify which of k or h is being restricted.
  3. [§2, Definition 2.2] The definition of homomorphism density is given for loopless multigraphs, but Lemma 2.4 and Lemma 2.5 refer to simple graphs and multigraphs respectively; the distinction is clear, yet the text would benefit from an explicit sentence confirming that t(F,W) is defined for all loopless multigraphs used later, including T^{k,h}_2 and T C^{k,h}.
  4. [Appendix A, Lemma A.1 proof] In the proof of Lemma A.1, the notation m is used both for the number of edges |E| and later in the sentence 'm||W − W'||_1'; using |E| consistently would avoid confusion.
  5. [§3.1, Theorem 3.11] In the covariance formula (3), the factor 1/c^{(k+h)/2 - v(T)} is correct from the derivation, but the notation j ≤ l/2 appears in the proof where l should be h; this typo should be fixed.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the CLT derivations are independent moment computations, with only a minor non-load-bearing self-citation in Corollary 3.12.

full rationale

The derivation chain is not circular. The main theorems (Theorems 3.8, 3.9, 3.11, 3.16, 3.17, 3.19) are proved by direct moment computations: the covariance of the rescaled traces is expanded into sums over closed-walk tuples, and a graph-counting argument identifies which union graphs survive in each sparsity regime. The limiting covariance is then expressed as homomorphism densities t(·,W) or β_T; these are the assumed convergence parameters of the variance-profile graphons, but the theorems determine which finite graphs appear and with which coefficients, so the output is not identical to the input by construction. No parameter is fitted to the target CLT. The only self-citation occurs in the proof of Corollary 3.12, where [68, Lemma 4.1] is used as a published black box to verify β_T = 1 under condition (4); this corollary is not load-bearing for the main theorems, which assume Assumption 3.4 directly. The technical concern that Lemmas 2.4 and 2.6 are stated for [0,1]-valued graphons while Assumption 3.1 only bounds W_n by an absolute constant C is a rigor or correctness gap, not circularity: the same continuity proofs extend to uniformly bounded graphons, but that extension is not written out. Overall circularity score is 1.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The main theorems depend on the Bernoulli random graph model with a variance profile that converges in a graphon sense. No parameters are fitted to data; the only constants are the sparsity parameter p and its limit c, which are part of the model. The graph classes introduced (T^{k,h}_1, etc.) are combinatorial objects used to index the covariance sums, not free parameters. Standard graphon theory lemmas and the moment method are the background axioms.

assumptions (5)
  • domain assumption Entries of the adjacency matrix are independent Bernoulli up to symmetry with a_ii=0 and a_ij ~ Ber(p s_ij).
    Assumption 3.1 defines the model. All theorems are stated for this model.
  • domain assumption The variance profile matrix S_n has a graphon limit in one of the senses: δ1, δ□, or tree homomorphism density convergence (Assumptions 3.2-3.4).
    The limiting covariance formulas are expressed via homomorphism densities to the graphon W. Without such convergence the results do not apply.
  • domain assumption Uniform boundedness of the variance profile: 0≤s_ij≤C and sup_ij(p s_ij)≤1.
    Assumption 3.1(3) is used throughout the proofs to bound contributions of union graphs; it also ensures probabilities are well-defined.
  • standard math Graphon theory lemmas: Lemma 2.4 (Lovász, Theorem 11.5) characterizing cut metric convergence via simple graph homomorphism densities, and Lemma 2.6 (proved in Appendix A) on continuity of multigraph homomorphism densities under δ1.
    Used to pass from convergence of W_n to convergence of t(G,W_n) for the graph classes in the covariance formulas.
  • standard math Wick's formula for Gaussian moments and the method of moments (Section 4 and 5, Step 2).
    Used to show joint moments of the linear statistics converge to Wick pairings, yielding joint Gaussianity.

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Pith. "Pith review of Central limit theorems for linear spectral statistics of inhomogeneous random graphs with graphon limits." pith.science (2026). https://pith.science/paper/UCOFGFKW

@misc{pith2026241219352,
  author       = {Pith},
  title        = {Pith review of: Central limit theorems for linear spectral statistics of inhomogeneous random graphs with graphon limits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UCOFGFKW}},
  note         = {Machine review of arXiv:2412.19352}
}
abstract

We establish central limit theorems (CLTs) for the linear spectral statistics of the adjacency matrix of inhomogeneous random graphs across all sparsity regimes, providing explicit covariance formulas under the assumption that the variance profile of the random graphs converges to a graphon limit. Two types of CLTs are derived for the (non-centered) adjacency matrix and the centered adjacency matrix, with different scaling factors when the sparsity parameter $p$ satisfies $np = n^{\Omega(1)}$, and with the same scaling factor when $np = n^{o(1)}$. In both cases, the limiting covariance is expressed in terms of homomorphism densities from certain types of finite graphs to a graphon. These results highlight a phase transition in the centering effect for global eigenvalue fluctuations. For the non-centered adjacency matrix, we also identify new phase transitions for the CLTs in the sparse regime when $n^{1/m} \ll np \ll n^{1/(m-1)}$ for $m \geq 2$. Furthermore, weaker conditions for the graphon convergence of the variance profile are sufficient as $p$ decreases from being constant to $np \to c\in (0,\infty)$. These findings reveal a novel connection between graphon limits and linear spectral statistics in random matrix theory.

Figures

Figures reproduced from arXiv: 2412.19352 by the authors.

Figure 1
Figure 1. We give an example of k = 8 and h = 6. The left figure is an element in T k,h 1 constructed by a rooted tree one of length 4 with edges {1, 2}, {2, 3}, {3, 4}, {4, 5} and the root vertex 3, and another rooted tree of length 3 with edges {4, 5}, {5, 6}, {6, 7} and a root vertex 6, and one overlapping edge {4, 5}. The right figure is a corresponding element in T k,h 2 constructed by two rooted pla￾nar trees, one of le… view at source ↗
Figure 2
Figure 2. We given an example of k = 12, h = 10 and r = 4. The left figure are two graphs G1 ∈ T Ck r and G2 ∈ T Ch r . The right graph shows two possible graphs in T Ck,h . Cov(Xk, Xh) =    p P G∈T Ck,h t(G, W′ ) if k, h ≥ 3 are odd, P T∈T k,h 1 t(T, W′ ) − 4p P T∈T k,h 2 t(T, W′ ) + p P G∈T Ck,h t(G, W′ ) if k, h ≥ 2 are even, 0 otherwise. When p → 0 and np → ∞, a different CLT holds under a weaker assumption on … view at source ↗
Figure 3
Figure 3. Given the rooted trees T1 and T2 in the left graph, the right graph shows two examples in P#(T1, T2), where T1 and T2 overlap in the blue part for each case. In Theorem 3.9, the only nontrivial covariance term arises from gluing two rooted planar trees by sharing a single overlapping edge. For the bounded expected degree regime np → c ∈ (0,∞), a more general tree gluing is required. Definition 3.10 (Tree gluing). Fo… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: We give an example of k = 4 and h = 5. The left figure is an example of F k,h 1 constructed by cycle one of length 4 with E = {{1, 2}, {2, 3}, {3, 4}, {4, 1}} and cycle two of length 5 with E = {{2, 3}, {2, 7}, {7, 6}, {6, 5}, {5, 3}} having one overlapping edge {2, 3}…
Figure 5
Figure 5. Figure 5: We give an example of h = 5. The left figure is an example of Ch and the right figure is an example of C2,h. In particular, the blue part in the left figure is an example of K2 and the blue part in the right figure is an example of C2. To describe the limiting variance…

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