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Directed inhomogeneous random graphs get a non-uniform circular law

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arxiv 2607.08696 v1 pith:THVMMBN2 submitted 2026-07-09 math.PR

Spectrum of Directed Inhomogeneous Random Graphs

Rajat Subhra Hazra , Giacomo Passuello This is my paper
classification math.PR MSC 60B2005C80
keywords circular lawdirected random graphsinhomogeneous random graphsChung–Lu modelspectral outliersfree probabilityS-transformBauer–Fike theorem
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper studies what the eigenvalues of a large directed random graph look like when different vertices have different connection patterns — a model that captures degree heterogeneity in real networks. For the directed Chung–Lu model, where each vertex carries an in-weight and an out-weight governing its connection probabilities, the authors prove that the bulk spectrum of the adjacency matrix (after scaling) converges to a deterministic distribution in the complex plane. Unlike the classical circular law, which spreads eigenvalues uniformly over a disk, this limit is radially symmetric but non-uniform: its radial profile depends on the joint distribution of in- and out-weights through an S-transform from free probability. When the weight product takes a Marchenko–Pastur distribution, the density takes an explicit rational form. The authors also prove that a finite-rank mean structure — such as a stochastic block model — produces outlier eigenvalues separated from the bulk, and that these outliers fluctuate jointly as a Gaussian vector at scale sqrt(s_n/n), with a covariance that depends on the weight profiles. The key mechanism throughout is a combination of Girko's Hermitization (reducing non-Hermitian eigenvalue problems to singular value problems), Bauer–Fike perturbation theory (replacing the unavailable Weyl inequalities for non-symmetric matrices), and a fixed-point expansion of the eigenvalue equation that isolates a leading linear random term amenable to a Lindeberg central limit theorem.

Core claim

The central discovery is that directed inhomogeneous random graphs with diverging average degree exhibit a three-level spectral structure: a non-uniform circular-law bulk whose radial density is determined by the S-transform of the limiting weight-product distribution, a set of outlier eigenvalues pinned by the finite-rank expectation matrix, and Gaussian fluctuations of those outliers at scale sqrt(s_n/n) with an explicitly computable covariance. The non-uniformity of the bulk is the key new feature: vertex heterogeneity deforms the uniform disk of the classical circular law into a radially symmetric but non-uniform distribution, and this deformation is exactly characterized by free-probabi

What carries the argument

Girko's Hermitization trick (reducing complex eigenvalue convergence to singular value control of shifted matrices), the Bauer–Fike theorem (providing eigenvalue perturbation bounds for diagonalizable non-Hermitian matrices), high-trace moment bounds on the centered adjacency matrix (controlling spectral norm via combinatorial path counting with Catalan/Dyck word structures), Lindeberg replacement principle (swapping Bernoulli entries for Gaussian ones with matching variance profiles), asymptotic freeness and R-diagonal operators from free probability (identifying the limiting singular value distribution as a free multiplicative convolution), and an eigenvalue fixed-point expansion (expresss

Load-bearing premise

In the sparse regime, the least singular value bound requires a technical symmetry condition pairing the first and second halves of the weight sequence, which the authors themselves flag as an artifact of their folding argument and expect to be removable. Without this condition, the bulk convergence proof is incomplete for the sparsest graphs covered, though the denser regime is unaffected.

What would settle it

If the boundedness condition on the variance profile (Assumption 2.1, requiring all weights to lie in [c, C]) is violated — for instance with power-law weight distributions having unbounded support — the deterministic equivalent results from free probability used to identify the limiting bulk distribution may fail, and the S-transform formula for the radial density would no longer apply in its current form.

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If this is right

  • The non-uniform circular law provides a template for detecting community structure or degree heterogeneity in directed networks: deviations of the empirical spectral distribution from the classical uniform disk reveal the weight profile.
  • The explicit Gaussian fluctuation covariance for outlier eigenvalues could enable statistical inference about the rank and structure of the mean matrix in directed random graph models, analogous to signal detection in undirected settings.
  • The spectral gap estimate for the transition matrix (Theorem 2.12) implies that simple random walks on such directed graphs mix at rate O(1/sqrt(s_n)), connecting spectral theory to mixing time bounds for non-reversible Markov chains.
  • The framework extends to directed stochastic block models, providing both the bulk shape and the fluctuation theory for community-detecting eigenvalues in the directed regime.

Where Pith is reading between the lines

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

  • The condition that s_n diverges faster than log^ξ(n) for ξ > 4 is likely an artifact of the proof technique (high-trace moment bounds and very-high-probability estimates) rather than a genuine phase transition; the true threshold for the bulk law and outlier separation may be closer to log(n), matching strong connectivity thresholds.
  • The bounded average degree regime (s_n = O(1)) should produce a qualitatively different spectral picture — potentially with atoms at the origin and non-circular bulk shapes — analogous to recent results for homogeneous sparse matrices, but the inhomogeneous case remains open.
  • The fixed-point expansion technique for eigenvalue fluctuations could extend to complex-valued finite-rank perturbations, provided outlier eigenvalues are separated in modulus, potentially yielding fluctuation results for non-real outlier pairs in directed models with complex mean structure.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. This paper studies the spectrum of the adjacency matrix $A_n$ of directed inhomogeneous random graphs on $n$ vertices with independent entries and diverging average degree scale $s_n$. The framework encompasses directed Chung--Lu graphs and directed stochastic block models. The authors establish three main types of results: (1) a non-homogeneous circular law for the rank-one Chung--Lu model (Theorem 2.5, Corollary 2.6), identified via Girko Hermitization and free probability (R-diagonal operators, S-transforms); (2) existence of spectral outliers separated from the bulk at scale $O(1/√{s_n})$ for both the adjacency matrix (Theorems 2.9, 2.15) and the random-walk transition matrix (Theorem 2.12), proved via Bauer--Fike perturbation theory and high-trace moment bounds; (3) Gaussian fluctuations of outlier eigenvalues at scale $√{s_n/n}$ with explicit covariance, for both rank-one (Theorem 2.13) and finite-rank (Theorem 2.16) models, obtained through a fixed-point expansion and Lindeberg CLT. The bulk analysis requires least singular value estimates (Theorems 3.3, 4.1), intermediate singular value control (Theorem 3.4), and a Lindeberg invariance principle to replace Bernoulli entries by Gaussian ones (Lemma 3.8). The sparser regime (Assumption 2.3) requires an additional symmetry condition on the weight profile.

Significance. The paper extends the spectral theory of directed random graphs to the inhomogeneous setting with degree heterogeneity and finite-rank mean structure, a natural and previously underdeveloped combination. The explicit radial description of the limiting bulk distribution via the S-transform (Corollary 2.6), including worked examples such as the Girko sombrero distribution (Example 2.17) and the Marchenko--Pastur case (Example 2.18), provides concrete and falsifiable predictions. The fluctuation result for finite-rank models (Theorem 2.16) with an explicit covariance formula is a substantive contribution that extends the undirected inhomogeneous theory of [CCH20] to the non-Hermitian setting. The use of Bauer--Fike in place of Weyl interlacing for the outlier analysis is a methodologically appropriate adaptation. The results are purely mathematical derivations from standard tools; no circularity concerns arise.

major comments (2)
  1. §6.2, Lemma 6.2 (Eqs. 6.12–6.21): The sequential Bauer–Fike argument for the rank-$r$ fluctuation proof requires uniform condition-number bounds on the intermediate matrices $S^{(ℓ)}_n$ ($ℓ=1,2,3$). The bound $∥P^{(ℓ)}_n∥∥(P^{(ℓ)}_n)^{-1}∥≲2$ is justified by citing [GEJ95] and asserting that 'the same diagonal approximation holds' for each $S^{(ℓ)}_n$, meaning eigenvectors are approximately canonical basis vectors. However, $S^{(1)}_n=∑_{k=0}^L V_{k,n}/λ_l^k$ involves random matrices whose off-diagonal entries are controlled only in expectation via Lemma 5.3, and the deterministic determinant bound $det(X_n)=O(1)$ from the rank-one case (Eq. 6.1) does not directly transfer to the perturbed eigenvector matrices of $S^{(ℓ)}_n$. The condition number of these perturbed eigenvector matrices is not controlled by the deterministic bi-orthogonality of $v_i^±$ alone. The authors should either (a)
  2. §4 (Theorem 4.1 adaptation): The least singular value estimate in the sparse regime (Assumption 2.3) requires the symmetry condition $w_x^+=w_{x+⌊n/2⌋}^+$ for $x≤⌊n/2⌋$, which the authors flag as technical (Remark 2.4). The adaptation of [BR18, Theorem 11.3] via the folding trick (Eqs. 4.5–4.6) is explained for the homogeneous case and stated to extend to the inhomogeneous setting under this symmetry, but the verification that the inhomogeneous variance profile preserves the key probabilistic estimates (particularly the Lévy concentration function bound in Eq. 4.11) is only sketched. Since this is the only gap between the two sparsity regimes and the authors themselves expect the condition to be removable, a more detailed justification or a clearer statement of what specifically fails without the symmetry would strengthen the result.
minor comments (6)
  1. §2.1.1, Corollary 2.6: The notation $ν^2$ and $√ν$ for push-forward measures is introduced but the notation $√{ρ̄}$ used in Theorem 2.7 (Eq. 2.16) could be confused with the square root of the measure rather than the push-forward. A brief clarifying remark would help.
  2. §3.3.2, Lemma 3.8: The Lindeberg replacement principle is applied to real and imaginary parts of the resolvent trace. The derivative bounds are stated to be 'the standard resolvent bounds and same as in [Coo19, Lemma 8.2]' but the specific form of the third-moment contribution $O(s_n^{-1/2})$ should be briefly justified, as the inhomogeneous variance profile introduces $x,y$-dependent third moments.
  3. §5.3, Lemma 5.7 (Eq. 5.25): The combinatorial bound involves Catalan numbers, Dyck words, and path counting. The step from Eq. (5.25) to the bound $3·4^m∑E_{m,l}$ uses the inequality $∑_{p=2}^{l+1}(2(l+1-p))^{l+1-p}n^p l^{l+1-p}p_{max}^l ≤ 2n^{l+1}$, which should be verified more explicitly or referenced.
  4. §5.5, Eq. (5.52): The variance computation uses $p_{x,y}=s_n v_x^+ v_y^-$ but the final expression in Eq. (2.24) is written in terms of integrals against $ρ(x^+,x^-)$. The convergence of the discrete sums to the integrals is standard but should be stated explicitly for completeness.
  5. Figure 1: The caption mentions a logarithmic scale on the $x$-axis after threshold 1, but the axis labels are not clearly readable. Improving the figure resolution or adding explicit axis labels would aid interpretation.
  6. References: The arXiv preprint [AT26] is cited for eigenvector localization in directed Erdős–Rényi graphs but appears to be from 2026; the authors should verify the citation details.

Circularity Check

0 steps flagged

No circularity: pure mathematics paper with self-contained proofs from standard external results

full rationale

This is a pure mathematics paper proving spectral theorems for directed inhomogeneous random graphs. The main results (Theorems 2.5, 2.7, 2.9, 2.12, 2.13, 2.15, 2.16) are derived from first principles using standard external tools: Girko's Hermitization, the circular law framework from [TV08, TV10, BR19], deterministic equivalents from [Coo+18], free probability (S-transforms, R-diagonal operators) from [Voi87, BV93, HL00, RS07, AP09], Bauer-Fike perturbation theory [BF60], and Lindeberg CLT. The self-citations [BP25, BPQ26] appear only in the motivation paragraph about random walk cutoff and are not load-bearing for any proof. The covariance formula (Eq. 2.29) is computed from the variance of a sum of independent centered Bernoulli variables (Eq. 6.28), not fitted to data. The fixed-point expansion (Eqs. 5.34-5.42) is a genuine perturbative derivation where the leading term is identified and remainder terms are bounded. The condition-number concern raised in the skeptic's headline is a correctness/completeness issue about whether the Bauer-Fike condition number bounds are adequately justified for intermediate matrices in Lemma 6.2 — this is a mathematical gap question, not circularity. The paper does not fit parameters to data and then call the fit a prediction, does not define quantities in terms of their own claimed outputs, and does not invoke self-authored uniqueness theorems to forbid alternatives. The derivation chain is self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 0 invented entities

The paper introduces no new physical entities, particles, or forces. All mathematical objects (weight distributions, S-transforms, free probability operators) are standard in random matrix theory. The models (Chung-Lu, SBM) are well-established in the literature. The axioms are domain assumptions about the graph model (bounded weights, diverging degree) and standard mathematical results from prior literature.

axioms (7)
  • domain assumption Boundedness of weight profile: c ≤ w^±_x ≤ C for all x, n (Assumption 2.1, Eq. 2.2)
    Ensures variance profile is uniformly bounded above and below; needed for deterministic equivalent results from [Coo+18] and for the least singular value bounds.
  • domain assumption Diverging average degree: s_n ~ n^α (Assumption 2.2) or log²(n) ≪ s_n ≪ n with weight symmetry (Assumption 2.3)
    Controls sparsity. The logarithmic lower bound ensures strong connectivity and sufficient concentration; the weight symmetry in 2.3 is needed only for the folding trick in the sparse least singular value proof.
  • domain assumption Stronger growth for outliers: log^ξ(n) ≪ s_n ≪ n with ξ > 4 (Assumption 2.8)
    Needed for very-high-probability estimates required in the fluctuation analysis (Remark 2.11). The exponent 4 arises from the moment method in Proposition 5.6.
  • domain assumption Convergence of empirical weight distribution to a compactly supported measure ρ (Assumption 2.1, Eq. 2.3)
    Identifies the limiting spectral distribution; needed to express the S-transform and covariance matrix as integrals against ρ.
  • standard math Bi-orthogonality of left/right eigenvectors: (v^+_i)^t v^-_j = δ_{ij} (Section 2.2, before Eq. 2.25)
    Standard linear algebra condition ensuring the expectation matrix is diagonalizable with distinct eigenvalues; needed for Bauer-Fike perturbation theory.
  • standard math Results from [Coo+18] on deterministic equivalents for non-Hermitian matrices with variance profiles (invoked in proof of Proposition 3.9)
    Provides the Schwinger-Dyson equations and admissibility conditions that yield the logarithmic integrability of the limiting symmetrized singular value distribution.
  • standard math Least singular value bound from [TV08, Theorem 2.9] for sparse matrices with κ-controlled second moment (invoked in Theorem 3.3)
    Provides the polynomial lower bound on σ_n(A_n + M_n) needed to control the logarithmic potential near the origin in Girko's Hermitization.

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Pith. "Pith review of Spectrum of Directed Inhomogeneous Random Graphs." pith.science (2026). https://pith.science/paper/THVMMBN2

@misc{pith2026260708696,
  author       = {Pith},
  title        = {Pith review of: Spectrum of Directed Inhomogeneous Random Graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/THVMMBN2}},
  note         = {Machine review of arXiv:2607.08696}
}
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We study the spectrum of the adjacency matrix $A_n$ of directed inhomogeneous random graphs on $n$ vertices. We assume that $A_n$ has independent entries and diverging average degree scale $s_n$. This framework includes, as special cases, the directed Chung--Lu random graph and directed stochastic block models. Assuming boundedness of the variance profile and that $s_n$ diverges faster than a suitable logarithmic function of $n$, we show that the rank-one Chung--Lu model satisfies a non-homogeneous version of the circular law, which in some situations allows for an explicit expression. Moreover, under mild conditions, we identify the asymptotic singular value distribution using tools from free probability. Finally, for finite-rank directed models, we prove the existence of eigenvalues outside the bulk and establish their joint Gaussian fluctuations at the scale $\sqrt{s_n/n}$, with an explicit covariance matrix. These results extend the theory of spectral outliers and their fluctuations to directed inhomogeneous random graphs.

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