{"id":"590fc290-aa26-42b3-9109-503c64042ce6","arxiv_id":"2608.11321","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Communities in graphs with inhomogeneous latent geometry can be recovered by density-based spectral clustering on deeper eigenspaces, with a proven almost-exact recovery guarantee.","lead":"This paper shows that hidden communities can be recovered from eigenvectors deep in the spectrum when low-dimensional geometry confounds the leading eigenvectors. It develops a density-based spectral clustering algorithm, DBSPEC, with a consistency guarantee and tests it on real networks.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The core theorem is conditional on an oracle set B; the paper's only concrete recipe for B (eq. 7.2) is unproved and is fitted in-sample, so the real-world bridge from Theorem 4.1 is missing.","rationale":"The reader's weakest_assumption is the main obstruction and I agree with it. The mathematical core of the paper, Propositions 3.1 and 3.2 as an operator approximation and the DBSCAN consistency argument in Section 6, is coherent; I do not see an internal contradiction in Theorem 4.1 once B is assumed to be valid and well-separated from 0. The conditional theorem is a genuine contribution. However, the paper is presented not only as a theorem but as a method with a real-world validation, and that is where the argument is least secure. Eq. (7.2) is not derived with error bounds; it uses parameters estimated with access to the ground truth; the DBLP A result required post-hoc data modification; and the reported accuracies have no error bars. Because the abstract explicitly claims predictions notably align with observations, the missing certified or held-out localization is load-bearing. There is also a smaller proof gap in Proposition 3.1 regarding sets whose boundary contains 0, which reinforces the need to state B with a positive distance from 0. These concerns do not justify rejection: the conditional theorem appears sound and the algorithmic idea is novel. They do justify keeping the reader's CONDITIONAL verdict: accept only if the authors either prove that a suitable B can be constructed from observable quantities, or re-run the real-data evaluation with held-out parameters and error bars.","tokens_in":26144,"tokens_out":18261,"duration_ms":181456,"concrete_test":"Run a Monte-Carlo version of the Section 7 pipeline inside the Section 2 model: choose X=[0,1]^2, K=1{||x-y||<=R} with R=1/(2pi), a=0.75, b=0.2, rho_n=log^2(n)/n, and n in {10^3, 10^4, 10^5}. For each n, estimate a_hat, b_hat from the true labels as in Table 1, form B_hat_n = [lambda_hat_max tau_hat_min/tau_hat_max - eps_n, lambda_hat_max tau_hat_min/tau_hat_max + eps_n] with eps_n = c/sqrt(n rho_n), and record whether the true informative eigenvalue n rho_n (a-b)kappa*/2 lies in B_hat_n and whether 0 is excluded, with probability tending to 1. Then repeat the Table 2 evaluation with a_hat, b_hat estimated on a random half of vertices and accuracy measured on the held-out half. If containment fails at the tested bandwidths, or if the held-out accuracy gain over classical spectral clustering disappears, then eq.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 4.1 guarantees almost exact recovery only for an input Borel set B that contains the ideal eigenvalue (a-b)kappa*/2 and excludes 0. The algorithm's eigenspace is defined by Lambda_n = {n rho_n y : y in B}, so if B is misplaced the embedding can miss the informative psi^-_* component entirely. Sections 3-6 do not construct such a B from data; they take it as an oracle input. The only concrete construction is the real-data estimator (7.2), lambda_hat_* = lambda_hat_max tau_min / tau_max, where tau_min and tau_max are built from the empirical densities a1, a2, b and community sizes in Table 1. These quantities are computed with access to the ground-truth labels that the method is supposed to recover, no confidence interval or bandwidth is specified, and no theorem states that an interval around lambda_hat_* contains n rho_n (a-b)kappa*/2 while staying away from 0. The experiments do not close this gap: in DBLP A the predicted index is 1, giving no improvement, and the match is obtained only after manually removing the two largest cliques (DBLP A*); for LiveJournal and DBLP B the predicted index is close to but not equal to the optimal index (77% vs 85% and 74% vs 75%). A related technical caveat is that the proof of Proposition 3.1 asserts that nu(dB)=0 yields uniform epsilon-separation from dB, which is false when 0 is in dB because the kappa_i accumulate at 0; the theorem statement should require a positive gap between B and 0. None of this invalidates the conditional theorem, but it means the paper's claim that the theory notably aligns with observations is not backed by a certified localization procedure.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies spectral clustering in a block latent-space model in which vertices have random positions on a compact set X and edge probabilities are ρ_n K(X_v,X_w) times a 2×2 block matrix P. This generalizes the soft geometric block model from toroidal to arbitrary compact latent geometry under an L2-continuity assumption on K. The authors prove that the empirical eigenvalues of the rescaled adjacency matrix converge to the spectrum of a limiting integral operator (Propositions 3.1 and 3.2), identify an informative eigenvector whose sign tracks the community labels, and propose DBSPEC, a DBSCAN-based algorithm using an eigenspace whose eigenvalues lie in a user-supplied set Λ_n. Theorem 4.1 states that, conditional on a Borel set B containing the ideal eigenvalue (a−b)κ*/2 and excluding 0, DBSPEC achieves almost exact recovery with o(n) errors under superlogarithmic average degree. The paper also reports real-world experiments on Political Blogs, LiveJournal, and DBLP, using the estimator λ̂* = λ̂max τmin/τmax in Eq. (7.2) to select a higher-order eigenvector, and reports that DBSPEC improves over sign-based clustering after normalization for degree fluctuations.","tokens_in":26566,"tokens_out":5123,"duration_ms":46755,"significance":"If the mathematical claims hold after the gaps noted below are fixed, the paper makes a substantive contribution: it replaces the torus-specific tracial moment analysis of prior work with an operator-theoretic framework, extends the model to inhomogeneous and non-toroidal latent spaces, and gives a consistency guarantee under average degree just superlogarithmic. The paper also provides a useful algorithmic insight: multidimensional eigenspaces are more robust than sign-based clustering when the informative eigenvalue is non-simple or poorly separated, and the proof strategy via Bernstein concentration, Weyl/Davis–Kahan perturbation, and Jentzsch positivity is standard and mostly convincing. The real-data experiments are suggestive, but the claimed alignment between theory and observations is currently weaker than the abstract suggests because the eigenvalue-location estimator is fitted in-sample and no theorem guarantees that it produces a valid set B for Theorem 4.1.","major_comments":[{"comment":"The abstract and Section 7 claim that theoretical predictions for the ideal eigenvalue location align with real-world observations, but this claim is not established by the current experiments. The quantities a1, a2, b, n1, n2 entering Eq. (7.2) are computed from the same graphs whose ground-truth labels are used both to evaluate accuracy and to estimate these parameters, and no confidence interval, bandwidth, or alternative data split is provided. More importantly, no theorem states that an interval around λ̂* satisfies the hypotheses of Theorem 4.1 (contains nρ_n(a−b)κ*/2 and excludes 0). Table 2 shows that the predicted index equals the optimal index only in DBLP A*, which is obtained by manually removing the two largest cliques; in LiveJournal and DBLP B the predicted index is not the optimal one (3 vs 4 and 4 vs 15, respectively), and in DBLP A it is 1, giving no improvement over classical spectral clustering. To support the predictive claim, the paper should either provide a provable construction of a set B from observable data, or state the empirical claim in a substantially more qualified form.","section":"§7, Eq. (7.2) and Table 2"},{"comment":"The proof of Proposition 3.1 asserts that ν(∂B)=0 implies the existence of ε>0 such that every eigenvalue of the limiting operator has distance at least ε from ∂B, and consequently that B contains only finitely many eigenvalues. This implication is false when B is not separated from 0: since the κ_i accumulate at 0, a Borel set with 0 in the closure of ∂B but 0∉B can have no uniform spectral gap and can contain infinitely many eigenvalues. Proposition 3.2 and Lemma 6.8 require the matrix Ψ_B to have finite dimension ν(B), and the Davis–Kahan step requires a positive spectral gap near B. The statements should replace the condition 0∉B with the stronger condition inf{|y|: y∈B}>0, or otherwise justify that the convergence in rearrangement distance alone suffices for unbounded-count indicator sets. As written, the proof of Propositions 3.1 and 3.2 has a load-bearing technical gap.","section":"§3, proof of Proposition 3.1 (Section 5) and Proposition 3.2"},{"comment":"The estimator (7.2) for unbalanced clusters is asserted without a derivation. Remark 4.4 and Theorem A.2 provide a sufficient condition of the form {τ_j κ*: τ_j≠0}⊆B, but they do not prove that the informative eigenvalue for clustering is given by λ̂max τmin/τmax, nor that this quantity is the analog of (a−b)κ*/2 when community sizes and internal densities differ. Since all datasets in Table 1 violate the equal-size/equal-density assumption of the main theorem, the empirical predictions in Table 2 depend on this unproved generalization. The paper should provide a derivation of Eq. (7.2) from the k-cluster analysis, or explicitly mark it as a heuristic and adjust the strength of the empirical conclusions accordingly.","section":"§7.1, Eq. (7.1)–(7.2), and Appendix A (Remark 4.4)"}],"minor_comments":[{"comment":"The journal name in reference [29] is misspelled as \"Bernouilli\"; it should be \"Bernoulli\".","section":"References"},{"comment":"The sentence \"There exists some ε0>0 such that such that asymptotically almost surely\" contains a duplicated \"such that\" and should be reworded.","section":"Lemma 6.9"},{"comment":"In the line defining r, the expression |(ρ_j)_s−(ρ_j)_s| should read |(ρ_j)_s−(ρ_j)_t|.","section":"Appendix A.3, proof of Lemma A.5"},{"comment":"The definition of P̃ in Eq. (7.1) would benefit from an explicit statement that it corresponds to P diag(n1/n, n2/n) from Remark 4.4, and the connection between n1,n2 and the q_j of the appendix should be stated in one place.","section":"Eq. (7.1) and surrounding text"},{"comment":"The notation in the proof of Lemma 6.12 uses w(0),...,w(k) while the text also refers to w(ℓ) and w(m); the indexing is slightly confusing and should be cleaned up for readability.","section":"§6, Lemma 6.12"}],"recommendation":"major_revision","confidential_remarks":"The mathematical core of this paper is attractive and likely correct after tightening the separation condition for B. The main reason for major revision rather than minor revision is that the paper's advertised real-world validation currently rests on an unproved and in-sample estimator of the ideal eigenvalue location; this is not a fatal flaw because the empirical claims can be weakened or supplemented with a provable construction, but it is load-bearing for the paper's stated contribution. I would be happy to see a revised version that addresses the three major comments."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis paper is worth taking seriously. It extends the soft geometric block model from a uniform torus to an arbitrary compact latent space with an L2-continuous kernel, and it analyzes the adjacency spectrum through a limiting integral operator rather than tracial moments. That is a real step forward. The consistency guarantee for the new DBSPEC algorithm in Theorem 4.1 is a substantial piece of work, and the proof—using Bernstein, Weyl, Davis–Kahan, and Jentzsch—hangs together. The synthetic example showing that a multi-eigenvector embedding beats sign-based clustering when the ideal eigenvalue is not simple is also convincing.\n\nThe main gap is the bridge from theorem to practice. Theorem 4.1 assumes a Borel set B containing the informative eigenvalue and excluding 0; neither the theorem nor the algorithm says how to find B from data. The only concrete recipe is equation (7.2), built from cluster densities a1,a2,b and the observed largest eigenvalue. Those quantities are estimated on the same graph, often using ground-truth labels the method is supposed to recover, and no theorem, confidence interval, or even a solid heuristic shows the resulting interval lands in a valid B. The experiments expose the problem: for DBLP A the predicted index is 1, no better than classical spectral clustering, and the improvement appears only after manually removing the two largest cliques. For LiveJournal and DBLP B the predicted eigenvalue is close but not equal to the empirically optimal one. So the abstract's claim that 'theoretical predictions notably align with observations' is not backed by a certified localization procedure.\n\nThere is also a small technical flaw in Proposition 3.1: the proof says ν(∂B)=0 gives uniform separation between ∂B and the limiting spectrum, but that is false when 0 lies in the closure of B because the κ_i accumulate at 0. The fix is easy—require dist(B,0)>0—but the statement as written is not quite correct.\n\nNone of this undermines the conditional theorem. The integral-operator approach is a genuine contribution and DBSPEC is a sensible algorithm. I would send this to a serious referee with the expectation of major revision: either prove a data-driven construction of B works under some assumptions, or clearly mark (7.2) as a heuristic and soften the empirical language. I would bring the paper to our reading group and would likely cite the spectral characterization even while treating the localization recipe as open.","headline":"Solid spectral theory for inhomogeneous latent geometry, but the practical eigenvalue-localization recipe is unproved and in-sample, so the algorithm's guarantee is conditional on an oracle set B.","tokens_in":27067,"tokens_out":3719,"would_cite":true,"duration_ms":77310,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C80","60B20","62H30","05C82"],"pacs":[],"model":"deepseek-v4-flash","headline":"Communities that latent geometry hides in the leading eigenvectors can still be recovered almost exactly from deeper eigenvalues, given only an approximate window around the informative eigenvalue, the paper proves.","keywords":["block latent-space model","almost exact recovery","density-based spectral clustering","geometric random graph","latent geometry","integral operator","higher-order spectral clustering","DBSCAN"],"falsifier":"Simulate the block latent-space model with a known kernel on, say, $\\mathcal{X} = [0,1]^2$ and compute the count of eigenvalues of $A/(n\\rho_n)$ inside a window $B$, comparing it with the predicted count $\\nu(B) = \\sum_i (\\mathbf{1}\\{(a+b)\\kappa_i/2 \\in B\\} + \\mathbf{1}\\{(a-b)\\kappa_i/2 \\in B\\})$; a discrepancy that persists as $n$ grows under $\\rho_n = \\omega(\\ln n/n)$ would refute Proposition 3.1. Independently, hand DBSPEC a window $B$ that provably contains $(a-b)\\kappa^*/2$ and excludes $0$, and measure the misclassification rate: Theorem 4.1 predicts $o(n)$ errors, so any linear fraction of errors would refute the algorithmic claim.","tokens_in":25926,"feed_emoji":"🕸️","tokens_out":22800,"duration_ms":174916,"temperature":0.7,"pith_summary":"Many real networks are shaped by two forces at once: where nodes sit in a latent space (physical or topical proximity) and which community they belong to, and the geometry can dominate the leading eigenvectors of the adjacency matrix so completely that standard spectral clustering fails. The paper shows this is not fatal: in a block latent-space model with an arbitrary compact latent space, an $L^2$-continuous kernel, and average degree growing faster than $\\log n$, the community information survives in an eigenvector whose eigenvalue sits near $(a-b)\\kappa^*/2$, generally not at the top of the spectrum. Its main theorem guarantees that DBSPEC, a density-based spectral clustering algorithm that needs only an approximate window around that eigenvalue, recovers the two clusters with only $o(n)$ errors. The spectral structure behind this is explicit: the limiting spectrum of the adjacency matrix equals that of the integral operator $\\tfrac{1}{2}P \\otimes \\mathbb{K}$, with eigenfunctions $(\\varphi_i, \\pm\\varphi_i)$, which predicts where the informative eigenvalue is. On real collaboration and social networks the prediction matches the empirically best eigenvalue, and higher-order eigenvectors lift accuracy from near-random to 76–99 percent.","feed_headline":"Deeper eigenvectors recover communities that geometry hides","feed_subtitle":"The community signal is buried mid-spectrum; a derived formula locates the right eigenvalue in real networks.","key_machinery":"The load-bearing object is the limiting integral operator $\\mathbb{A} = \\tfrac{1}{2}P \\otimes \\mathbb{K}$, acting on pairs of functions on the latent space by $(g_1,g_2)(x) = \\tfrac{1}{2} P \\int K(x,y)(f_1(y),f_2(y))\\,d\\mu(y)$; it is the mean-field limit of the rescaled adjacency matrix, and its eigenfunctions are $(\\varphi_i,\\varphi_i)$ and $(\\varphi_i,-\\varphi_i)$ with eigenvalues $(a\\pm b)\\kappa_i/2$. Two of its properties carry the proof. First, a Jentzsch-type theorem makes the principal eigenfunction $\\varphi^*$ strictly positive, so the vector $\\psi^-_* \\propto \\sigma_v\\varphi^*(X_v)$ has sign equal to the community labels, which is why its eigenvalue is the ideal location $(a-b)\\kappa^*/2$. Second, every nonzero eigenfunction of $\\mathbb{K}$ has a continuous representative, which forces the spectral embedding of each cluster to be a connected, uniformly dense point cloud while the two clusters sit a uniform distance apart; DBSCAN's density-connected components then isolate exactly the two clusters. A corollary of this operator picture is robustness: including a few extra eigenvectors near the window preserves the within-cluster continuity, so the algorithm tolerates approximate eigenvalue localization and eigenvalue multiplicities, whereas sign-based methods do not.","core_discovery":"The central claim is that community labels remain almost recoverable from mid-spectrum eigenvectors even when the latent geometry is arbitrary and inhomogeneous, and that the right eigenvalue is located by an explicit operator calculation. Concretely, for links realized with probability $\\rho_n K(X_v,X_w)P(\\sigma_v,\\sigma_w)$, the rescaled adjacency matrix $A/(n\\rho_n)$ mimics, in its eigenvalues and eigenspaces, the integral operator $\\mathbb{A} = \\tfrac{1}{2}P \\otimes \\mathbb{K}$ on two copies of the latent space: the limiting spectrum is the superposition of the sequences $\\tfrac{a+b}{2}\\kappa_i$ and $\\tfrac{a-b}{2}\\kappa_i$, where $\\kappa_i$ are the eigenvalues of the kernel operator $\\mathbb{K}$, and the limiting eigenvectors are $(\\psi^+_i, \\psi^-_i)$ with $\\psi^\\pm_i \\propto (\\pm 1)^{\\sigma_v}\\varphi_i(X_v)$. Because the principal eigenfunction $\\varphi^*$ is strictly positive, $\\psi^-_*$ has signs that match the cluster labels exactly, which makes it the informative eigenvector, with eigenvalue $(a-b)\\kappa^*/2$. Theorem 4.1 states that whenever the eigenvalue window $B$ contains that value and excludes $0$, DBSPEC — embed the vertices using all eigenvectors with eigenvalues in $n\\rho_n B$, then run DBSCAN with radius $\\Theta(1/\\sqrt{n})$ and minimum points $\\Theta(n)$ — outputs two clusters of size $n/2 - o(n)$ with $o(n)$ misclassified vertices, asymptotically almost surely. A density-based step succeeds where sign thresholds fail because within a cluster the embedding coordinates are continuous functions of the latent position and hence density-connected, while across clusters the $\\psi^-_*$ component leaves a uniform gap of order $1/\\sqrt{n}$.","pith_inferences":["A natural transfer test the paper does not run: the same limiting-operator mechanism should carry the informative component into related spectra, such as eigenvectors of the graph Laplacian or the normalized adjacency matrix, so a cheap check would be to run the windowed density-based scheme on those spectra of the same datasets and compare the recovered labels.","The step the theorem needs but does not supply is a provably valid, fully data-driven construction of the window $B$; the DBLP A versus DBLP A* comparison shows how sensitive the fitted estimator (7.2) is to outliers such as large cliques, so one testable remedy is an adaptive window that widens with the local eigenvalue gap around $\\hat\\lambda^*$, with a check that accuracy degrades gracefully ra","If the limiting-operator picture is right, the rescaled ideal eigenvalue $(a-b)\\kappa^*/2$ depends on the geometry only through the kernel's principal eigenvalue, not on the fine shape of the latent measure, which could be tested by keeping the kernel fixed and varying the latent distribution from uniform to boundary-concentrated while watching whether the best eigenvector's location stays put.","The four-cluster structure DBSPEC reveals inside DBLP A* (a dataset planted with two labels) suggests the algorithm can surface genuine substructure beyond the stated communities; a stability check across random halves of the network would test whether those subclusters replicate or are artifacts of the embedding."],"forward_implications":["Standard spectral clustering that thresholds the second eigenvector performs poorly (56 percent on DBLP A and LiveJournal, 65 percent on DBLP B) when geometry dominates; selecting the eigenvector nearest the predicted location raises accuracy to 74–77 percent, and the best higher-order eigenvector reaches 76–85 percent.","The recovery guarantee holds at superlogarithmic average degree, a much sparser regime than the linear average degree required by the earlier torus-based analysis, and the authors note that known regularization techniques could push almost-exact recovery even further.","The method is stable at degenerate eigenvalues where sign-based clustering fails: on the circle with parameters tuned so the ideal eigenvalue is non-simple in the limit, DBSPEC attains 100 percent accuracy while the sign-based predecessor attains 64 percent.","The same operator argument extends to $k \\ge 2$ clusters with unequal sizes and a general symmetric connectivity matrix: the limiting spectrum is the union over the eigenvalues $\\tau_j$ of $\\widetilde{P} = P\\,\\mathrm{diag}(q_1,\\dots,q_k)$ of the sequences $\\tau_j\\kappa_i$, and recovery holds whenever the window contains all nonzero $\\tau_j\\kappa^*$.","Degree fluctuations, visible as radial streaks in spectral embeddings of skewed networks, can be neutralized by normalizing each row to the unit sphere; for LiveJournal, DBSPEC then achieves 99.3 percent accuracy on 99.6 percent of the vertices even when the embedding uses the three leading eigenvectors."],"supporting_citations":[{"why":"Introduces the two-cluster soft geometric block model on a torus and the sign-based HOSC baseline that this paper generalizes to arbitrary latent spaces and compares against in experiments.","marker":"[5]"},{"why":"Supplies the theorem that eigenvalues of matrices sampled from an integral operator converge to the operator's spectrum, the eigenvalue half of Proposition 3.1.","marker":"[29]"},{"why":"Supplies the companion control of spectral projections and eigenvectors of such sampled matrices, the basis of Proposition 3.2.","marker":"[28]"},{"why":"Defines DBSCAN and its density-connectivity properties, which the proof of Theorem 4.1 uses to convert geometric separation of embeddings into cluster recovery.","marker":"[15]"},{"why":"Provides the Jentzsch-type theorem used to show the principal eigenfunction of the kernel operator is strictly positive, making the sign of the informative eigenvector equal to the cluster labels.","marker":"[25]"},{"why":"Provides the matrix Bernstein inequality used in Lemma 5.1 to concentrate the adjacency matrix around its conditional expectation at superlogarithmic average degree.","marker":"[40]"},{"why":"Provides the Davis–Kahan perturbation bound that carries the eigenvector approximation from the conditional expectation to the observed adjacency matrix.","marker":"[43]"}],"fun_headline_variants":["Mid-spectrum eigenvectors reveal communities hidden by geometry","Dig deeper in spectrum when geometry masks communities","DBSPEC finds community eigenvectors despite latent geometry","Mid-spectral eigenvectors expose clusters geometry hides"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The theorem presupposes that the algorithm receives a set of eigenvalues containing the ideal value $(a-b)\\kappa^*/2$ and excluding $0$, and the paper's practical recipe for finding such a set, equation (7.2), is fitted from the same graph's cluster densities and largest eigenvalue, with no proof that the estimate lands in a valid window.","fun_headline_variants_meta":{"raw":{"variants":["Mid-spectrum eigenvectors reveal communities hidden by geometry","Dig deeper in spectrum when geometry masks communities","DBSPEC finds community eigenvectors despite latent geometry","Mid-spectral eigenvectors expose clusters geometry hides"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000631,"raw_usage":{"total_tokens":2959,"prompt_tokens":1033,"completion_tokens":1926,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":649,"completion_tokens_details":{"reasoning_tokens":1870}},"tokens_in":649,"tokens_out":1926,"duration_ms":14077,"temperature":1.0,"reasoning_tokens":1870,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:12:36.959010+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the block latent-space model with a known kernel on, say, $\\mathcal{X} = [0,1]^2$ and compute the count of eigenvalues of $A/(n\\rho_n)$ inside a window $B$, comparing it with the predicted count $\\nu(B) = \\sum_i (\\mathbf{1}\\{(a+b)\\kappa_i/2 \\in B\\} + \\mathbf{1}\\{(a-b)\\kappa_i/2 \\in B\\})$; a discrepancy that persists as $n$ grows under $\\rho_n = \\omega(\\ln n/n)$ would refute Proposition 3.1. Independently, hand DBSPEC a window $B$ that provably contains $(a-b)\\kappa^*/2$ and excludes $0$, and measure the misclassification rate: Theorem 4.1 predicts $o(n)$ errors, so any linear fraction of errors would refute the algorithmic claim.","supporting_citations":[{"cited_title":"Avrachenkov, A","cited_arxiv_id":null,"evidence_quote":"Introduces the two-cluster soft geometric block model on a torus and the sign-based HOSC baseline that this paper generalizes to arbitrary latent spaces and compares against in experiments."},{"cited_title":"Horiguchi and Y","cited_arxiv_id":null,"evidence_quote":"Supplies the companion control of spectral projections and eigenvectors of such sampled matrices, the basis of Proposition 3.2."},{"cited_title":"Ester, H.-P","cited_arxiv_id":null,"evidence_quote":"Defines DBSCAN and its density-connectivity properties, which the proof of Theorem 4.1 uses to convert geometric separation of embeddings into cluster recovery."},{"cited_title":"Gaudio, X","cited_arxiv_id":null,"evidence_quote":"Provides the Jentzsch-type theorem used to show the principal eigenfunction of the kernel operator is strictly positive, making the sign of the informative eigenvector equal to the cluster labels."},{"cited_title":"Lovász.Large networks and graph limits","cited_arxiv_id":null,"evidence_quote":"Provides the matrix Bernstein inequality used in Lemma 5.1 to concentrate the adjacency matrix around its conditional expectation at superlogarithmic average degree."}],"review_version":1}