{"id":"3fb43e29-ccc6-48fc-a4e9-a4a897185f91","arxiv_id":"2507.06508","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"A matrix-based private subgraph counting framework, NAM, yields new triangle, quadrangle, and 2-star counting algorithms under edge-LDP.","lead":"This paper introduces a Noisy Adjacency Matrix approach for counting triangles, quadrangles, and 2-stars under edge-local differential privacy. It claims lower time complexity and the first quadrangle counting algorithm under pure edge-LDP.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Second-round sensitivity in TriTR/QuaTR is undercounted: adding one neighbor alters up to d_u clamped inner sums, so sensitivity is d_u·Δf_u, not Δf_u; pure ε-edge LDP fails.","rationale":"The reader's weakest assumption focused on the CLT approximation in Theorem 11 and the resulting β-probability of sensitivity bound violation. My stress-test confirms the general area but identifies a more concrete and decisive flaw: even if the CLT bound held deterministically, the sensitivity of the clamped sum in Algorithms 7 and 9 is not Δf_u but up to d_u·Δf_u, because the outer sum ranges over many inner sums, each of which can change when one neighbor is added. This is a deterministic miscomputation, not an approximation error. It directly invalidates the pure ε-edge LDP guarantees for TriTR (Theorem 5) and QuaTR (Theorem 9), which are central contributions of the paper: TriTR is claimed to achieve optimal accuracy, and QuaTR is the first pure edge-LDP quadrangle counting algorithm. Without these privacy guarantees the accuracy claims and the MSE bounds in Theorems 12 and 14 are moot. TriOR and TriMTR may still be salvageable (TriMTR's second round has sensitivity Δf_u due to single-term clamping, though its accuracy analysis still depends on the probabilistic bound), but the main headline results for two-round triangle and quadrangle counting are not established. Therefore the paper in its current form should be rejected, or at minimum undergo major revision with a corrected sensitivity analysis and re-derived privacy/accuracy guarantees. I give 'partial' agreement with the reader because the reader's CLT concern is real but secondary; the primary defect is the sensitivity calculation, which also explains why the CLT approach cannot yield pure ε-DP: the quantity being bounded by a quantile is not the function's sensitivity.","tokens_in":35265,"tokens_out":10357,"duration_ms":115984,"concrete_test":"Construct a graph where user u is connected to nodes 2,...,d+1, and the noisy matrix satisfies â_{i,1}=Δf_u for all i=2,...,d+1, with all other â entries zero. For the adjacency list without neighbor 1, sumu=0; with neighbor 1, each inner sum for i≥2 gains â_{i,1}=Δf_u, so sumu=d·Δf_u. The sensitivity is therefore d·Δf_u. Implement Algorithm 7 with ε2-edge LDP claim, evaluate the mechanism on these two adjacent lists, and compute the ratio Pr[output∈S]/Pr[output'∈S]. If for some output the ratio exceeds e^{ε2}, the privacy claim is refuted; this should occur whenever the data are as above and the Laplace noise has scale Δf_u/ε2.","verdict_should_be":"REJECT","load_bearing_attack":"The two-round privacy proofs fail because the global sensitivity of sumu is miscalculated. In Algorithm 7 (TriTR), sumu = Σ_{i∈Neiu} clamp(Σ_{j∈Neiu, j<i} â_{ij}, Δf_u). If a neighbor k is added, the inner sum for every existing neighbor i with i>k gains the term â_{ik}, and a new clamped term appears for i=k. The proof of Theorem 12 (Appendix E.14) claims the maximum increase is Σ_{j∈Neiu} â_{kj} and hence bounded by Δf_u after clamping, but this ignores the â_{ik} contributions to existing i's inner sums. The total change can be as large as d_u·Δf_u (e.g., set â_{i,1}=Δf_u for all i≥2, with u connected to all i and adding neighbor 1, giving a change of (d_u)·Δf_u). Algorithm 9 (QuaTR) has the identical structure and the same flaw. Thus the Laplace noise scale Δf_u/ε2 is too small by a factor of d_u, and the claimed pure ε2-edge LDP in Theorems 5, 9, 12, and 14 does not hold. This is not merely a CLT approximation issue; it is a deterministic sensitivity undercount, independent of β. Note that Algorithm 8 (TriMTR) clamps single matrix entries, so its per-user sensitivity is indeed Δf_u, but its accuracy guarantees still rely on the probabilistic clamping bound.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a Noisy Adjacency Matrix (NAM) framework for subgraph counting under edge local differential privacy. It proves that powers of an unbiased, symmetric, independently perturbed adjacency matrix give unbiased estimates of path counts (Theorem 1), and then derives five algorithms: TriOR (one-round triangle counting), TriTR and TriMTR (two-round triangle counting), QuaTR (two-round quadrangle counting), and 2STAR (2-star counting from a private degree-projection step). The paper claims pure edge-LDP guarantees for the one-round algorithm and (ε0+ε1+ε2)-edge LDP for the two-round algorithms, together with MSE bounds and experimental comparisons. The algebraic unbiasedness result is clean, but the privacy analysis of the second-round randomizers in Algorithms 7 and 9 is based on an incorrect sensitivity computation, and the stated total-MSE bounds omit bias from clamping and GraphProjection.","tokens_in":35588,"tokens_out":15674,"duration_ms":172231,"significance":"The NAM construction is an elegant idea and Theorem 1 is a genuinely useful observation: it provides a modular way to turn unbiased private edge perturbations into unbiased subgraph-count estimates via matrix powers. The one-round TriOR result, with time O(n^2.371866) and MSE O(nd_max^3+n^3), is a plausible contribution. However, the two-round algorithms are the main claimed advances, especially QuaTR as the first quadrangle counter under pure edge-LDP, and their privacy guarantee is not established. The 2STAR theoretical analysis also does not match the implemented estimator. If the sensitivity issue were repaired, the required noise scale and the accuracy bounds would change substantially, so the current theoretical and experimental comparisons do not support the headline claims.","major_comments":[{"comment":"The sensitivity analysis for the second-round randomizer in Algorithm 7 is incorrect. In line 2, sumu = Σ_{i∈Neiu} clamp(Σ_{j∈Neiu,j<i} â_ij, Δf_u). If a neighbor k is added, then for each existing neighbor i with i>k the i-th inner sum gains the term â_ik, and a new clamped term appears for i=k. The proof in E.14 correctly identifies Σ_{j∈Neiu} â_kj as the change in the unclamped sum, but it then incorrectly concludes that clamping each inner sum bounds the total change by Δf_u. Because clamping is 1-Lipschitz, each of the d_u affected inner sums can change by up to Δf_u, so the total change can be as large as d_u·Δf_u, not Δf_u. Consequently, the Laplace noise Lap(Δf_u/ε2) is too small by up to a factor of d_u, and the claimed ε2-edge LDP guarantee in Theorems 5 and 12 does not hold. Algorithm 9 and Theorem 14 have the identical structure and the same flaw. This is a deterministic sensitivity undercount, independent of the CLT approximation in Theorem 11.","section":"§5.2, Algorithm 7; Appendix E.14"},{"comment":"The probabilistic bound that calibrates the clamp range Δf_u is not established. Theorem 11 is stated only as an approximation via the CLT, yet the subsequent accuracy claims treat it as a rigorous β-quantile guarantee. Moreover, the proof asserts that 'After Graph Projection on graph G, for any vertex u ∈ [n], we have ˆdu ≥ du,' which is false: Algorithm 6 sets ˆdu = ⌊α + max{du + Lap(1/ε0), 0}⌋, and if the Laplace noise is sufficiently negative, ˆdu < du and edges are then removed. Thus Δf_u can be too small, and even the approximate probability bound Pr[λu > Δf_u] < β is unsupported. Since the total-MSE bounds in Theorems 12–14 depend on the clamping error being bounded by a β-fraction of data, this is a load-bearing gap in the accuracy analysis.","section":"Appendix E.13, Theorem 11, Algorithm 6"},{"comment":"The claimed total-MSE upper bounds in Theorems 12–14 are not proven. The proofs state that the total MSE has four components—GraphProjection error, clamping error, the variance of the unclamped sum, and the variance of the second-round Laplace noise—but then they bound only the last two components and assert that the first two have 'relatively small impact' without any quantitative estimate. Because clamping introduces bias and GraphProjection removes edges, the clamped estimator is not unbiased, and the displayed O(·) expressions are not valid upper bounds on the total MSE. The relative-error results in Theorem 15 inherit this problem, as they are derived from these MSE bounds.","section":"Theorems 12–14, Appendix E.14–E.16"},{"comment":"The theoretical guarantees for 2STAR do not apply to the algorithm as written. Theorem 16 analyzes the idealized estimator Σ_u[(du+Yu)(du-1+Yu)-2/ε0^2] with Yu ~ Lap(1/ε0), whereas Algorithm 10 uses the GraphProjection output (˜du-α)(˜du-α-1)-2/ε0^2, where ˜du is truncated, rounded, and followed by edge removal when ˜du<du. The paper acknowledges the discrepancy but provides no bias bound. The claims that 2STAR is unbiased and achieves RE=O(1/√#2-star) are therefore unsupported.","section":"§7.2, Theorem 16 vs. Algorithm 10"}],"minor_comments":[{"comment":"The sentence immediately before Theorem 7 reads 'TriTR holds the following guarantees,' but the surrounding algorithm is TriMTR; the label should be corrected.","section":"§5.3"},{"comment":"In the proof of Theorem 9, the name 'QuaMTR' is used; it should be 'QuaTR'.","section":"Appendix E.11"},{"comment":"The third term in the TriMTR relative-error bound is printed as '1/(ε2_1 d_avg^{3/2})'; based on the derivation in E.17 and the analogous term in Eq. (21), this should be '1/(ε1^2 d_avg^{3/2})'.","section":"Eq. (22)"},{"comment":"The caption is confusing: it says the solid and dashed blue lines correspond to 'the variance obtained with the Laplace mechanism and the curve obtained by doubling the variance,' but the red line is labeled as the Laplace variance in the text; the intended comparison between RR and Laplace should be restated clearly.","section":"Figure 2 caption"},{"comment":"The experimental section reports results averaged over 20 runs but provides no error bars or standard deviations. Given that several comparisons (e.g., Figure 8) are visually close, the reported accuracy differences would be more convincing with variance information.","section":"§8.1"}],"recommendation":"reject","confidential_remarks":"The core privacy theorems for the two-round algorithms are invalid because of the deterministic sensitivity undercount in Algorithms 7 and 9. This is not a presentation issue but a fundamental flaw in the main contribution. The NAM matrix-power result is nice, but the two-round privacy and accuracy claims would need to be reworked, and the experimental comparisons would likely change, before this could be considered for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the NAM framework is a genuinely nice idea, and the matrix-power estimators are clean. But the privacy proofs for the two-round algorithms TriTR and QuaTR have a load-bearing flaw: the second-round sensitivity is undercounted by a factor of the user's degree, so the claimed pure epsilon-edge LDP does not follow. TriMTR's privacy argument survives this particular objection, but the MSE bounds throughout omit clamping bias and GraphProjection bias. This is fixable, but it needs a real revision.\n\nWhat's genuinely new and good: the NAM formulation is a unifying way to see subgraph counting under edge-LDP, and Theorem 1 (unbiasedness of matrix powers) is proven correctly. TriOR gives an unbiased, one-round triangle estimate with O(n^2.37) time via fast matrix multiplication, which is a real improvement over cubic enumeration. QuaTR is the first quadrangle-counting algorithm under edge-LDP, and 2STAR is a neat byproduct of GraphProjection. The experiments qualitatively support the theory, even without error bars.\n\nNow the soft spots. I checked the stress-test concern against Algorithm 7 and the proof of Theorem 12, and it holds up. Adding one neighbor k changes the clamped inner sums for every existing neighbor i > k, not just the new term. The total change to sum_u can be roughly d_u times Delta_f_u, not Delta_f_u. The proof in E.14 only counts the new term; it ignores the contributions to existing inner sums. The same issue appears in Algorithm 9 and Theorem 14. This is a deterministic sensitivity miscalculation, independent of the CLT approximation. So the pure epsilon2-edge LDP claim for TriTR and QuaTR is not established. TriMTR clamps individual matrix entries, so its per-user sensitivity is indeed Delta_f_u; its second-round privacy is in better shape. However, the stated MSE bounds in Theorems 12-14 explicitly wave off the bias from clamping and GraphProjection, so those bounds are not actually proven as written. The experiments set beta = 0.01 without tying it to a delta budget, which matters if the authors move to (epsilon, delta)-DP.\n\nWho is this for? People in private graph analytics, especially those using matrix-based estimators or needing quadrangle counts under edge-LDP. The NAM idea deserves attention, and QuaTR is worth having once the privacy proof is fixed.\n\nMy recommendation: send it to a serious referee, but with clear guidance to focus on the second-round sensitivity analysis. The authors need to either derive an exact per-user sensitivity bound or convert the guarantee to (epsilon, delta)-DP with delta tied to beta. The core idea is solid; the privacy proofs are not there yet.","headline":"The NAM matrix-power idea is sound and QuaTR is a real first, but the second-round privacy proofs undercount sensitivity by a factor of d_u, so TriTR and QuaTR don't yet have the claimed edge-LDP guarantee.","tokens_in":36124,"tokens_out":4176,"would_cite":false,"duration_ms":49195,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68P27","05C30","05C85"],"pacs":[],"model":"deepseek-v4-flash","headline":"Powers of a noisy adjacency matrix give unbiased, edge-private subgraph counts, including the first pure edge-LDP quadrangle algorithm.","keywords":["edge local differential privacy","noisy adjacency matrix","subgraph counting","triangle counting","quadrangle counting","2-star counting","randomized response","graph privacy"],"falsifier":"Fix the parameters used in the experiments ($\\varepsilon$, $\\beta=0.01$) on a real social network, generate $\\hat{A}$ through GNAM, and directly measure the change in the clamped sums of Algorithms 7–9 when a single neighbor is added to or removed from a user's list; if the fraction of users whose change exceeds $\\Delta f_u$ is comparable to $\\beta$ or larger, the pure edge-LDP step in Theorems 12–14 fails.","tokens_in":35048,"feed_emoji":"🕸","tokens_out":12759,"duration_ms":130243,"temperature":0.7,"pith_summary":"The paper's central claim is that a graph's privacy-preserving subgraph statistics can be carried by a single object: the Noisy Adjacency Matrix (NAM), a symmetric matrix built from independent, zero-mean, debiased perturbations of each edge under edge-local differential privacy (edge-LDP). Because the diagonal of the cube of the true adjacency matrix counts triangles, the key identity $E[\\mathrm{tr}(\\hat{A}^3)] = \\mathrm{tr}(A^3)$ turns private triangle counting into one matrix-power computation, and similar identities cover quadrangles and 2-stars. On this basis the paper presents five algorithms — TriOR, TriTR, TriMTR, QuaTR, and 2STAR — and claims MSE bounds for each, with TriOR running in $O(n^{2.371866})$ time via fast matrix multiplication and QuaTR being the first quadrangle-counting algorithm under pure edge-LDP. The significance is practical: subgraph counts from social or transactional networks can be obtained without a trusted third party, with download cost reduced to $O(n)$ in the TriMTR variant, and the framework extends to directed and weighted graphs.","feed_headline":"Noisy matrix powers unlock private, unbiased subgraph counts","feed_subtitle":"One debiased edge-private matrix counts triangles, quadrangles, and 2-stars in one or two rounds.","key_machinery":"The Noisy Adjacency Matrix (NAM) is a symmetric $n\\times n$ matrix $\\hat{A}$ with $E[\\hat{A}]=A$, zero diagonal, and mutually independent off-diagonal entries; its key property (Theorem 1) is that the expectation of the diagonal of $\\hat{A}^3$ equals the diagonal of $A^3$, and the off-diagonal entries of $\\hat{A}^2$ are unbiased for $A^2$. The machinery is the vanishing of the cross terms in $(A+X)^3$ under symmetry, independence, and $E[X]=0$, together with the debiasing built into GNAM: each node reports its randomized adjacency list with entries for higher-index nodes blanked so that each edge is uploaded exactly once, and each reported bit is rescaled (e.g., $y = (x(e^{\\varepsilon}+1)-1)/(e^{\\varepsilon}-1)$ under randomized response) so the estimator is unbiased. These two pieces — the power identity and the one-upload debiasing — carry every counting algorithm in the paper.","core_discovery":"The discovery is that the expected value of products of independent zero-mean symmetric edge noises vanishes in exactly the terms that matter: writing $\\hat{A} = A + X$ with $E[X]=0$, $X$ symmetric, independent off-diagonal entries, and zero diagonal, the cross terms — $E[(AX)_{ii}]$, $E[(XAX)_{ii}]$, and so on — all vanish, leaving $E[(\\hat{A}^3)_{ii}] = (A^3)_{ii}$ and, for $i \\neq j$, $E[(\\hat{A}^2)_{ij}] = (A^2)_{ij}$. These identities make $\\mathrm{tr}(\\hat{A}^3)/6$ an unbiased estimator of the number of triangles, make $\\hat{A}^2$ an unbiased estimator of two-step path counts, and consequently allow quadrangle counts by subtracting the single 'return path' term $\\hat{b}_{ij} - 1$; the divisors 6 and 8 correct for the orientation multiplicities of triangles and quadrangles. The construction that generates $\\hat{A}$ (GNAM) lets each node randomize its own adjacency list once, blanks entries above its own index so that every edge is reported by exactly one endpoint, and the collector debiases the reports, which yields $\\varepsilon$-edge LDP. On this basis the paper claims its five algorithms improve on prior one-round and two-round triangle counters in accuracy or communication, and that QuaTR is the first quadrangle counter under pure edge-LDP.","pith_inferences":["The same power identity should generalize: any subgraph count that is a linear combination of entries of $A^k$ can be estimated by the same combination of $\\hat{A}^k$, so longer cycles, paths, and other motifs could inherit the one-round or two-round scheme without new per-motif privacy analyses.","The probabilistic sensitivity bound in Section 6 is the step most likely to need a privacy patch; a rigorous version would either state the guarantee as $(\\varepsilon,\\delta)$-DP with $\\delta$ tied to $\\beta$, or replace the central-limit quantile with a deterministic bound that holds for every input.","Because GNAM is deliberately mechanism-agnostic, the framework can be instantiated with other unbiased DP mechanisms; the variance and privacy trade-offs of, say, a Gaussian or Hadamard-response instantiation follow from the same proof template but are not analyzed in the paper."],"forward_implications":["Triangle counting becomes a one-round matrix-power computation with collector-side time $O(n^{2.371866})$ via fast matrix multiplication, replacing the $O(n^3)$ triplet enumeration of earlier one-round edge-LDP algorithms.","Two-round triangle counting reaches MSE $O(nd_{\\max}^3)$ (TriTR) or $O(nd_{\\max}^3 + n^2 d_{\\max})$ with download cost $O(n)$ (TriMTR), so accuracy and communication can be traded off.","QuaTR is the first quadrangle-counting algorithm under pure edge-LDP, with MSE $O(nd_{\\max}^5 + n^2 d_{\\max}^3)$, a gap left open by prior shuffle-model methods.","A single two-round pipeline yields simultaneous unbiased estimates of triangle, quadrangle, and 2-star counts, with 2STAR reaching relative error $O(1/\\sqrt{\\#2\\text{-star}})$.","Because the NAM proof does not depend on a specific mechanism or on undirectedness, the framework transfers to directed graphs, weighted graphs, and to f-DP or $(\\varepsilon,\\delta)$-DP by swapping the local randomizer."],"supporting_citations":[{"why":"Supplies the definition of $(\\varepsilon,\\delta)$-DP and the Laplace mechanism that all algorithms build on.","marker":"[16]"},{"why":"Provides the sensitivity-calibrated mechanism and composition argument used to prove GNAM's $\\varepsilon$-edge LDP and the two-round bounds.","marker":"[15]"},{"why":"Warner's randomized response is the default local randomizer in GNAM; Proposition 3 gives the debiasing transformation for RR.","marker":"[57]"},{"why":"Defines the prior one-round triangle algorithm RR$\\triangle$ and two-round $2R_\\triangle$ baselines whose complexity, download cost, and accuracy the paper compares against.","marker":"[22]"},{"why":"Supplies the communication-efficient two-round baselines 2R-Large$\\triangle$ and 2R-Small$\\triangle$ and the edge-sampling trade-offs the paper refines.","marker":"[23]"},{"why":"Shuffle-model algorithms Wshuffle$\\triangle$ and Wshuffle$\\square$, the prior state of the art for quadrangle counting that QuaTR aims to surpass.","marker":"[24]"},{"why":"Fast matrix multiplication with exponent 2.371866 that TriOR and TriMTR use to reduce collector-side time complexity.","marker":"[13]"},{"why":"Introduces f-DP and Gaussian DP, the framework the paper invokes to argue NAM is compatible with mechanisms other than RR and Laplace.","marker":"[12]"}],"fun_headline_variants":["Private subgraph counts from one noisy matrix","Edge-LDP subgraph counting via debiased matrix powers","Unbiased triangles, quads, and stars under edge privacy","Noisy adjacency matrix: private subgraph counts","Five algorithms, one matrix: private subgraph counting"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper bets that a user's rounded-up contribution can be bounded by a number computed from a normal approximation; if that number is wrong, the second-round noise is too small to deliver the promised privacy.","fun_headline_variants_meta":{"raw":{"variants":["Private subgraph counts from one noisy matrix","Edge-LDP subgraph counting via debiased matrix powers","Unbiased triangles, quads, and stars under edge privacy","Noisy adjacency matrix: private subgraph counts","Five algorithms, one matrix: private subgraph counting"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0007,"raw_usage":{"total_tokens":3257,"prompt_tokens":1137,"completion_tokens":2120,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":753,"completion_tokens_details":{"reasoning_tokens":2043}},"tokens_in":753,"tokens_out":2120,"duration_ms":14267,"temperature":1.0,"reasoning_tokens":2043,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:01:30.991070+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix the parameters used in the experiments ($\\varepsilon$, $\\beta=0.01$) on a real social network, generate $\\hat{A}$ through GNAM, and directly measure the change in the clamped sums of Algorithms 7–9 when a single neighbor is added to or removed from a user's list; if the fraction of users whose change exceeds $\\Delta f_u$ is comparable to $\\beta$ or larger, the pure edge-LDP step in Theorems 12–14 fails.","supporting_citations":[{"cited_title":"The algorithmic founda- tions of differential privacy","cited_arxiv_id":null,"evidence_quote":"Supplies the definition of $(\\varepsilon,\\delta)$-DP and the Laplace mechanism that all algorithms build on."},{"cited_title":"Calibrating noise to sensitivity in private data anal- ysis","cited_arxiv_id":null,"evidence_quote":"Provides the sensitivity-calibrated mechanism and composition argument used to prove GNAM's $\\varepsilon$-edge LDP and the two-round bounds."},{"cited_title":"Using randomized response for differential privacy preserving data col- lection","cited_arxiv_id":null,"evidence_quote":"Warner's randomized response is the default local randomizer in GNAM; Proposition 3 gives the debiasing transformation for RR."},{"cited_title":"Lo- cally differentially private analysis of graph statistics","cited_arxiv_id":null,"evidence_quote":"Defines the prior one-round triangle algorithm RR$\\triangle$ and two-round $2R_\\triangle$ baselines whose complexity, download cost, and accuracy the paper compares against."},{"cited_title":"983–1000","cited_arxiv_id":null,"evidence_quote":"Supplies the communication-efficient two-round baselines 2R-Large$\\triangle$ and 2R-Small$\\triangle$ and the edge-sampling trade-offs the paper refines."},{"cited_title":"{Communication-Efficient} triangle counting under lo- cal differential privacy","cited_arxiv_id":null,"evidence_quote":"Shuffle-model algorithms Wshuffle$\\triangle$ and Wshuffle$\\square$, the prior state of the art for quadrangle counting that QuaTR aims to surpass."},{"cited_title":"Faster matrix mul- tiplication via asymmetric hashing","cited_arxiv_id":null,"evidence_quote":"Fast matrix multiplication with exponent 2.371866 that TriOR and TriMTR use to reduce collector-side time complexity."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces f-DP and Gaussian DP, the framework the paper invokes to argue NAM is compatible with mechanisms other than RR and Laplace."}],"review_version":1}