{"id":"846ef055-a1c4-481e-9ac4-af5da9e2635a","arxiv_id":"2508.00010","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A stochastic geometry framework that generates paired planar and spherical point processes, quantifies their relative error, and gives an optimal planar altitude for approximating non-terrestrial networks.","lead":"This paper finds when a flat-Earth model can replace a curved-Earth model in satellite and high-altitude platform network analysis. It pairs the two random point processes, measures the error between them, and derives the best altitude to place the flat model.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The error metric couples the planar and spherical PPs through shared random draws and tunes the planar altitude h_opt below the actual deployment altitude, so the reported relative errors are best-case rather than the true planar-versus-spherical modeling gap.","rationale":"The reader's weakest assumption already names the coupling and tuning issue. My independent calculation of Proposition 1 for the paper's own case studies makes the stakes concrete: h_opt - R⊕ is approximately 13.7 km for HAPs and 389 km for LEO, versus nominal altitudes of 20 km and 550 km, respectively. Thus the planar model whose error is reported is not the same network; it is an altitude-tuned surrogate. This matters because the abstract and Section V present the numbers as a decision rule for 'planar or spherical.' The topology metric (t1) is particularly sensitive to the shared-draw coupling, since Earth-mover distance between identical-rank paired samples is far smaller than between independent samples. The paper's homogeneity proof and asymptotic-similarity result remain useful, and the optimal-altitude expression is a legitimate design tool if framed as such. The missing baseline—nominal altitude and independent realizations—is exactly what a conditional acceptance should require. My verdict therefore remains conditional on that validation.","tokens_in":14413,"tokens_out":14348,"duration_ms":158787,"concrete_test":"Rerun Algorithm 2 for all six metrics with (i) independent uniform draws for the planar and spherical PPs and (ii) the planar altitude fixed at h_p = R_s (nominal altitude), for HAP 20 km and LEO 550 km and across the Fig. 4/5 parameter grid. Compare the resulting relative-error curves to Figs. 3, 6, and 7; if the errors exceed the 0.1% threshold or change the HAP/LEO modeling recommendations, the tuned-coupling numbers are an optimistic upper bound and the modeling-choice conclusions must be rederived.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The mathematical core—homogeneity of Algorithm 1's PPs and their asymptotic similarity (Theorems 1–2)—is internally plausible. The load-bearing weakness is external validity of Algorithm 2's error measure. Step (6) of Algorithm 1 draws the same u(n), v(n) for both PPs, so each planar point is generated from the same uniforms as its spherical partner. Consequently, topological metrics, especially (t1) Earth-mover distance, measure a transport cost between paired configurations that are already nearly matched; independent realizations would yield substantially larger errors. In addition, the outer loop of Algorithm 2 chooses h_p to minimize the error for each metric, and Sec. V uses this optimized altitude rather than the actual deployment altitude h_s = R_s - R⊕. For LEO with h_s=550 km, h_opt is about 389 km altitude; for HAP with 20 km, about 13.7 km. Thus the reported relative errors in Figs. 3–7 and the 0.1% thresholds are minima over an extra tuning parameter, not the error a practitioner incurs by replacing the spherical model with a planar model at the nominal altitude. The central recommendation therefore rests on a best-case comparison.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies whether planar or spherical stochastic geometry models should be used for non-terrestrial networks (NTNs). It proposes Algorithm 1, which generates paired planar and spherical binomial point processes from the same uniform random draws, and proves that both are homogeneous and asymptotically similar as the sphere radius grows with the cap area fixed. It then defines six similarity metrics (three topology-based, three system-level), develops Algorithm 2 to estimate the relative error between the two models, and derives a closed-form optimal planar altitude for the squared-distance metric. Numerical results for HAP and LEO altitudes and for different beamwidths and deployment areas lead to recommended error thresholds and modeling choices.","tokens_in":14712,"tokens_out":12041,"duration_ms":122229,"significance":"The mathematical core of the paper—the homogeneity of Algorithm 1's point processes and their asymptotic similarity—is plausible and, apart from typos in the appendix, correctly derived. The explicit algorithm and the closed-form optimal altitude in Proposition 1 are useful and reproducible, and the paper would fill a real gap by giving a quantitative alternative to the empirical 20-km rule for choosing between planar and spherical models. However, the central quantitative claims depend on two protocol choices that make the reported relative errors best-case: the planar and spherical processes are generated from the same random draws, and the planar altitude is tuned to minimize the error rather than being set to the nominal deployment altitude. These issues must be addressed before the practical recommendations can be accepted.","major_comments":[{"comment":"The same random variables u(n), v(n) are used to generate both the spherical and the planar point processes. For metric (t1), the Earth-mover distance is therefore computed between configurations that are paired point-by-point by construction; this is not the transport distance between two independent realizations of the two models and will systematically underestimate the modeling gap. For the other metrics the coupling acts as a variance-reduction device, but it means that the relative errors in Figs. 3–7 do not reflect the error a practitioner would see when drawing an independent planar or spherical configuration. Please repeat the comparison with independently generated planar and spherical processes from the same parameters, or justify explicitly why the paired comparison is the correct definition of the model gap, and quantify the difference between the two protocols.","section":"Algorithm 1, steps 3–5; Algorithm 2, step 8"},{"comment":"The outer loop of Algorithm 2 selects hp to minimize the relative error, and the numerical results in Sec. V use this optimized altitude rather than the nominal deployment altitude hs = Rs − R⊕. For the LEO case (hs = 550 km), the optimized altitude hopt − R⊕ is approximately 389 km; for the HAP case (hs = 20 km), it is roughly 14 km. A planar model placed at the actual deployment altitude would incur a larger error than the figures report, so the 0.1% thresholds and the conclusion that planar modeling is often sufficient for HAPs are best-case statements. Please report the relative error at hp = Rs − R⊕ alongside the optimized value, and discuss whether hopt is meant as a physical altitude or as a tuned proxy parameter.","section":"Algorithm 2, lines 2–13; Sec. V"},{"comment":"The case-study conclusions (HAP versus LEO, beamwidth thresholds) are based on relative errors at the metric-specific optimal altitudes found by Algorithm 2, not on a single recommended altitude. Proposition 1 is derived only for metric (t2), and its generalization to other metrics is validated in Fig. 2 only for the LoS cap θmax = arccos(R⊕/Rs). The paper should either derive or empirically verify the optimal-altitude behavior for the beam-angle and fixed-area scenarios, or present the sensitivity of the conclusions to the choice of altitude within the admissible range.","section":"Sec. V.C; Fig. 2"},{"comment":"No error bars or confidence intervals are provided for any Monte Carlo curve, and the values of Nin, Nout, and the number of channel realizations for the system-level metrics are not stated. Since the paper makes quantitative recommendations based on small relative errors (e.g., the 0.1% threshold), the statistical uncertainty of these estimates must be quantified before the thresholds can be considered reliable.","section":"Sec. V, Figs. 3–7"}],"minor_comments":[{"comment":"The text 'When ρmax = Rs cos θmax' should read 'ρmax = Rs sin θmax'; otherwise the subsequent algebra does not follow. Also, Eq. (13) is missing a factor of 4 in the second term, although the term vanishes in the limit and does not affect the conclusion.","section":"Appendix B, near Eqs. (13)–(14)"},{"comment":"The spherical-cap area should be A = 2π R_s^2 (1 − cos θmax), not 2πR_s(1 − cos θmax); Eq. (6) is consistent with the corrected formula, so this appears to be a typographical error.","section":"Appendix E, Eq. (26)"},{"comment":"The input list of Algorithm 2 omits ρmax; the paper should state explicitly that ρmax = Rs sin θmax is used, as required by Theorem 2.","section":"Algorithm 2"},{"comment":"The sentence 'the y-axis is in units of %, for example, 100% = 0.01' is confusing; it should presumably read '1% = 0.01'.","section":"Sec. V.A"},{"comment":"The figure uses θmax = arccos(R⊕/Rs), but this is not stated in the caption; clarify which cap geometry applies to each figure.","section":"Fig. 2"},{"comment":"The reference for Slivnyak's theorem [26] is a general probability text; a standard point-process reference would be more appropriate.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The core mathematical lemmas are sound, but the paper's quantitative claims rest on a comparison protocol that is best-case. I would ask the editor to require the independent-realization and nominal-altitude experiments before publication. There is also a question of scope fit: the paper is more a methodology/measurement paper than a traditional stochastic-geometry analysis, which is acceptable provided the comparison is made rigorous."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I've read through Wang et al.'s NTN modeling paper. It's a useful, careful piece of work that addresses a real problem: when can you replace a spherical SG model with a planar one without losing accuracy? The authors are the first, to my knowledge, to propose a paired generation algorithm that produces both PPs from the same random draws, and they prove both are homogeneous and asymptotically similar. The distance CDFs and the optimal altitude formula in Proposition 1 are derived correctly, and the relative-error framework across six metrics is a sensible way to make the \"planar vs. spherical\" choice quantitative rather than reliant on the usual hand-wavy 20 km rule.\n\nThat said, the paper has a significant soft spot that the authors don't address. The relative error reported in Figs. 3–7 is computed after optimizing the planar altitude hp (via Algorithm 2's outer loop), and the planar and spherical point sets are generated from the same uniforms. This makes the comparison a best-case scenario, not the actual gap a practitioner faces when using a planar model at a nominal altitude (e.g., 550 km for LEO). The shared random draws particularly reduce the Earth-mover distance, and tuning hp removes the main error that would come from just placing the planar model at the orbital altitude. The 0.1% thresholds are therefore optimistic lower bounds on the real modeling error. I'd like to see the same metrics computed with independent planar and spherical BPPs and with hp set to the actual deployment altitude.\n\nOther minor issues: no error bars on the Monte Carlo curves, no code or data, and a typo in Appendix B (ρmax = Rs cos θmax should be sin θmax). The homogeneity proofs and the optimal altitude result are solid, so the math is not the problem.\n\nOverall, this is a good methods paper that deserves a serious referee. The central recommendation—that you can use planar modeling for HAPs and for LEO only with narrow beams—might be qualitatively correct, but the quantitative thresholds need validation against independent realizations. I'd suggest the authors add that validation before publication, but I'd send it to review rather than desk reject.","headline":"A useful quantitative framework for choosing planar vs. spherical SG models, but the reported error numbers are best-case because the planar altitude is optimized and the two point processes share random draws.","tokens_in":15182,"tokens_out":3433,"would_cite":true,"duration_ms":37261,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60D05","60G55"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper gives a quantitative rule for choosing between flat and curved network models, built on a paired point-process construction and an optimal altitude formula.","keywords":["non-terrestrial networks","stochastic geometry","binomial point process","planar approximation","spherical model","relative error","optimal altitude","coverage probability"],"falsifier":"For a fixed LEO geometry with $R_s = R_\\oplus + 550$ km and $\\theta_{\\max} = \\arccos(R_\\oplus/R_s)$, generate independent spherical and planar binomial point processes of equal intensity, not the coupled pair from Algorithm 1, and compute the relative error of the coverage probability at the paper's optimal altitude $h_{\\mathrm{opt}}$. If the uncoupled error exceeds the paper's reported value or the 0.1 percent threshold by more than a small factor, the coupling and the altitude formula understate the true modeling gap.","tokens_in":14223,"feed_emoji":"🛰️","tokens_out":6024,"duration_ms":59294,"temperature":0.7,"pith_summary":"The paper takes sides in a long-running modeling debate: when can a flat (planar) stochastic-geometry model stand in for a curved (spherical) one in non-terrestrial networks? It answers with a quantitative relative-error framework, pairing a spherical and a planar binomial point process that share identical random draws, so both processes are homogeneous and asymptotically identical as Earth's radius grows. It then defines six metrics, three topological and three system-level, for measuring the relative error, and derives a closed-form planar altitude that minimizes that error. The upshot is a practical guideline: for a given altitude and beamwidth, the heat maps show whether planar modeling stays under a chosen error threshold. The paper demonstrates the rule on high-altitude platforms at 20 km (planar modeling suffices) and low-Earth-orbit satellites at 550 km (spherical modeling is needed for wide beams).","feed_headline":"One formula picks the altitude where flat models match round satellites","feed_subtitle":"Quantifies the flat-vs-round gap so engineers know when planar network models are safe.","key_machinery":"The load-bearing object is the paired-generation scheme of Algorithm 1: each point is a single uniform draw $(u,v)$, simultaneously realized as a spherical-cap location and a planar-disk location. The spherical point receives polar angle $\\theta_s = \\arccos(1 - u(1-\\cos\\theta_{\\max}))$ and azimuth $2\\pi v$, while the planar point receives radius $\\rho_p = \\sqrt{u}\\,\\rho_{\\max}$ and the same azimuth. This coupling is what guarantees homogeneity (Theorem 1), asymptotic similarity (Theorem 2), and a deterministic, non-random relative error between the two models. The second mechanism is the relative-error estimator (Algorithm 2), which sweeps the planar altitude $h_p$ and records the minimum relative error and the altitude $h_{\\mathrm{opt}}$; Proposition 1 supplies $h_{\\mathrm{opt}}$ in closed form by equating the expected $\\mathrm{(t2)}$ metrics in the limit $N_s \\to \\infty$.","core_discovery":"On the paper's own terms, the central discovery is that a spherical binomial point process and a planar one can be constructed from the same random draws so that both are homogeneous and, as the Earth's radius diverges, they coincide point for point. This construction (Algorithm 1) makes the relative error between the two models a deterministic function of altitude, region, and metric rather than a Monte Carlo artifact. The paper then proves that for one of the six metrics, the discrete Wasserstein distance to the typical user, the planar altitude that exactly zeroes the expected relative error is $h_{\\mathrm{opt}} = R_\\oplus + \\sqrt{R_\\oplus^2 - \\tfrac{1}{2}\\rho_{\\max}^2 - (1+\\cos\\theta_{\\max}) R_s R_\\oplus + R_s^2}$, with $R_s = R_\\oplus + h_s$, and numerically shows the same altitude stays near-optimal, within about 8.5 percent, for the other five metrics. The consequence is a quantitative, tunable threshold: below a chosen relative error, say 0.1 percent, planar modeling is recommended; above it, spherical modeling is necessary.","pith_inferences":["The coupled-sampling trick is a form of common random numbers; on independently generated planar and spherical deployments the same error metric would likely report larger gaps, so the recommended thresholds should be treated as optimistic until validated on uncoupled processes.","The $h_{\\mathrm{opt}}$ formula can be read as the tangent-plane altitude that best compensates Earth's curvature over a given spherical cap; it could be reused for other spherical-cap planar projections, for example in radar coverage or atmospheric-science modeling.","The paper's future-work suggestion of weighting single-layer errors points to a natural multi-layer extension: choose $h_{\\mathrm{opt}}$ per layer independently and combine the relative errors in a weighted sum, giving a decomposition of total modeling error into per-layer contributions.","A metric-independent bound on the relative error might be derivable from the distance CDFs in Lemma 1 and Lemma 2, since both CDFs are explicit functions of altitude and cap size; the paper does not attempt this."],"forward_implications":["With $h_{\\mathrm{opt}}$ from Proposition 1, the outer search loop of Algorithm 2 can be skipped, cutting the complexity of relative-error estimation from $O(N_{\\mathrm{in}} N_{\\mathrm{out}})$ to $O(N_{\\mathrm{in}})$.","For high-altitude platforms at 20 km, planar modeling keeps the relative error under 0.1 percent even for a receiver main lobe as wide as $\\psi = \\pi/2$; for LEO satellites at 550 km, spherical modeling becomes necessary once the main lobe exceeds about $\\pi/12$.","Because the six metrics can disagree, the thresholds are metric-specific: an acceptable planar error for average SINR does not automatically imply an acceptable error for coverage probability or achievable rate.","Holding the deployment area fixed, the relative error peaks between roughly 100 and 200 km altitude, so the planar approximation is not simply monotone in altitude.","The relative error grows as the receive beamwidens, so directional antennas at the user side are a key enabler of planar approximation at higher altitudes."],"supporting_citations":[{"why":"Maps spherical HAP coordinates to cylindrical coordinates, an earlier planar mapping that the paper argues is neither homogeneous nor asymptotically similar, motivating Algorithm 1.","marker":"[7]"},{"why":"Projects spherical ships onto a tangent plane with underestimated distances; the paper cites this as a technical flaw that the new generation algorithm avoids.","marker":"[17]"},{"why":"Uses Wasserstein distance for error estimation in LEO models; the paper adapts this as metric (t2) and bases Proposition 1 on it.","marker":"[23]"},{"why":"Uses contact distance for error estimation; the paper adopts it as metric (t3).","marker":"[22]"},{"why":"Supplies the aerial-to-ground channel model used in the numerical evaluations for altitudes up to 1000 km.","marker":"[8]"},{"why":"Supplies the space-to-ground channel model used in the numerical evaluations for altitudes of 10,000 km and above.","marker":"[30]"},{"why":"Slivnyak's theorem is invoked to justify rotating coordinates so the typical user sits at $(R_\\oplus,0,0)$ without loss of generality.","marker":"[26]"},{"why":"Proposes an area-preserving mapping from planar to spherical point processes, a related mapping strategy the paper contrasts with its own.","marker":"[25]"}],"fun_headline_variants":["Formula finds altitude where flat Earth models beat round ones","Quantifying when planar models suffice for satellite networks","Relative error picks the altitude for safe planar NTN modeling","Optimal altitude derived for flat-vs-round network models"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The reported error thresholds assume that the gap between planar and spherical models is fairly measured by comparing two point processes that share the same random draws and that use the tuned optimal planar altitude, rather than independently generated processes or a planar process placed at the actual deployment altitude.","fun_headline_variants_meta":{"raw":{"variants":["Formula finds altitude where flat Earth models beat round ones","Quantifying when planar models suffice for satellite networks","Relative error picks the altitude for safe planar NTN modeling","Optimal altitude derived for flat-vs-round network models"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0007,"raw_usage":{"total_tokens":3186,"prompt_tokens":993,"completion_tokens":2193,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":609,"completion_tokens_details":{"reasoning_tokens":2129}},"tokens_in":609,"tokens_out":2193,"duration_ms":16119,"temperature":1.0,"reasoning_tokens":2129,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:36:40.245654+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a fixed LEO geometry with $R_s = R_\\oplus + 550$ km and $\\theta_{\\max} = \\arccos(R_\\oplus/R_s)$, generate independent spherical and planar binomial point processes of equal intensity, not the coupled pair from Algorithm 1, and compute the relative error of the coverage probability at the paper's optimal altitude $h_{\\mathrm{opt}}$. If the uncoupled error exceeds the paper's reported value or the 0.1 percent threshold by more than a small factor, the coupling and the altitude formula understate the true modeling gap.","supporting_citations":[{"cited_title":"Spectrum sharing for high altitude platform networks,","cited_arxiv_id":null,"evidence_quote":"Maps spherical HAP coordinates to cylindrical coordinates, an earlier planar mapping that the paper argues is neither homogeneous nor asymptotically similar, motivating Algorithm 1."},{"cited_title":"Performance analysis of end-to-end LEO satellite-aided shore-to-ship communications: A stochastic geometry approach,","cited_arxiv_id":null,"evidence_quote":"Projects spherical ships onto a tangent plane with underestimated distances; the paper cites this as a technical flaw that the new generation algorithm avoids."},{"cited_title":"Evaluating the accuracy of stochastic geometry based models for LEO satellite networks analysis,","cited_arxiv_id":null,"evidence_quote":"Uses Wasserstein distance for error estimation in LEO models; the paper adapts this as metric (t2) and bases Proposition 1 on it."},{"cited_title":"Modeling and Analysis of GEO Satellite Networks","cited_arxiv_id":"2312.15924","evidence_quote":"Uses contact distance for error estimation; the paper adopts it as metric (t3)."},{"cited_title":"Coverage and rate analysis for vertical heterogeneous networks (VHetNets),","cited_arxiv_id":null,"evidence_quote":"Supplies the aerial-to-ground channel model used in the numerical evaluations for altitudes up to 1000 km."},{"cited_title":"Stochastic geometry-based analysis of LEO satellite communica- tion systems,","cited_arxiv_id":null,"evidence_quote":"Supplies the space-to-ground channel model used in the numerical evaluations for altitudes of 10,000 km and above."},{"cited_title":"Feller, An introduction to probability theory and its applications, Volume 2","cited_arxiv_id":null,"evidence_quote":"Slivnyak's theorem is invoked to justify rotating coordinates so the typical user sits at $(R_\\oplus,0,0)$ without loss of generality."},{"cited_title":"Meta distribution of the SIR in a narrow-beam LEO uplink,","cited_arxiv_id":null,"evidence_quote":"Proposes an area-preserving mapping from planar to spherical point processes, a related mapping strategy the paper contrasts with its own."}],"review_version":1}