F4 — xmin sensitivity sliding-window scan (W7-D)¶

Date: 2026-05-15 Reviewer concern (W5-A §3.8 / §4.1): Standard powerlaw.Fit with KS-distance minimization is correct in principle but has documented edge-case behavior on small n and on distributions with curvature near the candidate xmin regime. The paper does not report sensitivity of alpha to perturbations in xmin. Recommended: for each phase, add a supplementary figure showing alpha(xmin) on a sliding window across the support, with the chosen KS-minimum xmin marked.

Implementation¶

paper/figures/methodology/generate_F4.py — sweeps xmin in log-space across [baseline × 0.5, baseline × 2.0] in 20 steps. At each xmin, fits a fixed-xmin Clauset MLE alpha on the tail subset and records the point estimate.

Baseline xmin per system is taken from the published Clauset fit in the phase's *_results.json (e.g. powerlaw_fit.xmin or clauset_fit.xmin). Earthquake uses energy-domain median (Aki MLE doesn't expose a Clauset xmin in the energy units).

Systems swept (8 of 13)¶

Drift classification (heuristic, applied per F4 script): - range < 0.2 -> robust - 0.2 <= range < 0.5 -> mild_drift - range >= 0.5 -> substantial_drift

Five-system not covered¶

DeFi compound, DeFi maker, mouse cortex (Phase 4), and power-grid (Phase 7) are deferred to a follow-up round: - Compound / maker use the same Aave-style debt-to-cover field and would require the same fitting fix as defi_aave. - Mouse cortex (Phase 4) doesn't have a single canonical alpha — it has alpha_T and tau across 16-fold bin sweeps, requiring a different sensitivity framing. - Power grid (Phase 7) is n=123 literature meta, baseline xmin not directly applicable.

Findings interpretation¶

Robust (3 of 8): wildfire, solar, bank_failure¶

Alpha varies by < 0.2 across a 4×-multiplicative xmin range. These three systems' Clauset MLE alpha is well-determined and not artifactually dependent on the xmin choice. The published verdicts are defensible.

Mild drift (2 of 8): earthquake, wikipedia¶

Alpha varies by 0.27-0.28 across the sweep. For earthquake this likely reflects the b-value vs Clauset-energy-fit discrepancy (paper notes the 3-sigma divergence between alpha = 1 + b/1.5 ≈ 1.72 and Clauset alpha ≈ 1.79). For wikipedia (which the scholar review already flagged as weak — Vuong-LN beats PL by 6.3 sigma, top-1000 truncation), the drift is a third independent reason to demote the verdict label from "CONFIRMED" to "consistent within caveats."

Substantial drift (2 of 8): stockmarket, github_stars¶

Alpha varies by > 0.7 across the sweep. This is the largest new finding from F4 — both systems where the Clauset fit climbs to the alpha=3.0 power-law ceiling at the upper xmin range (i.e., the upper-tail-only sub-fit returns a canonical inverse-cubic). This is not necessarily a defect: it's consistent with the scholar-reviewer (§4.1) observation that PL/LN coexistence in finite samples (Mitzenmacher 2004) shows up as xmin-sensitivity. Both systems have Vuong-LN p > 0.1, i.e., the Vuong test could not pick a winner.

Action: in the C1 v0.3 manuscript, alpha estimates for stockmarket and github_stars should be reported with their drift range (e.g., "alpha = 3.00 ± 0.04 (Clauset) with xmin-sensitivity range alpha in [2.29, 3.00] across 0.5x-2x xmin sweep"). This is the honest reading.

defi_aave fitting failure¶

DeFi Aave V2 liquidation debt_to_cover_raw (wei-denominated integers spanning many decades) trips powerlaw.Fit's edge-case handling — alpha clamps to ~1.0 with warnings about discrete=False on integer-valued data. The published Clauset fit (alpha = 1.68 in multiprotocol_results.json) uses a different code path (multiprotocol_results.json reports per-protocol fits not the raw input). For F4, this means the defi_aave row in the figure should be read as N/A. A follow-up F4-bis with discrete=True and proper xmax handling would close this gap.

Output¶

• paper/figures/methodology/F4_xmin_sensitivity.pdf — 8-panel grid
• paper/figures/methodology/F4_xmin_sensitivity.png — same in PNG
• paper/figures/methodology/F4_xmin_sensitivity_data.json — raw alpha(xmin) data per system

Recommended manuscript edit¶

Add as supplementary figure to C1 v0.3 with a paragraph:

Figure S4. Clauset alpha estimator sensitivity to xmin choice across ±2× of the KS-minimum xmin. Three of the 8 phases tested (wildfire, solar, bank failure) show robust alpha variation < 0.2 across the sweep. Two phases (earthquake energy, Wikipedia) show mild drift (0.27-0.28), consistent with their published b-vs-Clauset discrepancy and top-1000 truncation caveats respectively. Two phases (S&P 500 returns, GitHub stars) show substantial drift > 0.7 with alpha climbing to the inverse-cubic ceiling at large xmin — consistent with Mitzenmacher (2004) PL/LN coexistence in finite samples and with the Vuong LR inconclusive verdicts already reported. We add the drift range to Table 1 columns.

References¶

• Clauset A, Shalizi CR, Newman MEJ (2009). "Power-law distributions in empirical data." SIAM Rev. 51, 661-703.
• Deluca A, Corral A (2013). "Fitting and goodness-of-fit test of non- truncated and truncated power-law distributions." Acta Geophys. 61, 1351.
• Voitalov I et al. (2019). "Scale-free networks well done." Phys. Rev. Res. 1, 033034.
• Mitzenmacher M (2004). "A brief history of generative models for power-law and lognormal distributions." Internet Math. 1, 226-251.