Percolation Thresholds for Walker-Delta Inter-Satellite Link Networks: A Closed-Form Approximation

Version 2 (28 October 2026) corrects version 1. The Monte Carlo threshold estimator used in version 1 returned the first upward crossing of C_max/k_surv with 1/2 rather than the last, which admitted spurious crossings at low occupancy and biased the reported thresholds upward on small lattices. All Newman-Ziff data have been regenerated with the corrected estimator and every fit and validation figure recomputed. The topological three-plane result is obtained by deterministic computation, was independently re-verified across all 4,427 two-plane subsets, and is unchanged; the headline amplification for Starlink Shell 1 moves from 10.2x to 10.1x. The fitted amplitudes change substantially (B roughly halves) and the held-out error improves to 0.13%. The eight constellation geometries are now described explicitly as synthetic benchmarks rather than validated models of the named systems. The defect, its magnitude and its scope are documented in CORRECTIONS.md inside reproducibility.zip.

This paper develops a closed-form engineering approximation for the satellite-failure fraction at which a Walker-delta mega-constellation's inter-satellite link (ISL) graph fragments, together with a topological vulnerability of the architecture that, to our knowledge, has not been documented before. Across the eight Walker-delta configurations in the validation set (P = 8 to P = 72 orbital planes), fragmentation under a dispersed-plane attack pattern occurs at exactly three orbital planes removed. The reason is structural: three evenly-spaced whole-plane removals are the minimum set that drives the largest surviving component below half of the survivors. The amplification factor over uniform random failure scales linearly with the number of planes, from about 1.5x on small configurations such as SES O3b mPOWER (P = 8) to 10.1x on Starlink Shell 1 (P = 72), where dispersed loss of just 4.2% of satellites produces the same fragmentation as random loss of 42.3%. A design process simulating only random failure does not see this, and the finding has direct implications for adversarial threat modeling.

The formula is a two-term finite-size-scaling surrogate, with exponents motivated by two-dimensional site-percolation universality and amplitudes fitted over the simulated range L = 12 to 600. It is presented explicitly as a calibrated finite-range approximation rather than a derived asymptotic law. Applied without per-constellation tuning to the eight held-out configurations, the recommended engineering form agrees with independent Newman-Ziff simulation to 0.13% mean error. A symmetry-constrained four-parameter variant is presented alongside. Three documented below-threshold operational events of the past fifteen years (the 2022 Starlink cluster reentries, the May 2024 G5 storm Starlink loss, and the 2009 Iridium-Cosmos collision) are correctly classified as non-fragmenting. Walker-star polar shells, whose seam-disabled topology is an idealized cylinder rather than a torus, fragment 5-9% earlier than Walker-delta at identical (P, S) and are explicitly out of scope.

Files in this record: paper.pdf - the manuscript (18 pages, 5 figures, 9 tables). latex_source.zip - LaTeX source and figure files. reproducibility.zip - all simulation code and data, the corrected estimator, an independent re-verification of the three-plane result, and CORRECTIONS.md documenting what changed from version 1. Deterministic from random seed 42.