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

Abstract

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 constellations 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 grows with the number of planes, from about 1.7x on small constellations such as SES O3b mPOWER (P = 8) to 10.2x on Starlink Shell 1 (P = 72), where dispersed loss of just 4.2% of satellites produces the same fragmentation as random loss of 43%. 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, which covers every operational constellation. It is presented explicitly as a calibrated finite-range approximation rather than a derived asymptotic law. Applied without per-constellation tuning to the eight validation constellations (Starlink Shells 1 and 4, OneWeb, SES O3b mPOWER, Telesat Lightspeed, Amazon Kuiper, Globalstar-2, and Guowang), the recommended engineering form agrees with independent Newman-Ziff simulation to under 1% 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 3-9% earlier than Walker-delta at identical (P, S) and are explicitly out of scope. The closed form evaluates in microseconds and supports design-space sweeps and scenario analyses that direct Monte Carlo makes impractical. Files in this record: paper.pdf - the manuscript (16 pages, 5 figures, 9 tables)latex_source.zip - LaTeX source and figure filesreproducibility.zip - all simulation code and data, with a README mapping every figure and table to the script that generates it. All results are deterministic from random seed 42.