F-Rosette Framework for LEO Networks

• F-Rosette is a scalable, fractal-based framework that leverages recursive geometry to maintain full-Earth coverage with deterministic, time-invariant connectivity.
• The framework employs a hierarchical addressing scheme mapped to IPv6, reducing IP churn and enabling efficient local routing even under high satellite mobility.
• Empirical results demonstrate low latency, optimal routing paths, and resource efficiency, confirming F-Rosette’s effectiveness for reliable space-ground communication.

The F-Rosette framework is a stable, scalable space-ground network structure designed for low Earth orbit (LEO) satellite mega-constellations. By combining recursive fractal geometry with a time-invariant hierarchical addressing and routing scheme, F-Rosette provably mitigates the instability, frequent IP address changes, and routing re-convergence endemic to conventional IP-based LEO networks. The framework achieves deterministic, globally time-invariant connectivity and enables efficient, local routing in the presence of high satellite mobility and dynamic many-to-many space-ground mappings (Li et al., 2021).

1. Fractal Recursion over Rosette Constellation

F-Rosette generalizes the classic single-layer "Rosette" constellation [Ballard ’80], whose $N$ satellites cover Earth via circular, Earth-repeat orbits. In its base case $F\text{-Rosette}_0 = (N,m)$, each satellite $S_i$ (with inclination $\beta$ and right-ascension $\alpha_i = \tfrac{2\pi i}{N}$) ensures full-Earth coverage, with minimum satellite count determined by: $$\sec R = \sqrt{3}\,\tan\left(\tfrac{\pi}{6}\,\tfrac{N}{N-2}\right)$$ where $R$ is determined by altitude $H$ and minimum elevation $\phi$.

F-Rosette introduces fractal layering: for $k \geq 1$,

$F\text{-Rosette}_0 = (N,m)$0

This recursive procedure produces $F\text{-Rosette}_0 = (N,m)$1 satellites after $F\text{-Rosette}_0 = (N,m)$2 fractal layers, with each satellite's degree scaling as $F\text{-Rosette}_0 = (N,m)$3 and the global satellite graph remaining strictly time-invariant. All layers retain the "ground-track repeat" property, guaranteeing persistent, uniform Earth coverage at every resolution. For example, with $F\text{-Rosette}_0 = (N,m)$4, $F\text{-Rosette}_0 = (N,m)$5, $F\text{-Rosette}_0 = (N,m)$6, $F\text{-Rosette}_0 = (N,m)$7-Rosette$F\text{-Rosette}_0 = (N,m)$8 yields a 512-satellite constellation maintaining the original Rosette's coverage and symmetry.

2. Hierarchical, Time-Invariant Network Addressing

The framework's deterministic and unchanging topology enables a natural, hierarchical addressing scheme. Each satellite is assigned a base-$F\text{-Rosette}_0 = (N,m)$9 $S_i$0-digit address: $S_i$1 where each digit $S_i$2 denotes membership in the corresponding fractal layer.

For ground terminals, the Earth's surface is partitioned into a recursive hierarchy of static “cells” tiled by satellite ground tracks. At layer $S_i$3, each cell is subdivided into $S_i$4 subcells (Theorem 3). Cells and their encapsulating satellites are referenced by similarly structured codes: $S_i$5 Address-to-geocoordinate mapping leverages “cell tables” based on precomputed geometry, supporting efficient $S_i$6 reconstruction.

This static, hierarchical coding maps directly into IPv6 prefix space, ensuring address stability: addresses only change if a user physically moves across a static cell boundary, which occurs rarely (on the order of hours). This eliminates the frequent IP reallocation seen in legacy LEO architectures.

3. Geographical-to-Topological Routing

F-Rosette implements a geographical-to-topological routing embedding that requires no global re-convergence and no BGP/OSPF. Forwarding toward satellite $S_i$7 proceeds digit-by-digit: at each fractal level, the routing mechanism corrects the relevant digit with a clockwise or counter-clockwise step, always choosing the minimal direction (Theorem 4). This operation is provably local ($S_i$8 using arithmetic and prefix matching), with routing tables per satellite bounded by $S_i$9 (Theorem 5).

For inter-ground traffic, source and destination cell codes are mapped to covering satellites, and routing "lifts" to the satellite graph using analogous digit-wise correction, then delivers directly when the destination cell is covered (Algorithm 6). No global messaging or topology dissemination is required; shortest-hop paths are guaranteed (Theorem 4), with hop stretch

$\beta$0

Redundancy is embedded: there are always $\beta$1 node-disjoint shortest paths connecting any satellite pair (Theorem 6).

4. Theoretical Guarantees

F-Rosette's core theorems formalize its performance, connectivity, and scalability properties:

• Theorem 1 (Full-Earth Coverage): $\beta$2-Rosette$\beta$3 with $\beta$4 satellites achieves guaranteed full-Earth coverage if satellite altitude $\beta$5 exceeds

$\beta$6

• Theorem 2 (Stable Topology): The ISL graph is invariant if $\beta$7, where $\beta$8 is the longest relevant ISL.
• Theorem 3 (Cell Hierarchy): Each layer-$\beta$9 cell subdivides into $\alpha_i = \tfrac{2\pi i}{N}$0 layer-$\alpha_i = \tfrac{2\pi i}{N}$1 subcells; total cells are $\alpha_i = \tfrac{2\pi i}{N}$2.
• Theorem 4 (Shortest Satellite Path): The digit-correction procedure is hop-optimal without global messaging.
• Theorem 5 (Routing Table Bound): Each FIB stores at most $\alpha_i = \tfrac{2\pi i}{N}$3 entries.
• Theorem 6 (Disjoint Multipaths): $\alpha_i = \tfrac{2\pi i}{N}$4 disjoint shortest-hop paths connect each satellite pair, facilitating fault tolerance.

All routing and addressing decisions are local and time-invariant, making F-Rosette immune to high mobility and re-convergence churn typical of LEO networks.

5. Implementation and Empirical Evaluation

A $\alpha_i = \tfrac{2\pi i}{N}$5K-line C prototype, running atop Linux/Quagga and orchestrated via StarPerf's [Lai ’20] orbit engine, demonstrates the F-Rosette approach on commodity hardware. Ground-station emulation leverages Mininet and real-world datasets (TLEs; NASA population grids).

Operational findings include:

• Routing: First-packet FIB installation: $\alpha_i = \tfrac{2\pi i}{N}$6 ms; subsequent lookups: $\alpha_i = \tfrac{2\pi i}{N}$7 ms; CPU load $\alpha_i = \tfrac{2\pi i}{N}$8, extra memory $\alpha_i = \tfrac{2\pi i}{N}$9 MB.
• Network Throughput: $$\sec R = \sqrt{3}\,\tan\left(\tfrac{\pi}{6}\,\tfrac{N}{N-2}\right)$$0 Gbps links saturated, achieving $$\sec R = \sqrt{3}\,\tan\left(\tfrac{\pi}{6}\,\tfrac{N}{N-2}\right)$$1 Mbps user throughput (comparable to IPv6/OSPF baselines).
• Addressing: Up to $$\sec R = \sqrt{3}\,\tan\left(\tfrac{\pi}{6}\,\tfrac{N}{N-2}\right)$$2-bit addresses for $$\sec R = \sqrt{3}\,\tan\left(\tfrac{\pi}{6}\,\tfrac{N}{N-2}\right)$$3 ($$\sec R = \sqrt{3}\,\tan\left(\tfrac{\pi}{6}\,\tfrac{N}{N-2}\right)$$4 satellites); $$\sec R = \sqrt{3}\,\tan\left(\tfrac{\pi}{6}\,\tfrac{N}{N-2}\right)$$5 MB for global cell-to-geocoordinate tables at $$\sec R = \sqrt{3}\,\tan\left(\tfrac{\pi}{6}\,\tfrac{N}{N-2}\right)$$6.
• Routing Optimality: Empirical hop-count stretch $$\sec R = \sqrt{3}\,\tan\left(\tfrac{\pi}{6}\,\tfrac{N}{N-2}\right)$$7 (always shortest), RTT stretch $$\sec R = \sqrt{3}\,\tan\left(\tfrac{\pi}{6}\,\tfrac{N}{N-2}\right)$$8 even under ISL jitter of $$\sec R = \sqrt{3}\,\tan\left(\tfrac{\pi}{6}\,\tfrac{N}{N-2}\right)$$9 ms for intercontinental routes (Beijing–New York, $R$0, $R$1 satellites).

The emulator confirms low-latency, stable performance with minimal resource overheads, suitable for resource-constrained LEO platforms.

6. Implications, Limitations, and Open Challenges

F-Rosette eliminates global routing convergence downtime and minimizes address churn, stabilizing the control plane for LEO mega-constellations in which satellites orbit at $R$2 km/s. No re-addressing occurs unless ground users physically traverse cell boundaries (a reduction in address churn by 80% over Starlink-style IP-over-LEO designs). The architecture supports incremental, orbit-by-orbit deployment and natively interworks with terrestrial IPv6 ASes.

However, several constraints remain intrinsic:

• All satellites must conform exactly to prescribed fractal geometry and altitude, implying high-precision deployment requirements.
• Cell-boundary effects may induce minimal detour stretch for last-mile users if serving satellite links fail.
• ISL outages, particularly on long-range equatorial hops, require sufficient altitude provisioning (per Theorem 2).
• Open avenues for research include optimizations trading satellite count for increased ISL complexity, and dynamic linking to further lower path stretch without sacrificing network invariance.

F-Rosette thus constitutes the first framework integrating fractal constellation design, hierarchical geographic addressing, and local π-space routing to yield provably stable, performant, and scalable space-ground IP networking for LEO mega-constellations (Li et al., 2021).