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 $k \geq 1$,

$$F\text{-Rosette}_k = \bigcup_{i=0}^{N-1} \left(\mathrm{Shift}(F\text{-Rosette}_{k-1}, \tfrac{2\pi}{N^k} i)\right) \cup \{\text{links between layer-}k{-}1\text{ copies: } S_i^i \rightarrow S_{i+1}^{i+1}\}$$

This recursive procedure produces $N^{k+1}$ satellites after $k$ fractal layers, with each satellite's degree scaling as $2(k+1)$ 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 $N=8$, $m=6$, $k=2$, $F$-Rosette$_2$ 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-$N$ $(k+1)$-digit address: $$S \equiv s_0.s_1.\ldots.s_k, \quad s_i \in \{0,\ldots,N-1\}$$ where each digit $s_i$ 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 $i$, each cell is subdivided into $N^2$ subcells (Theorem 3). Cells and their encapsulating satellites are referenced by similarly structured codes: $$C \equiv c_0.c_1.\ldots.c_k, \quad c_0\in\{0,\ldots,(N-m)^2-1\},\;c_i\in\{0,\ldots,N^2-1\}$$ Address-to-geocoordinate mapping leverages “cell tables” based on precomputed geometry, supporting efficient $O(k)$ 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 $D=d_0.d_1.\ldots.d_k$ 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 ($O(1)$ using arithmetic and prefix matching), with routing tables per satellite bounded by $2(k+1)\lceil\log_2(N/2)\rceil$ (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

$\leq \tfrac{(k+1)N}{2}$

Redundancy is embedded: there are always $2(k+1)$ 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): $F$-Rosette$_k$ with $N^{k+1}$ satellites achieves guaranteed full-Earth coverage if satellite altitude $H$ exceeds

$$H_{\min} = R_E\left(\frac{1}{\cos R-\sin R\,\tan\phi}-1\right), \quad R = \sec^{-1}\!\left(\sqrt3\,\tan\left(\frac{\pi}{6}\frac{N^{k+1}}{N^{k+1}-2}\right)\right)$$

• Theorem 2 (Stable Topology): The ISL graph is invariant if $$H > \max\{\frac{R_E}{\cos(r_{\max}/2)}-R_E, H_{\min}\}$$, where $r_{\max}$ is the longest relevant ISL.
• Theorem 3 (Cell Hierarchy): Each layer-$(k-1)$ cell subdivides into $N^2$ layer-$k$ subcells; total cells are $(N-m)^2 N^{2k}$.
• 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 $2(k+1)\lceil\log_2(N/2)\rceil$ entries.
• Theorem 6 (Disjoint Multipaths): $2(k+1)$ 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 $13$K-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: $0.048\text{--}0.058$ ms; subsequent lookups: $0.006\text{--}0.008$ ms; CPU load $<1\%$, extra memory $<1.3$ MB.
• Network Throughput: $1$ Gbps links saturated, achieving $867$ Mbps user throughput (comparable to IPv6/OSPF baselines).
• Addressing: Up to $32$-bit addresses for $N=16,\,k=8$ ($2^{32}$ satellites); $2$ MB for global cell-to-geocoordinate tables at $N=16,\,k=3$.
• Routing Optimality: Empirical hop-count stretch $= 1$ (always shortest), RTT stretch $< 1.4\%$ even under ISL jitter of $<10$ ms for intercontinental routes (Beijing–New York, $k=1$, $256$ 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 $7$ 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).