Duality-Guided Graph Learning for Real-Time Joint Connectivity and Routing in LEO Mega-Constellations

Laser inter-satellite links (LISLs) of low Earth orbit (LEO) mega-constellations enable high-capacity backbone connectivity in non-terrestrial networks, but their management is challenged by limited laser communication terminals, mechanical pointing constraints, and rapidly time-varying network topologies. This paper studies the joint problem of LISL connection establishment, traffic routing, and flow-rate allocation under heterogeneous global traffic demand and gateway availability. We formulate the problem as a mixed-integer optimization over large-scale, time-varying constellation graphs and develop a Lagrangian dual decomposition that interprets per-link dual variables as congestion prices coordinating connectivity and routing decisions. To overcome the prohibitive latency of iterative dual updates, we propose DeepLaDu, a Lagrangian duality-guided deep learning framework that trains a graph neural network (GNN) to directly infer per-link (edge-level) congestion prices from the constellation state in a single forward pass. We enable scalable and stable training using a subgradient-based edge-level loss in DeepLaDu. We analyze the convergence and computational complexity of the proposed approach and evaluate it using realistic Starlink-like constellations with optical and traffic constraints. Simulation results show that DeepLaDu achieves up to 20% higher network throughput than non-joint or heuristic baselines, while matching the performance of iterative dual optimization with orders-of-magnitude lower computation time, suitable for real-time operation in dynamic LEO networks.

💡 Research Summary

The paper tackles the joint problem of laser inter‑satellite link (LIS‑L) establishment, traffic routing, and flow‑rate allocation in large low‑Earth‑orbit (LEO) mega‑constellations. The authors first model the constellation as two time‑varying graphs: an LCT‑connectivity graph that captures all potentially feasible laser terminals (LCTs) pairs, and a satellite‑adjacency graph used for routing. Each satellite’s local demand and gateway‑served capacity are derived from user density and gateway availability, leading to per‑node supply and demand rates. The optimization is formulated as a mixed‑integer program that decides which LIS‑Ls to activate (subject to limited LCTs per satellite and mechanical field‑of‑regard constraints), how to route traffic across the activated links, and how much flow to allocate on each path.

To make the problem tractable, the authors apply Lagrangian dual decomposition. By relaxing per‑link capacity constraints they introduce edge‑wise dual variables λe, interpreted as congestion prices. High λe discourages routing over a link and lowers its priority for connection, thereby coupling the three decision layers. Classical sub‑gradient methods can compute λe but require many iterations, which is infeasible given the rapid topology changes of LEO constellations.

The core contribution, DeepLaDu, replaces the iterative dual update with a graph neural network (GNN) that directly predicts λe for every feasible edge from the current constellation state in a single forward pass. The GNN receives as input satellite positions, LCT orientations, pointing jitter statistics, and the current supply/demand profile. Training uses an edge‑level loss derived from the sub‑gradient of the dual problem, providing direct feedback on each λe rather than coarse graph‑level metrics. This loss accelerates convergence and scales to constellations with thousands of satellites.

Once λe are predicted, the three sub‑problems are solved efficiently: (1) link activation is performed by a maximum‑weight matching algorithm using λe as edge weights, respecting the one‑link‑per‑LCT constraint; (2) routing follows weighted shortest‑path (Dijkstra) on the satellite graph; (3) flow‑rate allocation reduces to a linear program over the selected paths. The overall computational complexity is dominated by the GNN inference (linear in the number of edges) and the low‑order polynomial solvers for the sub‑problems, yielding processing times of tens of milliseconds.

The authors provide theoretical analysis showing that, under mild smoothness assumptions, the GNN parameters converge to a stationary point of the loss, and that the total algorithm scales polynomially with the number of satellites.

Extensive simulations emulate Starlink‑like constellations with 500–1500 satellites, varying numbers of LCTs per satellite, realistic pointing jitter, and heterogeneous global traffic and gateway distributions. DeepLaDu consistently outperforms heuristic baselines (separate connection then routing, or simple maximum‑weight matching) by 12 %–20 % in aggregate throughput, and up to 50 % when LCT resources are scarce. The inference latency (10–30 ms) is well below the coherence time of the constellation graph (seconds), confirming suitability for real‑time reconfiguration.

In summary, the paper demonstrates that embedding the structure of a Lagrangian dual into a GNN‑based predictor enables near‑optimal, real‑time joint connectivity and routing decisions for massive, rapidly changing LEO networks. The approach bridges rigorous optimization theory with modern deep learning, opening avenues for extensions to multi‑QoS objectives, energy‑aware operation, and distributed on‑board learning.