Hybrid communication systems: forming a multilevel mathematical concept of routing

This article develops mathematical routing models for hybrid communication systems that integrate ground, stratospheric, and space segments. The study addresses networks where topology, demand matrices, link capacities, loss levels, and delay characteristics vary at the same time. The paper aims to formulate a multilevel mathematical routing concept that combines three core ideas: fractional multicommodity flow, path-limited routing, and delay minimization. The study uses multicommodity flow models on directed graphs, path-based formulations with a bounded number of routes per demand, convex delay-aware objectives, and an analysis of modern approximation algorithms. The results show that fractional multicommodity flow defines the upper level for estimating throughput, fairness, and priority-aware service; path-limited formulations translate this solution into engineering policies that a routing plane can install and maintain; and delay-oriented models account for quality-of-service requirements and temporal dynamics. The paper also shows how this concept links routing with radio-resource allocation, structural adaptation of the network, and routing-information dissemination. The results support a multistage routing logic in which a fractional formulation estimates the theoretical upper bound, a path-limited model compresses this solution into an installable routing policy, and a delay-oriented stage refines the decision for hybrid-network operation. The proposed concept applies to the design and control of communication systems that link spacecraft, airborne platforms, and terrestrial infrastructure. The article concludes that this concept can provide a theoretical basis for routing in hybrid communication systems and can naturally extend to lossy transmission models, dynamic network scenarios, and integrated network-control problems.