In recent years, the emergence of large-scale Low-Earth-Orbit (LEO) satellite constellations has introduced unprecedented opportunities for global connectivity. However, routing efficiency and inter-shell communication remain key challenges in multi-shell architectures. This paper investigates the structural properties and network dynamics of a representative six-shell mega-constellation composed of 10,956 satellites and 198 gateway stations (GSs). Leveraging tools from complex network analysis, we identify several critical findings: (1) the constellation exhibits strong small-world characteristics, enabling efficient routing despite large network diameters; (2) GS relays play a pivotal role in enhancing inter-shell connectivity by bridging otherwise disconnected components; (3) feeder links significantly reduce average path length, making long-haul communication more feasible; (4) betweenness analysis reveals load imbalances among GSs, indicating the need for traffic-aware management strategies; (5) the architecture offers excellent spatial coverage and resilience, maintaining connectivity and low routing costs even under GS failures. These insights not only explain the design rationale behind current mega-constellations like SpaceX Starlink, but also provide valuable guidance for the evolution of future satellite network infrastructures.