In modern network design, "efficiency" is often conflated with raw performance metrics like latency or aggregate throughput. This paper proposes a resource-centric definition of efficiency, isolating the hardware cost required to maintain a non-blocking throughput constraint. By modeling network cost as a function of the Traffic Multiplier (Hop Count) and Router Complexity (Radix), we demonstrate that the optimal topology is determined by the technological ratio between link interface costs ($α$), crossbar switching costs ($β$), and the network concentration ratio. We conclude that while high-radix direct networks optimize efficiency at small to medium scales, indirect networks (e.g., Fat Trees) are required to cap router complexity at massive scales. Furthermore, we posit that redundancy is most efficiently handled via parallel network instances (e.g., multi-plane Star networks) rather than intrinsic topological path diversity.

翻译：在现代网络设计中，“效率”常与原始性能指标（如延迟或聚合吞吐量）混为一谈。本文提出一种资源中心的效率定义，聚焦于维持非阻塞吞吐量约束所需的硬件成本。通过将网络成本建模为流量乘数（跳数）和路由器复杂度（基数）的函数，我们证明最优拓扑由链路接口成本（$α$）、交叉开关交换成本（$β$）与网络集中率之间的技术比率决定。我们的结论是：虽然高基数直接网络在中小规模下能优化效率，但间接网络（如胖树）在大规模场景中对于限制路由器复杂度是必需的。此外，我们认为冗余处理的最有效方式是通过并行网络实例（如多平面星型网络）而非依赖固有的拓扑路径多样性。