This study investigates the mathematical existence and asymptotic properties of Ulanowicz's structural resilience in complex systems such as supply chain networks. While ecological evidence suggests that sustainable systems gravitate toward an optimal state at $α= 1/\mathrm{e}$, the universality of this configuration in generalized networks remains theoretically unverified. We prove that while optimal resilience is unattainable in two-node networks due to structural over-determinacy, it exists for any directed graph with $N_\mathcal{V} \geq 3$. By constructing a symmetric network model with three types of link weights $(x, y, z)$ and uniform marginal distributions, we derive the governing equations for the optimal resilience configuration. Our analytical and numerical results reveal that as the network size $N_\mathcal{V}$ increases, the link weights required to maintain optimal resilience exhibit a power-law scaling behavior: the adjacent links scale as $O(N_\mathcal{V}^{-1})$, while the non-adjacent links scale as $O(N_\mathcal{V}^{-2})$, both accompanied by specific logarithmic corrections. This work establishes a rigorous mathematical foundation for the optimal resilience framework and provides a unified perspective on how entropy-based principles govern the robustness and evolution of large-scale complex networks, which may offer quantitative guidance for designing large-scale networked systems under robustness constraints.

翻译：本研究探讨了乌拉诺维奇结构弹性在供应链网络等复杂系统中的数学存在性与渐近性质。尽管生态学证据表明可持续系统倾向于收敛于 $α= 1/\mathrm{e}$ 的最优状态，但该构型在广义网络中的普适性在理论上尚未得到验证。我们证明：在双节点网络中，由于结构超定性，最优弹性无法实现；但对于任意节点数 $N_\mathcal{V} \geq 3$ 的有向图，最优弹性是存在的。通过构建具有三类边权重 $(x, y, z)$ 且满足均匀边际分布的对称网络模型，我们推导了最优弹性构型的主导方程。解析与数值结果表明：随着网络规模 $N_\mathcal{V}$ 增大，维持最优弹性所需的边权重呈现幂律标度行为：相邻边权重按 $O(N_\mathcal{V}^{-1})$ 标度，非相邻边权重按 $O(N_\mathcal{V}^{-2})$ 标度，且均伴有特定的对数修正项。本研究为最优弹性框架建立了严格的数学基础，并为基于熵的原理如何支配大规模复杂网络的鲁棒性与演化提供了统一视角，可为鲁棒性约束下的大规模网络化系统设计提供定量指导。