Si-Yao Wei, Wei-Xing Zhou
This study provides a foundational theoretical investigation into the mathematical existence and asymptotic properties of Ulanowicz's structural resilience. While ecological evidence suggests that sustainable systems gravitate toward an optimal efficiency-redundancy balance at $α= 1/\mathrm{e}$, the mathematical attainability of this configuration across broader network topologies remains unverified. We rigorously prove that while optimal resilience is structurally unattainable in two-node networks, there exists at least one optimal flow configuration within the feasible probability space for any weighted and directed network with the network size $N_\mathcal{V} \geq 3$ and no self-loops. To make the derivations analytically tractable, we introduce a parameterized symmetric network model with uniform marginal distributions. Using this stylized ansatz, our analytical and numerical results reveal that maintaining the optimal state requires distinct asymptotic scaling behaviors as $N_\mathcal{V}$ increases: adjacent primary links scale as $O(N_\mathcal{V}^{-1})$, whereas non-adjacent background links exhibit a steeper quadratic decay of $O(N_\mathcal{V}^{-2})$ with specific logarithmic corrections. Rather than serving as an immediate engineering tool, this work establishes a rigorous mathematical boundary for the optimal resilience framework, demonstrating analytically how an optimally resilient system differentiates into high-throughput primary channels and sparse redundancy pathways.