On the existence of Ulanowicz's optimal structural resilience in complex networks

TL;DR

The paper addresses whether Ulanowicz's entropy-based notion of structural resilience has a mathematical optimum in general directed networks, not just ecological ones. By modeling a symmetric multi-link network with three weights and deriving exact expressions for efficiency $e$ and redundancy $r$, it shows that the optimum $R = 1/\mathrm{e}$ is unattainable for two nodes but exists for any $N_\mathcal{V} \ge 3$; the optimum is realized by configurations where $\alpha = 1/\mathrm{e}$. The authors develop governing equations and analyze several symmetric cases (notably $x=y$ and $y=z$), proving existence results and obtaining precise asymptotic scalings: the dominant link weights decay as $x \sim 1/(\mathrm{e}+1) \cdot 1/N_\mathcal{V}$ and background weights as $z \sim (\mathrm{e}-1)/(\mathrm{e}+1) \cdot 1/N_\mathcal{V}^2$, with logarithmic corrections. They further show that, via a convex-combination construction between a complete graph and a ring, there exists a one-parameter path to achieve the optimal resilience for any $N_\mathcal{V} \ge 3$, and discuss how these results generalize to other constraint cases and offer guidance for designing large-scale robust networks.

Abstract

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.

Figures (9)

• Figure 1: The function $R=-\alpha\ln\alpha$.
• Figure 2: The network represented by Eq. (\ref{['Eq_Network_xyz']}).
• Figure 3: Four cases of governing equations. (a) $x=y$, (b) $x=z$, (c) $z=0$, and (d) $x=0$.
• Figure 4: The functions $f(z)$ for $N_\mathcal{V}=4,5,6$ (a), the values of $x$ (b), and $z$ (c) for different $N_\mathcal{V}$. The black dashed line denotes its first-order approximation and the red line denotes denotes its second-order approximation, where $a_1$ and $b_1$ are shown in Eq. (\ref{['Eq_x_equal_y_a1_b1']}).
• Figure 5: The values of $x$ (a) and $z$ (b) for different $N_{\mathcal{V}}$. The black dashed line denotes its first-order approximation and the red line denotes denotes its second-order approximation, where $a_1$ and $b_1$ are shown in Eq. (\ref{['Eq_y_equal_z_a1_b1']}).

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• proof