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

2026-01-21

AI Summary

This study rigorously validates the universality of Ulanowicz’s optimality condition for structural resilience (α = 1/e) in generalized complex networks, resolving the critical question of whether this optimal state—unattainable in two-node networks—can be achieved in larger systems. By constructing a symmetric directed network model with three distinct edge weights (x, y, z) and uniform marginal distributions, and employing tools from graph theory, variational analysis, asymptotic expansion, and numerical simulation, the work establishes for the first time the existence of an optimally resilient structure for networks with at least three nodes. It further reveals that edge weights scale with network size via power laws with logarithmic corrections: adjacent edges decay as O(N⁻¹), while non-adjacent edges decay as O(N⁻²). These findings provide a unified mathematical framework for entropy-driven network robustness, offering a theoretical foundation for the robust design of large-scale networks.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Social Networks and Social Media: Social media analysis through the lenses of networks
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $\alpha = 1/\mathrm{e}$, 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.

Problem

• structural resilience
• complex networks
• optimal configuration
• Ulanowicz
• network robustness

Innovation

• structural resilience
• optimal configuration
• complex networks
• power-law scaling
• entropy-based robustness