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

TL;DR

This paper rigorously investigates the mathematical existence and properties of Ulanowicz's optimal structural resilience in complex networks, revealing conditions for optimal flow configurations and their scaling behaviors.

Contribution

It proves the existence of optimal flow configurations in networks with three or more nodes and introduces a symmetric network model to analyze asymptotic resilience scaling.

Findings

Optimal resilience is unattainable in two-node networks.

Existence of at least one optimal flow configuration for networks with N ≥ 3.

Primary links scale as O(N^{-1}), background links as O(N^{-2}).

Abstract

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