On the existence of Ulanowicz's optimal structural resilience in complex networks
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910151550697472 |
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| author | Wei, Si-Yao Zhou, Wei-Xing |
| author_facet | Wei, Si-Yao Zhou, Wei-Xing |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_14747 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the existence of Ulanowicz's optimal structural resilience in complex networks Wei, Si-Yao Zhou, Wei-Xing Physics and Society Information Theory Numerical Analysis 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. |
| title | On the existence of Ulanowicz's optimal structural resilience in complex networks |
| topic | Physics and Society Information Theory Numerical Analysis |
| url | https://arxiv.org/abs/2601.14747 |