拥塞控制网络修复下的规范能量保持

中文摘要 AI 辅助

我们研究了伯努利边失效后局部网络修复下有限规范能量流的保持性。宏观需求网络记录待路由的终端对，而微观物理网络包含局部备用路径、绕行通道和共享走廊。我们证明了一个确定性的规范能量修复定理：若可用需求网络 $\mathcal{B}^{\sharp}$ 承载有限 $\Phi$-能量流 $\theta$，则修复后的物理网络 $H$ 承载一个提升的有限 $\Phi$-能量流 $\Theta$，且满足 $$ \mathcal{E}^\Phi_H(\Theta) \le L\\,\beta_\Phi(K)\\,\mathcal{E}^\Phi_{\mathcal{B}^{\sharp}}(\theta), $$ 其中 $L$ 为路由长度上界，$K$ 为路由拥塞上界，$\mathcal{E}$ 表示能量，$\beta_\Phi$ 为规范膨胀常数。随后我们将此比较转化为概率性修复准则：有限依赖的局部修复通过乘积测度的控制来处理，随机修复长度则通过与化学距离估计兼容的可变成本公式来处理。作为主要应用，我们证明了一个有限依赖的局部绕行定理：任何宏观网络，若其超临界渗流簇承载有限规范能量流，则在有界范围的局部加固后，只要局部修复概率足够高，该网络仍保持规范能量稳定。这产生了加固格点和楔形示例，并为超越树状或边不相交构造的随机网络修复提供了位势论框架。

英文摘要

We study preservation of finite gauge-energy flows under local network repair after Bernoulli edge failures. A macroscopic demand network records terminal pairs to be routed, while a microscopic physical network contains local backup routes, bypasses and shared corridors. We prove a deterministic gauge-energy repair theorem: if the usable demand network $\mathcal{B}^{\sharp}$ carries a finite $Φ$-energy flow $θ$, then the repaired physical network $H$ carries a lifted finite $Φ$-energy flow $Θ$ with $$ \mathcal{E}^Φ_H(Θ) \le L\,β_Φ(K)\,\mathcal{E}^Φ_{\mathcal{B}^{\sharp}}(θ), $$ where $L$ bounds route length, $K$ bounds routing congestion, $\mathcal{E}$ marks energy, and $β_Φ$ is the gauge dilation constant. We then convert this comparison into probabilistic repair criteria: finite-dependent local repair is handled via domination by product measures, and random repair lengths via a variable-cost formulation compatible with chemical-distance estimates. As a main application, we prove a finite-dependent local bypass theorem: any macroscopic network whose supercritical percolation cluster supports a finite gauge-energy flow remains gauge-energy stable after bounded-range local reinforcement, provided the local repair probability is sufficiently high. This yields reinforced lattice and wedge-type examples and provides a potential-theoretic framework for random network repair beyond tree-like or edge-disjoint constructions.

发表机构

• Peking University（北京大学）

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