用数学对偶性检测网络结构失衡，提前预警系统性风险。
Prism: Structural Symmetry Scanning via Duality-Constrained Laplacian Projection
- 通过拉普拉斯矩阵与对偶算子的反对易度量网络对称性破坏程度。
- 在真实股票数据中，缺陷值90天内从0.43升至0.73，早于传统指标发现风险。
- 无需训练数据，毫秒级计算，适合金融、社交等复杂网络监测。
我们提出Prism框架，用于复杂网络的结构对称性诊断。给定图拉普拉斯矩阵L和对偶算子P（对称对合），Prism计算双重要素缺陷δ(L,P) = ||LP - PL||_F / ||L||_F，一个衡量网络偏离结构自洽性的标量。当P反映网络真实对称性时，δ值随结构退化单调上升；任意P则产生噪声。我们证明最优满足[L', P] = 0的L'可通过闭式块对角投影获得，并提供基于费德尔向量的无监督交替优化算法学习P。合成网络实验表明，真实P下的缺陷值对结构退化的敏感度是索引反转基线的3.38倍，且高于模块度。在扎卡里空手道俱乐部网络中，5%边噪声下，Prism社区检测准确率达94.5%，优于原始拉普拉斯基线的76.6%。应用于2026年5月17日的实时标准普尔500数据，90天内缺陷值从0.43升至0.73，而表面相关性仍低——这是相关性方法无法捕捉的信号。历史回测覆盖2011–2020年五次重大危机事件显示，双重要素缺陷总在相关性飙升前升高，并在结构性脆弱期持续高位，而传统指标却将其视为平静期。该缺陷是基于第一性原理的结构可接受性条件，无需训练数据，可在毫秒内完成计算。
原文摘要 · Abstract (English)
We introduce \textbf{Prism}, a framework for structural symmetry diagnosis in complex networks. Given a graph Laplacian $L$ and a duality operator $P$ (a symmetric involution), Prism computes the \emph{duality defect} $δ(L,P) = \|LP - PL\|_F / \|L\|_F$ -- a scalar measuring how far the network deviates from structural self-consistency. When $P$ encodes the network's true symmetry, $δ$ starts near zero and rises monotonically as structure degrades; an arbitrary $P$ gives noise. We prove that the optimal $L'$ satisfying $[L', P] = 0$ is given by a closed-form block-diagonal projection, and provide an unsupervised alternating optimization that learns $P$ from the graph's own Fiedler vector. Experiments on synthetic networks show the true-$P$ defect is $3.38\times$ more sensitive to structural degradation than an index-reversal baseline and more sensitive than modularity. On Zachary's Karate Club with edge noise, Prism achieves $94.5\%$ community detection accuracy at $5\%$ noise versus $76.6\%$ for the raw Laplacian baseline. Applied to live S\&P~500 data (2026-05-17), Prism detects rising structural stress (defect $0.43 \to 0.73$ over 90 days) while surface correlations remain low -- a signal invisible to correlation-based methods. In a historical backtest spanning five major stress events (2011--2020), the duality defect exhibits a consistent pattern: it reaches elevated levels \emph{before} the correlation spike that accompanies each crisis, and sustains high readings during periods of structural fragility that conventional metrics classify as calm. The duality defect is a first-principles structural admissibility condition, requiring no training data and computable in milliseconds.
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