发表机构
Shiv Nadar Institution of Eminence; Jawaharlal Nehru University; SRM University - AP（希夫纳达尔卓越机构; 贾瓦哈拉尔·尼赫鲁大学; SRM大学-安得拉邦校区）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究采用随机矩阵理论分析随机混合图的谱与本征向量性质，揭示连接性和方向性对GOE-GUE交叉的影响，并将框架应用于标普500混合图，刻画了市场崩盘与复苏阶段的谱统计差异。
AI 中文摘要
我们采用随机矩阵理论（RMT）研究结合无向与有向相互作用的随机混合图的谱与本征向量性质。该网络由厄米邻接矩阵表示，无向边对应实数值，有向边对应纯虚共轭对，以保证谱为实数。网络密度由连接概率控制，而方向性设定有向边的比例。我们聚焦于GOE到GUE的交叉：在固定连接性下，增加方向性会破坏时间反演对称性，使谱统计从GOE转变为GUE。我们表明，该转变需要足够的连接性；在稀疏网络中，无论方向性如何，弱能级排斥都会产生泊松统计。在固定方向性下，增加连接性会驱动泊松到GUE的交叉；仅在稀疏阈值以上，方向性诱导的GOE到GUE转变才会显现。在谱密度遵循Wigner半圆律的稠密区域，该交叉通过间距分布、间距比和谱刚性来表征；在展开不可靠的稀疏网络中，间距比统计提供了无需展开的表征。我们通过多重分形维度和分量分布研究本征向量结构，同时Kullback-Leibler散度证实了转变的稳健性。将该框架应用于标普500混合图，结果显示在四次主要市场崩盘期间存在GOE到GUE的交叉，且较稠密的危机时期比较稀疏的复苏时期表现出更尖锐的交叉。这些结果为连接性与对称性破缺如何调控谱和本征向量普适性提供了统一图景，同时为探测金融市场结构的变化提供了透明的工具。
英文摘要
We study the spectral and eigenvector properties of random mixed graphs, combining undirected and directed interactions, using random matrix theory (RMT). The network is represented by a Hermitian adjacency matrix, with undirected links as real entries and directed links as purely imaginary conjugate pairs, ensuring a real spectrum. Network density is controlled by the connection probability, while directionality sets the fraction of directed edges. We focus on the GOE-to-GUE crossover: at fixed connectivity, increasing directionality breaks time-reversal symmetry and drives spectral statistics from GOE to GUE. We show that this transition requires sufficient connectivity. In sparse networks, weak level repulsion produces Poisson statistics regardless of directionality. At fixed directionality, increasing connectivity drives a Poisson-to-GUE crossover; only above a sparsity threshold does the directionality-induced GOE-to-GUE transition emerge. In the dense regime, where the spectral density follows the Wigner semicircle law, the crossover is characterized using spacing distributions, spacing ratios, and spectral rigidity. In sparse networks, where unfolding is unreliable, spacing-ratio statistics provide an unfolding-free characterization. Eigenvector structure is examined through multifractal dimensions and component distributions, while Kullback--Leibler divergence confirms the robustness of the transitions. Applied to S&P 500 mixed graphs, the framework reveals a GOE-to-GUE crossover across four major market crashes. Denser crisis periods show sharper crossovers than sparser recovery periods. The results provide a unified picture of how connectivity and symmetry breaking govern spectral and eigenvector universality, while providing a transparent probe of changing financial-market organization.
Comments25 pages, 24 figures
