复赋权图上的Ollivier Ricci曲率
Ollivier's Ricci Curvature on Complex-weighted Graphs
AI总结：
本研究将Ollivier Ricci曲率扩展至复赋权图（含有向图特例），确立其理论特性并开发计算方法，验证了其在有向网络社区检测中的实用性。
AI中文摘要：
理解复杂网络的几何特性对各领域的有效建模与分析至关重要。尽管离散Ricci曲率概念已成为表征网络局部与全局结构的有力工具，但现有公式大多局限于具有实值权重的无向网络，这限制了曲率分析在诸多应用场景中对有向和复赋权关系的使用，此类关系自然出现在从社会、生物系统到量子和信号处理网络等领域。本研究中，我们提出了将Ollivier Ricci曲率扩展到复赋权图的严谨方案，该方案将有向图作为特例包含在内。我们确立了这一新概念的基础理论特性，包括其与磁拉普拉斯算子的关系，以及将曲率与局部邻域环结构关联起来的组合上下界。我们还开发了曲率估计的计算方法，并在有向网络的社区检测中验证了其实用性。
英文摘要：
Understanding the geometry of complex networks is critical for effective modeling and analysis across domains. While discrete notions of Ricci curvature have emerged as powerful tools for characterizing both local and global network structure, existing formulations are largely confined to undirected networks with real-valued weights. This limits the use of curvature-based analysis of directional and complex-weighted relations that arise naturally in many applications, from social and biological systems to quantum and signal-processing networks. In this work, we introduce a principled extension of Ollivier's Ricci curvature to complex-weighted graphs, which encompasses directed graphs as a special case. We establish fundamental theoretical properties of this new notion, including relations to the magnetic Laplacian and combinatorial upper and lower bounds that relate curvature to cycle structure in local neighborhoods. We further develop computational methods for curvature estimation and demonstrate their utility in community detection on directed networks.
