用于聚类有向图和无向图的广义狄利克雷能量与图拉普拉斯算子
Generalized Dirichlet Energy and Graph Laplacians for Clustering Directed and Undirected Graphs
AI总结：
针对有向图聚类中边不对称导致经典谱方法失效的问题，提出广义狄利克雷能量（GDE）框架与广义谱聚类（GSC）方法，利用参数化顶点测度编码方向与密度，无需传送即可实现弱连通有向图聚类，在真实数据集上准确率与鲁棒性均优于现有方法。
摘要：
由于边连接的不对称性，有向图聚类仍是一个基本挑战，这限制了最初为无向图设计的经典谱方法的适用性。一种常见的变通方法是邻接矩阵对称化，但这通常会导致丢失关键的方向信息。在本研究中，我们引入了广义狄利克雷能量（GDE），这是一种新颖的能量泛函，将经典狄利克雷能量扩展以处理任意正顶点测度和马尔可夫转移矩阵。GDE提供了一个适用于有向图和无向图的统一框架，并与随机游走的扩散动力学密切相关。基于该框架，我们提出了广义谱聚类（GSC）方法，无需对随机游走转移矩阵引入传送即可实现弱连通有向图的原则性聚类。我们方法的一个关键组成部分是利用参数化顶点测度来编码图的方向性和密度。在真实点云数据集上的实验表明，GSC在聚类准确性和鲁棒性方面始终优于现有的谱聚类方法，为基于图的数据分析提供了强大的新工具。
英文摘要：
Clustering in directed graphs remains a fundamental challenge due to the asymmetry in edge connectivity, which limits the applicability of classical spectral methods originally designed for undirected graphs. A common workaround is to symmetrize the adjacency matrix, but this often leads to losing critical directional information. In this work, we introduce the generalized Dirichlet energy (GDE), a novel energy functional that extends the classical Dirichlet energy to handle arbitrary positive vertex measures and Markov transition matrices. GDE provides a unified framework applicable to both directed and undirected graphs, and is closely tied to the diffusion dynamics of random walks. Building on this framework, we propose the generalized spectral clustering (GSC) method that enables the principled clustering of weakly connected digraphs without resorting to the introduction of teleportation to the random walk transition matrix. A key component of our approach is the utilization of a parametrized vertex measure encoding graph directionality and density. Experiments on real-world point-cloud datasets demonstrate that GSC consistently outperforms existing spectral clustering approaches in terms of clustering accuracy and robustness, offering a powerful new tool for graph-based data analysis.
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