发表机构
Middle East Technical University（中东技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究从理论上分析网络划分聚类，探讨硬、软分配方案及四个关键模型，通过数学分析揭示模型结构特性，发现不同模型在聚类中心位置倾向等方面的差异，为设计高效算法提供新方向，对多领域有重要意义。
AI 中文摘要
本研究对网络上的划分聚类进行了理论分析，分析了具有不同目标函数的硬分配和软分配方案。聚类中心不限于顶点，也可位于边上。研究了四个关键模型：硬分配下的P - 中位数（PMP）和平方和聚类（SSC），以及软分配下的概率距离聚类（PDC）和模糊C - 均值（FCM）。通过数学分析，揭示了区分这些模型的结构特性，如分配瓶颈点的重要性和顶点受限解在确定最优聚类中心中的作用。研究发现，虽然SSC和FCM可沿边产生最优中心，但PMP和PDC本质上倾向于顶点放置，为网络聚类行为提供了见解。这些见解为设计高效算法提供了新方向，其影响范围从设施选址、网络设计到现代检索系统中支持相似性搜索的嵌入图上的聚类。
英文摘要
This study presents a theoretical analysis of partitional clustering on networks, analyzing both hard and soft assignment schemes with different objective functions. Cluster centers are not restricted to vertices but can also be located along the edges. We examine four key models: P-Median (PMP) and Sum of Squares Clustering (SSC) under hard assignment, and Probabilistic Distance Clustering (PDC) and Fuzzy C-Means (FCM) under soft assignment. Through mathematical analysis, we uncover structural properties that differentiate these models, such as the significance of assignment bottleneck points and the role of vertex-restricted solutions in determining optimal cluster centers. Our findings reveal that, while SSC and FCM can yield optimal centers along edges, PMP and PDC inherently favor vertex placement, leading to insights into clustering behavior on networks. These insights offer new directions for designing efficient algorithms and have implications ranging from facility location and network design to clustering on the embedding graphs that power similarity search in modern retrieval systems.
Comments40 pages, 16 figures
