为图上路径定义度量空间，支持路径间距离与相似性分析。
Metric spaces of walks and Lipschitz duality on graphs
- 将路径视为利普希茨序列，用加权度量衡量路径间距离。
- 提出路径相似性表示公式，支持在不同假设下精确构造。
- 适用于强化学习中的探索策略设计与网络结构回归建模。
我们研究图上行走的度量结构，将其理解为利普希茨序列。为此引入加权度量处理序列，基于逐步顶点距离与加权范数定义路径间的距离。分析这些度量空间的主要性质，为弱形式相对距离度量（即邻近性）提供基础。在不同假设下给出邻近性的表示公式，并提供具体构造方法。该度量框架可运用经典度量建模工具，如从路径子空间扩展利普希茨函数，通过上述表示保持邻近函数的基本性质。潜在应用包括邻近性估计及基于探索性行走的强化学习策略开发，为网络结构上的利普希茨回归提供稳健方法。
原文摘要 · Abstract (English)
We study the metric structure of walks on graphs, understood as Lipschitz sequences. To this end, a weighted metric is introduced to handle sequences, enabling the definition of distances between walks based on stepwise vertex distances and weighted norms. We analyze the main properties of these metric spaces, which provides the foundation for the analysis of weaker forms of instruments to measure relative distances between walks: proximities. We provide some representation formulas for such proximities under different assumptions and provide explicit constructions for these cases. The resulting metric framework allows the use of classical tools from metric modeling, such as the extension of Lipschitz functions from subspaces of walks, which permits extending proximity functions while preserving fundamental properties via the mentioned representations. Potential applications include the estimation of proximities and the development of reinforcement learning strategies based on exploratory walks, offering a robust approach to Lipschitz regression on network structures.
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