Extending graph metrics to hyperspaces of vertex sets: optimization and reinforcement learning applications
Abstract
An undirected connected graph can be interpreted as a metric space by considering the path distance between its vertices. Extending this perspective, we explore how walks in the graph, viewed as sequences of subsets of vertices V, fit into a more general framework where the path distance emerges as a special case of broader metrics that define hypertopologies on the space of subsets of V. In this paper, we propose a general framework to address this issue, motivated by the extension of recent reinforcement learning algorithms on graphs through Lipschitz function extensions. Our approach demonstrates how models for dynamical systems and related machine learning techniques can be adapted to (quasi-pseudo-metric) hyperspaces, offering a novel perspective for the analysis and modeling of dynamical systems. Finally, we illustrate the practical relevance of our theoretical contributions through a concrete graph optimization example, highlighting the utility and effectiveness of our proposed approach.
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BibTeX
@misc{jonardperez2026extending,
title = {{Extending graph metrics to hyperspaces of vertex sets: optimization and reinforcement learning applications}},
author = {Natalia Jonard-Pérez and Enrique A. Sánchez-Pérez and Sergi Sanjuan},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07860},
eprint = {2611.07860},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07860},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Extending graph metrics to hyperspaces of vertex sets: optimization and reinforcement learning applications
AU - Natalia Jonard-Pérez
AU - Enrique A. Sánchez-Pérez
AU - Sergi Sanjuan
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.07860
UR - https://arxiv.org/abs/2611.07860
AB - An undirected connected graph can be interpreted as a metric space by considering the path distance between its vertices. Extending this perspective, we explore how walks in the graph, viewed as sequences of subsets of vertices V, fit into a more general framework where the path distance emerges as a special case of broader metrics that define hypertopologies on the space of subsets of V. In this paper, we propose a general framework to address this issue, motivated by the extension of recent reinforcement learning algorithms on graphs through Lipschitz function extensions. Our approach demonstrates how models for dynamical systems and related machine learning techniques can be adapted to (quasi-pseudo-metric) hyperspaces, offering a novel perspective for the analysis and modeling of dynamical systems. Finally, we illustrate the practical relevance of our theoretical contributions through a concrete graph optimization example, highlighting the utility and effectiveness of our proposed approach.
ER -