Metric spaces of walks and Lipschitz duality on graphs

The suggested procedure involves integrating the proximity function null P as a mechanism to guide exploration on the space of walks. While the use of null P that we have explained has focused on classification and metric analysis, its geometric interpretation and ability to quantify similarity between walks suggest a broader applicability, particularly in settings where the reward landscape is sparse or the graph structure is too large for exhaustive exploration. We propose an improvement to the exploration strategy used in reinforcement learning algorithms that incrementally construct walks within graph-based environments. Traditionally, these algorithms alternate between exploitation (choosing the next node to maximize an estimated reward) and exploration (randomly selecting a new node). The novelty lies in replacing random exploration with a proximity-guided strategy using a function null P . Instead of sampling uniformly, the agent compares potential path extensions to a reference set of high-reward walks, prioritizing those that are most similar in structure. This approach introduces a more informed, data-driven method for exploration, focusing on areas of the graph that resemble previously successful trajectories.