Graph Labeling For Topological Data Analysis In Machine Learning

Abstract: Graph labeling is an approach to systematically labeling the vertices of a graph with numerical labels, thus shedding light on such basic structural and combinatorial features. At the same time, Topological Data Analysis (TDA) has emerged as a powerful paradigm for the extraction of high-dimensional data's topological invariants with tools like persistent homology. This paper introduces a new fusion of graph labeling techniques with the TDA paradigm to enhance the extraction and utilization of topological features in machine learning. A new family of labelings, specifically designed to preserve the topological features and obey the persistence structure of graph filtrations, is presented. Promising strategies, like edge-weighted magic labelings and cycle-preserving vertex labelings, are constructed and analyzed. Analytical findings determine the conditions under which these labelings produce persistence diagrams showing stability under data perturbations. Experiments on synthetic and state-of-the-art benchmark datasets demonstrate that topology-aware labelings dramatically enhance the discriminative ability of TDA-based machine learning models. This paper unifies discrete labeling theory and computational topology, thus offering an new paradigm for the extraction of stable topological features from structured data.