TOPOLOGICAL INVARIANTS OF WEIGHTED SIMPLICIAL COMPLEXES AND THEIR APPLICATIONS IN DATA CLUSTERING

Clustering algorithms built on distance thresholds or centroid geometry often fail when data lies on curved manifolds, contains multi-scale structure, or exhibits noise that distorts local density estimates. This paper investigates whether topological invariants derived from weighted simplicial complexes - specifically Betti numbers, persistence diagrams, and Euler characteristics computed across a filtration of edge weights - can serve as a more stable basis for cluster identification than conventional distance-based methods. A weighted Vietoris-Rips complex was constructed from five benchmark datasets spanning synthetic and real-world domains, and persistent homology was computed at successive filtration values to extract zeroth and first-order Betti numbers. These invariants were then used to define a topological clustering criterion, compared against kmeans, DBSCAN, and spectral clustering using adjusted Rand index and silhouette scores. Results indicate that topology-based clustering achieved a mean adjusted Rand index of 0.81 across the five datasets, compared with 0.64 for k-means and 0.71 for DBSCAN, with the largest gains observed on datasets containing non-convex or nested cluster shapes. Persistence diagrams also proved more resistant to Gaussian noise injection, retaining stable Betti-1 counts up to a noise variance of 0.15 where distance-based methods degraded sharply. These findings support the proposition, stated at the outset, that weighted simplicial complexes encode structural information about data shape that persists across scale and noise in ways centroid- or density-based methods cannot capture, and they suggest a practical pathway for incorporating topological data analysis into standard clustering pipelines without prohibitive computational cost for datasets under approximately 2,000 points.