International Journal of Science, Engineering and TechnologyAuthors: Assistant Professor Rahul Kaushik, Rajkumar Soni
Abstract: Algebraic topology has emerged as a powerful framework for extracting shape‐based features from complex datasets by mapping data points to topological spaces and computing invariants that persist across scales (Carlsson 255). However, traditional network metrics often fail to capture higher-order connectivity patterns beyond edges, leaving a gap in quantitative tools for analyzing cavities, cycles, and voids in graphs and real-world networks (Horak, Maletić, and Rajković). This paper argues that persistent homology and its extensions provide robust, multi-scale invariants that quantitatively characterize both empirical networks and theoretical graph models. We construct simplicial complexes—via Vietoris-Rips and clique filtrations—from weighted and unweighted graphs and apply standard persistent homology algorithms to compute Betti numbers and persistence diagrams (Zomorodian and Carlsson 249; Otter et al. 4). Applications span network science—where homological scaffolds reveal mesoscale brain connectivity and sliding-window embeddings track dynamic signal networks—and graph theory—where Betti distributions elucidate phase transitions in Erdős–Rényi models and persistence distortion defines novel graph distances (Petri et al.; Dey, Shi, and Wang). Our findings demonstrate that topological descriptors complement spectral and combinatorial measures by uncovering hidden structural features with provable stability and interpretable summaries (Cohen-Steiner, Edelsbrunner, and Harer 103). We conclude by outlining future directions, including real-time homology computation for streaming networks and integration with machine-learning pipelines to further bridge topology and data science.
