Persistent Homology as a Framework for Detecting Structural Phase Transitions in Dynamic Social Networks

Complex networks are commonly described using graph-theoretic measures such as degree distributions, clustering coefficients, modularity, and shortest path lengths. These quantities summarize important structural properties but provide only a partial description of network organization, particularly as networks evolve over time. Community formation, fragmentation, and large-scale reorganization often occur gradually across multiple scales, making it difficult to determine when a network undergoes a meaningful structural transition. Persistent homology offers an alternative mathematical framework by describing the topology of a network through features that persist across varying connectivity thresholds rather than at a single graph representation. Although persistent homology has been applied in computational topology and data analysis, its ability to characterize structural transitions in evolving social networks remains incompletely understood. This project asks whether topological invariants derived from persistent homology identify structural transitions in dynamic networks more effectively than conventional graph statistics. Temporal interaction networks will be constructed from publicly available longitudinal datasets, with simplicial complexes generated from weighted graph representations at successive time points. Persistent diagrams and Betti numbers will quantify the evolution of connected components and higher-order topological features throughout network development. The central question is whether changes in these topological descriptors consistently precede or better characterize shifts in community organization than traditional graph-theoretic metrics.