Topology of Data in Persistent Homology Analysis of Social Network Structures in Nigerian Urban Centres

This study applies topological data analysis, specifically persistent homology, to examine the structural properties of social network data from urban centres in Nigeria, contributing to the application of algebraic topology in the social and computational sciences in the West African context. Persistent homology provides a rigorous mathematical framework for extracting multi-scale topological features from high-dimensional data by tracking the birth and death of connected components, loops, and voids across a filtration of simplicial complexes, and its application to social network structure offers insights unavailable from conventional graph-theoretic metrics. Social network data are constructed from mobile phone communication metadata provided by a major Nigerian telecommunications operator, covering a sample of 50,000 anonymised subscribers in Lagos, Kano, and Port Harcourt. Vietoris-Rips filtrations are constructed from pairwise communication frequency matrices, and persistent homology computations are performed using the Ripser software library. Persistence diagrams and barcodes are generated for zeroth and first-order Betti numbers, and statistical comparisons of topological features across the three cities are conducted using bottleneck distances and persistence entropy metrics. Results reveal distinct topological signatures in the three cities' social network structures, with Lagos networks exhibiting more complex higher-order topological features consistent with a more extensively connected and diverse social fabric. Keywords: persistent homology, topological data analysis, social networks, Betti numbers, Nigerian cities