Algebraic Topology Approaches to the Analysis of Complex Networks in Nigerian Financial Systems

This dissertation applies algebraic topology, specifically persistent homology and sheaf theory on networks, to develop original analytical methods for characterising the structural complexity and systemic risk properties of the Nigerian financial system network, contributing a novel methodological framework to the mathematical finance and financial stability literature. The interconnectedness of financial institutions through credit exposures, interbank lending, and common asset holdings creates network structures whose topological properties determine the propagation dynamics of financial distress, yet existing network analyses of Nigerian financial systems have relied predominantly on graph-theoretic metrics that do not capture higher-order topological features. Using bilateral exposure data from the Central Bank of Nigeria's supervisory reporting system for the period 2015 to 2023, financial network snapshots are constructed as weighted directed graphs, and Vietoris-Rips and Cech filtrations are constructed from the network's connectivity data. Persistent homology computations in dimensions zero, one, and two are performed, and novel systemic risk indicators derived from persistence diagrams are proposed and validated against observed episodes of financial stress. A sheaf-theoretic framework is developed for analysing the consistency of bank-level risk signals propagated across the network, and obstructions to consistent global risk assessment are interpreted as systemic fragility indicators. The dissertation establishes original mathematical propositions relating topological features of the financial network to its robustness under targeted and random institution failures. Keywords: algebraic topology, persistent homology, financial networks, systemic risk, sheaf theory