  In $4$-dimensional pure compact $U(1)$ lattice gauge theory, we analyse
topological aspects of the dynamics of monopoles across the deconfinement phase
transition. We do this using tools from Topological Data Analysis (TDA). We
demonstrate that observables constructed from the zeroth and first homology
groups of monopole current networks may be used to quantitatively and robustly
locate the critical inverse coupling $\beta_{c}$ through finite-size scaling.
Our method provides a mathematically robust framework for the characterisation
of topological invariants related to monopole currents, putting on firmer
ground earlier investigations. Moreover, our approach can be generalised to the
study of Abelian monopoles in non-Abelian gauge theories.
