  Topological Data Analysis (TDA) has been praised by researchers for its
ability to capture intricate shapes and structures within data. TDA is
considered robust in handling noisy and high-dimensional datasets, and its
interpretability is believed to promote an intuitive understanding of model
behavior. However, claims regarding the power and usefulness of TDA have only
been partially tested in application domains where TDA-based models are
compared to other graph machine learning approaches, such as graph neural
networks. We meticulously test claims on TDA through a comprehensive set of
experiments and validate their merits. Our results affirm TDA's robustness
against outliers and its interpretability, aligning with proponents' arguments.
However, we find that TDA does not significantly enhance the predictive power
of existing methods in our specific experiments, while incurring significant
computational costs. We investigate phenomena related to graph characteristics,
such as small diameters and high clustering coefficients, to mitigate the
computational expenses of TDA computations. Our results offer valuable
perspectives on integrating TDA into graph machine learning tasks.
