We see in the first sublevel set M 0.05 that connected components start to form. These are representations are invariant to different types of deformation of the data. Specifically, we see that both the hydrogen bond network and water density manifold topology varies significantly over time, but their corresponding topologies (captured by ⟨χ(G ww )⟩ and EC curves) are similar. This result is of practical relevance, because it shows that the EC can be used to monitor the dynamics of topology (e.g., to determine when the system is undergoing a topological transition or has reached a steady-state). For example, Figure illustrates the topological convergence of χ(G ww ) for a simulation of a SAM with hydroxyundecanethiol endgroups. The topology of the hydrogen bonding networks converges after approximately 1500 picoseconds of simulation time; as such, this portion of the simulation is not included in the training/testing data. Before building a linear regression model that predicts HFE from the SAM structure, we first determined if there was indeed a relationship between the topology of the SAM interfacial water structure and the HFE. In Figure , we see that there is indeed a strong correlation between the topology of the time-averaged water density field at the SAM surface and its emergent HFE. Specifically, the local minima and maxima (i.e., critical points) of the density change in both shape and magnitude as the HFE for the SAM is changed.