Here, the individual water molecules can be considered vertices of a graph with noncovalent bonds (e.g., hydrogen bonds) representing edges between the vertices. Manifold representations for molecular simulations arise when there is a continuous function describing behavior over a space (or surface) of a simulation. An example of a manifold representation arises in the analysis of time-averaged spatial density, where density is computed at each spatial location resulting in a continuous function over the entire simulation space. Manifolds can be extended to more complex spaces, such as the surface of a molecule, nanoparticle, polymer, or protein. In these approaches, the surface is treated as a manifold and physical characteristics (e.g., hydrophobicity, charge, forces, curvature) represent functions on the manifold. However, graph and manifold representations of molecular simulation data can be high-dimensional and not directly amenable to common analysis tasks (e.g. classification, regression, visualization). Thus, there is need for simple and computationally efficient methods for reducing and quantifying these topological data representations. This characterization can be accomplished by performing a decomposition of the space into a set of independent topological bases that capture basic topological features such as holes, connected components, and voids. The EC is a scalar integer quantity that is defined as the alternating sum of the rank of these topological bases of an object. The EC is often combined with a data processing technique known as filtration, which enables the characterization of more complex topological objects such as matrices, images, fields/functions, and weighted graphs. The filtration process gives rise to the so-called EC curve, which is a function that summarizes how topological features emerge and disappear through the filtration process. Compared to descriptors obtained from persistent homology (e.g., persistence diagrams), the EC curve provides a quantifiable and easy-to-interpret descriptor of complex data objects.