This graph (network) representation can be used to understand how water is interacting both locally and globally, by quantifying specific topological features, such as the number of cycles and connected components of the graph. A cycle represents a path that traverses edges on a graph starting at a particular vertex v j and ending at that same vertex v j . Physical examples of graph cycles are found in tetramer, pentamer, and hexamer water structures. A connected component is a subset of a graph C(V C , E C ) ⊆ G(V, E) in which any vertex v i ∈ V C of the subgraph can reach any other vertex v j ∈ V C by traversing edges of the subgraph {v i , v j } ∈ E C , and is disconnected from all other subsets of the graph. In other words, the number of connected components is the number of connected partitions of a graph. In a hydrogen bonding network for water, connected components help us understand the physical state of the system; for example, in a condensed (or crystalline) state, there will be many hydrogen bonds present, reducing the total number of connected components but increasing the total number of molecules in each connected component. The opposite would hold true for a system acting as an ideal gas; here, no bonds are formed and thus each molecule exists as its own connected component. Data can also be encoded in a graph object (in nodes and edges) using functions f : V → R and f : E → R. Values attached to nodes or edges are typically called weights or features; as such, graphs that encode data are also known as weighted graphs. The graph topology is quantified by subtracting the total number of cycles from the total number of connected components in the graph (χ = 12 -2 = 10). (e) The manifold is quantified through a filtration. At multiple increasing density thresholds k i ∈ R, the EC χ i is computed by subtracting the total number of holes from the total number of connected components in the filtered manifold x ∈ M : f (x) ≤ k i . We note that filtered manifolds all originate from the same data object and that the vertical layout is meant to illustrate the topological changes as the filtration is performed. The paired values {k i , χ i } are used to construct an Euler characteristic curve.