  Recent advances in microscopy techniques make it possible to study the
growth, dynamics, and response of complex biophysical systems at single-cell
resolution, from bacterial communities to tissues and organoids. In contrast to
ordered crystals, it is less obvious how one can reliably distinguish two
amorphous yet structurally different cellular materials. Here, we introduce a
topological earth mover's (TEM) distance between disordered structures that
compares local graph neighborhoods of the microscopic cell-centroid networks.
Leveraging structural information contained in the neighborhood motif
distributions, the TEM metric allows an interpretable reconstruction of
equilibrium and non-equilibrium phase spaces and embedded pathways from static
system snapshots alone. Applied to cell-resolution imaging data, the framework
recovers time-ordering without prior knowledge about the underlying dynamics,
revealing that fly wing development solves a topological optimal transport
problem. Extending our topological analysis to bacterial swarms, we find a
universal neighborhood size distribution consistent with a Tracy-Widom law.
