Topological metric detects hidden order in disordered media

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 biofilms to tissues. 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 distance between disordered structures that compares local graph neighborhoods of the microscopic cell-centroid networks. Building on an efficient algorithmic implementation, we first show that this metric can reliably distinguish between random ellipsoid packings of various aspect ratios and polydispersivity. We then demonstrate the broad applicability of this framework to non-equilibrium systems by analyzing synthetic data from active Brownian particle simulations at different values of particle density and activity; in this case, our approach succeeds in reconstructing the two-dimensional activity-density phase-space from static simulation snapshots alone. Finally, by measuring the topological distance between unsorted experimental images of fly embryo wings in various developmental stages, we are able to recover their temporal ordering without prior knowledge about the underlying dynamics.