  In a system of many similar self-propelled entities such as flocks of birds,
fish school, cells and molecules, the interactions with neighbors can lead to a
"coherent state", meaning the formation of visually compelling aggregation
patterns due to the local adjustment of speed and direction. In this study, we
explore one of the open questions that arise in studying collective patterns.
When such entities, considered here as particles, tend to assume a coherent
state beginning from an incoherent (random) state, what is the time interval
for the transition? Also, how do model parameters affect this transition time
interval? Given the observations of particle migration over a given time period
as a point cloud data sampled at discrete time points, we use Topological Data
Analysis, specifically persistent homology, to infer the transition time
interval in which the particles undergo regime change. The topology of the
particle configuration at any given time instance is captured by the persistent
homology specifically Persistence Landscapes. We localize (in time) when such a
transition happens by conducting the statistical significance tests namely
functional hypothesis tests on persistent homology outputs corresponding to
subsets of the time evolution. This process is validated on a known collective
behavior model of the self-propelled particles with the regime transitions
triggered by changing the model parameters in time. As an application, the
developed technique was ultimately used to describe the transition in cellular
movement from a disordered state to collective motion when the environment was
altered.
