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.

翻译：在众多相似自推进实体（如鸟群、鱼群、细胞和分子）组成的系统中，个体与邻居的相互作用可导致“相干态”，即因速度和方向的局部调整而形成视觉上引人注目的聚集模式。本研究探讨集体模式研究中的一个开放性问题：当此类被视为粒子的实体从无序（随机）状态趋向相干态时，转变的时间间隔是多少？此外，模型参数如何影响这一转变时间间隔？基于给定时间段内粒子运动的离散时间点采样点云数据，我们利用拓扑数据分析（特别是持续同调）推断粒子发生状态转变的时间间隔。通过持续同调（具体为持续景观）捕捉任意时刻粒子构型的拓扑特征。我们通过对时间演化子集对应的持续同调输出进行统计显著性检验（即函数假设检验），定位（在时间上）此类转变发生的时刻。该过程在已知的自推进粒子集体行为模型上得到验证，其中通过随时间改变模型参数触发状态转变。作为应用，所开发的技术最终用于描述环境改变时细胞运动从无序态向集体运动的转变。