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A point-to-point process describes a dynamic network where a set of edge events are observed, each of which is associated with a time of occurrence and two vertices lying in their state spaces. This study intends to investigate one application of such processes, using NYC Taxi and Limousine Commission dataset that reports taxi trips between two locations at a certain time. Here a point-to-point process is formed with edge events being taxi trips and the vertices adjacent to the edge events are pick-up and drop-off locations, described by latitude and longitude pairs. The intensity of an edge event can have a temporal dependence in addition to being dependent on a latent, spatially-coherent community structure for the vertices. To this end, we have developed a methodology that estimates a spatially smoothed community structure and localizes temporal change-points for point-to-point processes. By applying this to our dataset, we can explore the spatio-temporal dynamics of the demand of taxi trips. More specifically, with reasonable assumptions, the latent community structure is estimated by spectral partitioning based on a low-rank reconstruction of aggregated taxi-trip network; and the temporal change-point localization can be carried out by solving a matrix group fused LASSO program.