  Capabilities for collecting and storing data on mobile objects have increased
dramatically over the past few decades. A persistent difficulty is summarizing
large collections of mobile objects. This paper develops methods for extracting
and analyzing hotspots or locations with relatively high levels of mobility
activity. We use kernel density estimation (KDE) to convert a large collection
of mobile objects into a smooth, continuous surface. We then develop a
topological algorithm to extract critical geometric features of the surface;
these include critical points (peaks, pits and passes) and critical lines
(ridgelines and course-lines). We connect the peaks and corresponding
ridgelines to produce a surface network that summarizes the topological
structure of the surface. We apply graph theoretic indices to analytically
characterize the surface and its changes over time. To illustrate our approach,
we apply the techniques to taxi cab data collected in Shanghai, China. We find
increases in the complexity of the hotspot spatial distribution during normal
activity hours in the late morning, afternoon and evening and a spike in the
connectivity of the hotspot spatial distribution in the morning as taxis
concentrate on servicing travel to work. These results match with scientific
and anecdotal knowledge about human activity patterns in the study area.
