Efficient and stable circular cartograms for time-varying data by using improved elastic beam algorithm and hierarchical optimization

The circular cartogram, also known as the famous DorlingMap, is widely used to visualize geographical statistics by representing geographical regions as circles. However, all existing approaches for circular cartograms are only designed for static data. While applying these approaches for time-varying data, the circle locations in each circular cartogram are recomputed separately and will result in low efficiency and low visual stability between sequential circle cartograms. To generate visually stable circular cartograms for time-varying data efficiently, we propose a novel approach by improving the elastic beam algorithm with a hierarchical optimization strategy. First, the time-varying data at different time points are grouped using a hierarchical clustering method based on their similarity, and a hierarchy is then built for their corresponding circular cartograms. Second, we generate intermediate circle locations level by level for clusters of circular cartograms according to the built hierarchy with an improved elastic beam algorithm iteratively. The elastic beam algorithm is improved in its proximity graph construction and force computation by considering that the algorithm will be applied to displace circles in a cluster of circular cartograms. The iterative process stops until we obtain satisfactory circular cartograms for each time point. The evaluation results indicate that the proposed approach can achieve a higher quality (184.85%up arrow and 265.69%up arrow) on visual stability, and a higher efficiency (58.54%up arrow and 73.96%up arrow) with almost the same quality on overlap avoidance and relation maintenance by comparing to the existing approaches. Project website: https:// github.com/TrentonWei/DorlingMap.