Data-Spatial Layouts for Grid Maps

Grid maps are a well-known technique to visualize data associated with spatial regions. A grid map assigns each region to a tile in a grid (often orthogonal or hexagonal) and then represents the associated data values within this tile. Good grid maps represent the underlying geographic space well: regions that are geographically close are close in the grid map and vice versa. Though Tobler’s law suggests that spatial proximity relates to data similarity, local variations may obscure clusters and patterns in the data. For example, there are often clear differences between urban centers and adjacent rural areas with respect to socio-economic indicators. To get a better view of the data distribution, we propose grid-map layouts that take data values into account and place regions with similar data into close proximity. In the limit, such a data layout is essentially a chart and loses all spatial meaning. We present an algorithm to create hybrid layouts, allowing for trade-offs between data values and geographic space when assigning regions to tiles. Our algorithm also handles hierarchical grid maps and allows us to focus either on data or on geographic space on different levels of the hierarchy. Leveraging our algorithm we explore the design space of (hierarchical) grid maps with a hybrid layout and their semantics

Reordering for Matrix Visualization with Reorder.js

International audienceEffective matrix visualizations rely heavily on the order of the vertices. The patterns that appear thanks to such an ordering provide information about the structure of the graph. However, implementing reordering algorithms is not trivial. Reorder.js is a JavaScript library that provides several algorithms for matrix reordering. This library has now been updated with the state-of-the-art measure Moran’s I, algorithms based on Moran’s I, and simultaneous reordering for multiple matrices. The poster shows the results of the new algorithms applied to several visualizations. Finally, our examples aim to remind practitioners of the importance of reordering in visualization, e.g., for Parallel Coordinate Plots and tables

Reordering for Matrix Visualization with Reorder.js

International audienceEffective matrix visualizations rely heavily on the order of the vertices. The patterns that appear thanks to such an ordering provide information about the structure of the graph. However, implementing reordering algorithms is not trivial. Reorder.js is a JavaScript library that provides several algorithms for matrix reordering. This library has now been updated with the state-of-the-art measure Moran’s I, algorithms based on Moran’s I, and simultaneous reordering for multiple matrices. The poster shows the results of the new algorithms applied to several visualizations. Finally, our examples aim to remind practitioners of the importance of reordering in visualization, e.g., for Parallel Coordinate Plots and tables

Simultaneous Matrix Orderings for Graph Collections

Undirected graphs are frequently used to model phenomena that deal with interacting objects, such as social networks, brain activity and communication networks. The topology of an undirected graph G can be captured by an adjacency matrix; this matrix in turn can be visualized directly to give insight into the graph structure. Which visual patterns appear in such a matrix visualization crucially depends on the ordering of its rows and columns. Formally defining the quality of an ordering and then automatically computing a high-quality ordering are both challenging problems; however, effective heuristics exist and are used in practice. Often, graphs do not exist in isolation but as part of a collection of graphs on the same set of vertices, for example, brain scans over time or of different people. To visualize such graph collections, we need a single ordering that works well for all matrices simultaneously. The current state-of-the-art solves this problem by taking a (weighted) union over all graphs and applying existing heuristics. However, this union leads to a loss of information, specifically in those parts of the graphs which are different. We propose a collection-aware approach to avoid this loss of information and apply it to two popular heuristic methods: leaf order and barycenter. The de-facto standard computational quality metrics for matrix ordering capture only block-diagonal patterns (cliques). Instead, we propose to use Moran's I, a spatial auto-correlation metric, which captures the full range of established patterns. Moran's I refines previously proposed stress measures. Furthermore, the popular leaf order method heuristically optimizes a similar measure which further supports the use of Moran's I in this context. An ordering that maximizes Moran's I can be computed via solutions to the Traveling Salesperson Problem (TSP); orderings that approximate the optimal ordering can be computed more efficiently, using any of the approximation algorithms for metric TSP. We evaluated our methods for simultaneous orderings on real-world datasets using Moran's I as the quality metric. Our results show that our collection-aware approach matches or improves performance compared to the union approach, depending on the similarity of the graphs in the collection. Specifically, our Moran's I-based collection-aware leaf order implementation consistently outperforms other implementations. Our collection-aware implementations carry no significant additional computational costs

Crossing Numbers of Beyond-Planar Graphs Revisited

Graph drawing beyond planarity focuses on drawings of high visual quality for non-planar graphs which are characterized by certain forbidden (edge) crossing configurations. A natural criterion for the quality of a drawing is the number of edge crossings. The question then arises whether beyond-planar drawings have a significantly larger crossing number than unrestricted drawings. Chimani et al. [GD'19] gave bounds for the ratio between the crossing number of three classes of beyond-planar graphs and the unrestricted crossing number. In this paper we extend their results to the main currently known classes of beyond-planar graphs characterized by forbidden edge configurations and answer several of their open questions