Contextual Matrix Orderings for Graph Collections

Visualizing a graph directly via its adjacency matrix is a common and effective technique. Such matrix visualizations rely crucially on a good ordering of the vertices to highlight intrinsic patterns in the graph. When analyzing collections of graphs, such as time varying sequences or connectivity information ranging over multiple specimens, the user currently needs to make the choice: either order each graph individually to optimize its ordering quality, or use a single, simultaneous ordering for all graphs in the collection, which necessarily reduces the ordering quality for the individual graphs. In this paper we explore the space of contextual orderings that lie between these two extremes. Intuitively, contextual orderings maintain a higher level of consistency than individual orderings and deliver a higher ordering quality than simultaneous orderings. To formally reason about contextual orderings we define a distance measure between orderings which is based on individual block moves (IBM). The IBM distance allows us to relate consistency within the context of the collection with ordering quality. Specifically, we define the consistency of an ordering as the IBM distance to the simultaneous ordering for the collection. Our experiments show that already at a small IBM distance to the simultaneous ordering we can find contextual orderings with significantly improved ordering quality. Furthermore, we can create orderings that are nearly as good as individual orderings, but exhibit considerably improved consistency. We hence believe that contextual orderings can enable a more fine-grained analysis of graph collections, by allowing the user to focus on individual graphs while maintaining a sense of the context they appear in.