  Undirected graphs are frequently used to model 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 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 exist as part of a collection of graphs on the same set of
vertices. 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. The popular leaf order method heuristically optimizes a
similar measure which supports the use of Moran's I in this context.
  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.
