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.