Distance Adjustment of a Graph Drawing Stress Model

TL;DR

This study proposes two different methods of adjusting the graph-theoretical distance matrix to a distance matrix suitable for graph drawing while preserving its structure and demonstrates that the proposed method improves some quality metrics, including the node resolution and the Gabriel graph property, when compared to conventional stress models.

Abstract

Stress models are a promising approach for graph drawing. They minimize the weighted sum of the squared errors of the Euclidean and desired distances for each node pair. The desired distance typically uses the graph-theoretic distances obtained from the all-node pair shortest path problem. In a minimized stress function, the obtained coordinates are affected by the non-Euclidean property and the high-dimensionality of the graph-theoretic distance matrix. Therefore, the graph-theoretic distances used in stress models may not necessarily be the best metric for determining the node coordinates. In this study, we propose two different methods of adjusting the graph-theoretical distance matrix to a distance matrix suitable for graph drawing while preserving its structure. The first method is the application of eigenvalue decomposition to the inner product matrix obtained from the distance matrix and the obtainment of a new distance matrix by setting some eigenvalues with small absolute values to zero. The second approach is the usage of a stress model modified by adding a term that minimizes the Frobenius norm between the adjusted and original distance matrices. We perform computational experiments using several benchmark graphs to demonstrate that the proposed method improves some quality metrics, including the node resolution and the Gabriel graph property, when compared to conventional stress models.

Figures (5)

• Figure 1: Drawing results of qh882 and dwt_1005 graphs using our propoed methods Low-rank SGD (LR-SGD), Distance-adjusted FullSGD (DAF-SGD), and Distance Adjuted SparseSGD (DAS-SGD). $p$ in LR-SGD and $k$ in DAF-SGD an DAS-SGD are parameters expressing the strength of distance adjustment. The stronger the ditance adjustment, the more the detailed structure of the graph is simplified and the overall rough structure is emphasized.
• Figure 2: Examples of distance relations and complete graph drawing of degree 4. In Case (a), the nodes that satisfy the given distance relationship cannot be arranged in a two-dimensional space. In (b), the distance relationship similar to that in (a) is possible in a three-dimensional space. In (c), the distance relationship cannot be satisfied, even in a high-dimensional space.
• Figure 3: Box plots showing th change in the quality metrics caused by the distance adjustment in the qh882, dwt_1005, 1138_bus, and USpowerGrid graphs.
• Figure 4:
• Figure 6: Heatmap of the number of graphs showing a quality metric improvement of 10% or more.