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.

翻译：应力模型是一种有前景的图绘制方法，它通过最小化每个节点对的欧氏距离与期望距离的加权平方误差和来实现节点布局。期望距离通常采用由全节点对最短路径问题计算得到的图论距离。在最小化应力函数的过程中，获得的节点坐标会受到图论距离矩阵的非欧几里得特性及高维性质的影响。因此，应力模型中使用的图论距离未必是确定节点坐标的最佳度量标准。本研究提出两种在保持距离矩阵结构的前提下，将其调整为更适合图绘制的方法：第一种方法是对距离矩阵的内积矩阵进行特征值分解，通过将绝对值较小的特征值置零获得新距离矩阵；第二种方法是在应力模型中添加调整后距离矩阵与原始距离矩阵之间的弗罗贝尼乌斯范数最小化项。通过多个基准图的数值实验表明，与经典应力模型相比，所提方法在节点分辨率与加布里埃尔图性质等质量指标上均有改善。