Readability criteria, such as distance or neighborhood preservation, are often used to optimize node-link representations of graphs to enable the comprehension of the underlying data. With few exceptions, graph drawing algorithms typically optimize one such criterion, usually at the expense of others. We propose a layout approach, Graph Drawing via Gradient Descent, $(GD)^2$, that can handle multiple readability criteria. $(GD)^2$ can optimize any criterion that can be described by a smooth function. If the criterion cannot be captured by a smooth function, a non-smooth function for the criterion is combined with another smooth function, or auto-differentiation tools are used for the optimization. Our approach is flexible and can be used to optimize several criteria that have already been considered earlier (e.g., obtaining ideal edge lengths, stress, neighborhood preservation) as well as other criteria which have not yet been explicitly optimized in such fashion (e.g., vertex resolution, angular resolution, aspect ratio). We provide quantitative and qualitative evidence of the effectiveness of $(GD)^2$ with experimental data and a functional prototype: \url{http://hdc.cs.arizona.edu/~mwli/graph-drawing/}.

中文翻译：

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通过梯度下降绘制图形，$(GD)^2$

可读性标准，例如距离或邻域保留，通常用于优化图的节点链接表示，以实现对基础数据的理解。除了少数例外，图形绘制算法通常会优化一个这样的标准，通常以牺牲其他标准为代价。我们提出了一种布局方法，Graph Drawing via Gradient Descent，$(GD)^2$，可以处理多个可读性标准。$(GD)^2$ 可以优化任何可以用平滑函数描述的标准。如果标准不能被平滑函数捕获，则该标准的非平滑函数与另一个平滑函数组合，或者使用自动微分工具进行优化。我们的方法很灵活，可以用来优化之前已经考虑过的几个标准（例如，获得理想的边长、应力、邻域保留）以及其他尚未以这种方式明确优化的标准（例如，顶点分辨率、角度分辨率、纵横比）。我们通过实验数据和功能原型提供了 $(GD)^2$ 有效性的定量和定性证据：\url{http://hdc.cs.arizona.edu/~mwli/graph-drawing/}。