IEEE Transactions on Visualization and Computer Graphics (Proc. IEEE VIS 2022), to appear

Taurus: Towards A Unified Force Representation and Universal Solver for Graph Layout

Mingliang Xue, Zhi Wang, Fahai Zhong, Yong Wang, Oliver Deussen, Yunhai Wang


Over the past few decades, a large number of graph layout techniques have been proposed for visualizing graphs from various domains. In this paper, we present a general framework, Taurus, for unifying popular techniques such as the spring- electrical model, stress model, and maxent-stress model. It is based on a unified force representation, which formulates most existing techniques as a combination of quotient-based forces that combine power functions of graph-theoretical and Euclidean distances. This representation enables us to compare the strengths and weaknesses of existing techniques, while facilitating the development of new methods. Based on this, we propose a new balanced stress model (BSM) that is able to layout graphs in superior quality. In addition, we introduce a universal augmented stochastic gradient descent (SGD) optimizer that efficiently finds proper solutions for all layout techniques. To demonstrate the power of our framework, we conduct a comprehensive evaluation of existing techniques on a large number of synthetic and real graphs. We release an open-source package, which facilitates easy comparison of different graph layout methods for any graph input as well as effectively creating customized graph layout techniques.



Figure 1: (a) Resultant force for two nodes with varying graph-theoretical and Euclidian distances generated by our balanced SM. (b) Layouts generated by SM and balanced SM (c) on the graph cluster_2000. Our method clearly separates different clusters. (d,e) The boxplots summarize the value ranges (same for all boxplots in this paper) of the stress error (d) in terms of different graph distances and neighborhood preservation (e) degrees for different ring sizes.


Figure 2: Comparison of layout results on the same data generated by the same methods using the existing implementation (left) and our implementation (right). (a,b) Results generated by FDP; (c,d) results generated by Maxent.

bsm visual

Figure 3: SM results of four graphs (top down: grid1000, btree513, cluster4463, and USpowergrid) with different structures in comparison to baseline methods (SM, FDP, LinLog, Maxent and FM3) implemented by Taurus.


This work was supported by the grants of the National Key Research and Development Program of China (2018AAA0101100), and NSFC (62132017, 62141217). Oliver Deussen was funded by the German Research Foundation (DFG) - Project-ID 2,51654672 - TRR 161.

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