Visualizing High-Dimensional Graph Embeddings via Informed Multi-View Projections

# Visualizing High-Dimensional Graph Embeddings via Informed Multi-View Projections
Source: [https://arxiv.org/html/2606.31119](https://arxiv.org/html/2606.31119)
\\onlineid

0\\vgtccategoryResearch

\\authororcidXuefeng Li0009\-0007\-6059\-9747\\authororcidTimo Brand0009\-0004\-3111\-2045\\authororcidJacob Miller0000\-0002\-0567\-785X\\authororcidPeng Zhang0009\-0002\-7858\-3691\\authororcidStephen Kobourov0000\-0002\-0477\-2724and\\authororcidYifan Hu0000\-0003\-2017\-924X

###### Abstract

Graphs are commonly visualized in 2D, where humans readily interpret spatial relationships, yet such layouts often distort higher\-dimensional structure\. We propose to embed graphs in high\-dimensional space and search for informative 2D viewpoints that optimize aesthetic and readability metrics \(e\.g\., edge crossings and angular resolution\), enabled by a novel differentiable surrogate for edge crossings\. Numerical experiments show that these viewpoints consistently outperform standard 2D layouts, and can even surpass methods explicitly designed to optimize these metrics\. We further introduce\\DataFly, an interactive system for exploring multiple candidate viewpoints through seamless navigation\. A usability study demonstrates that our approach reveals structural patterns that remain hidden in conventional 2D visualizations\.

###### keywords:

Graph visualization, dimension reduction

\\teaser![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x1.png)

Example visualizations produced from high\-dimensional projections\. \(Top\) shows a 7D hypercube embedded via\\sfdpin 10D and projected to optimize stress, t\-SNE score, edge crossings, angular resolution, and edge length variance, respectively\. \(Bottom\) shows the classic football network embedded via a spectral embedding in 10 dimensions, from five different axis\-aligned viewpoints\. The set of projections reveals that class memberships are spatially salient in some projections and disappear in others, highlighted by the circled cluster in the first two projections\.

\\shortauthortitle

Jiet al\.: Visualizing High\-Dimensional Graph Embeddings

\\authorfooter

Ya Ji, Xuefeng Li, Peng Zhang and Yifan Hu are with the Khoury College of Computer Sciences, Northeastern University, Seattle, WA, USA\. E\-mail: \{ji\.ya1 \| li\.xuefen \| zhang\.peng2, yif\.hu\}@northeastern\.edu\.

Timo Brand, Jacob Miller and Stephen Kobourov are with the School of Computation, Information and Technology, Technical University of Munich, Heilbronn, Germany\. E\-mail: \{timo\.brand \| jacob\.miller \| stephen\.kobourov\}@tum\.de\.

Introduction

Graph visualization converts abstract relationships into readable graphics, enabling users to understand complex relational data, uncover patterns, and make informed decisions\. Traditionally, graphs are visualized in 2D, reflecting human\-interpretable spatial relationships\. However, large real\-world graphs often exhibit characteristics reminiscent of high\-dimensional data: the small\-world phenomenon, in which most node pairs are connected by short paths, mirrors the distance concentration observed in high dimensions, and real\-world graphs, like high\-dimensional point clouds, tend to be locally dense and globally sparse\. Two\-dimensional layouts struggle to preserve these higher\-dimensional relationships\. Just as a cube cannot be projected onto a 2D plane with equal edge lengths, complex networks contain local connectivity patterns, such as dense cliques or overlapping communities, that no single 2D projection can faithfully represent\. It follows that a single 2D graph layout, although intuitive and accessible, risks distorting or obscuring the higher\-dimensional structure of the data\.

However, high\-dimensional layouts introduce a practical challenge: presenting them to human readers requires choosing a 2D projection, or viewpoint, at any given time\. Thus, high\-quality viewpoints are necessary for high\-dimensional layouts to be effective visualization tools, but this has received limited exploration in the literature\. Many methods rely on computing a single projection that is optimal under some criteria\[Gajer\_Goodrich\_Kobourov\_2000,PivotGraph,spectral\]\. Van Wageningen et al\.\[Wageningen2025\]investigate the viewpoint problem in the context of 3D drawings, but it remains unclear whether effective projections of higher\-dimensional layouts can be achieved\.

These observations motivate the following research question:Do high\-dimensional graph embeddings contain 2D projections that reveal structural patterns otherwise obscured by traditional static layouts?To investigate this, we develop a computational pipeline to systematically explore high\-dimensional graph layouts, which produces a family of candidate projections\. We further introduce a gradient\-based optimization scheme that finds near\-optimal projections for a given quality metric, such as stress, edge crossings, or angular resolution\. Our findings show that such projections can outperform state\-of\-the\-art algorithms that are explicitly designed to optimize these same metrics directly in 2D\. Throughout the paper, we use the term*optimal projection*as shorthand for the result of this gradient\-based optimization, noting that it converges to a local rather than global optimum\.

To make these high\-dimensional embeddings accessible and explorable, we provide\\DataFly, an interactive online tool for real\-time visualization of high\-dimensional graph layouts\.\\DataFlyenables principled exploration of viewpoints: users can inspect optimal projections for specific metrics, or transition between them by “flying over” the high\-dimensional space, observing how one projection morphs into another\. Because all viewpoints derive from a single shared embedding, diverse visual perspectives on the same network coexist within a coherent spatial structure, allowing users to discover structural patterns that remain hidden in any single 2D layout\.

Our main contributions are as follows:

- •We show that the optimal 2D projection of a high\-dimensional graph embedding can yield better metric values than several state\-of\-the\-art benchmarks that optimize such metrics directly in 2D\.
- •We propose an differentiable loss function, SigmoidX, which enables the effective minimization of edge crossings, and demonstrate that the optimal projection using this loss function achieves significantly fewer crossings thanSGD2\\text\{SGD\}^\{2\}and SmartGD\.
- •We provide a working prototype of\\DataFly, an interactive system for exploring high\-dimensional graph layouts and identifying viewpoints of interest, evaluated by a small usability study with both expert and non\-expert participants\.

While our primary focus is on visualizing graph data with straight\-line node\-link diagrams, the name\\DataFlyreflects the fact that our interactive multi\-view exploration approach naturally extends to general high\-dimensional data\. A video of the system in action can be found at[http://tiny\.cc/datafly\-video](http://tiny.cc/datafly-video)\.\\DataFlyis made publicly available at[https://datafly\.algo\.cit\.tum\.de](https://datafly.algo.cit.tum.de/)\.

In the remainder of the paper,DataFlyrefers to both the interactive system and the associated optimal\-projection framework\.

## 1Related Work

Since 1963\[tutte\_1963\], numerous graph drawing algorithms have been proposed\. A widely used class of methods for generating straight\-line drawings relies on physical models that minimize the energy of the system\. Examples include stress\-based models, which minimize the stress energy of springs connecting nodes\[kamada\_kawai\_1989,neato,zheng\-gd2\], and force\-directed systems of springs and electrical charges, in which springs along edges contract while electrical charges on nodes repel each other\[Eades\_1984,Fruchterman\_Reingold\_1991,kobourov\_2013\]\.

### 1\.1Layout algorithms for optimizing specific metrics

There are a number of layout methods that specifically target aesthetic metrics, such as minimizing edge crossings\[xing\-heuristic,spx\], maximizing crossing angles\[Argyriou,Bekos\-xangle\], or a combination\[didimo\]\. Two notable algorithms areSGD2\\text\{SGD\}^\{2\}\[sgd2\]and SmartGD\[wang\_2023\_smartgd\]\.

SGD2\\text\{SGD\}^\{2\}is a method that can jointly optimize multiple graph readability criteria\. In particular, it can optimize any criterion that can be described by a differentiable function\. When a criterion is not a differentiable function \(e\.g\., edge crossings\), a proxy objective is used instead\.SGD2\\text\{SGD\}^\{2\}supports optimization of angular resolution, edge crossings, and desired edge lengths, among other criteria\. SmartGD\[wang\_2023\_smartgd\]is a deep\-learning framework for graph drawing based on generative adversarial networks\. It can learn from examples of a “good” layout, where “good” may be defined by even non\-differentiable metrics, e\.g\., edge crossings\. WhileS​G​D2SGD^\{2\}and SmartGD are among the first graph embedding methods that can simultaneously optimize multiple criteria, neither works well beyond small graphs \(\|V\|≈100\|V\|\\approx 100\)\. The optimal projection algorithm proposed in this paper performs well on both small and large graphs \(over 1000 nodes\) while supporting multiple criteria\.

While joint optimization seeks to be effective at minimizing a given metric while preserving the aesthetic appeal of the layout, it will not necessarily generate a drawing with any one criterion optimal\. Of particular interest to the algorithmic community are algorithms that optimize a specific metric, possibly at the expense of readability\.

One such algorithm isVertex Movement\[radermacher2019geometric\]\(VM\), which optimizes edge crossings while ignoring other aesthetic criteria\. The algorithm starts with an initial layout, iterates over the vertices, and tries to find the crossing\-minimal position for the current vertex\. While not guaranteed to find an optimal drawing, it achieves strong performance in finding drawings with few edge crossings\. We use VM as a state\-of\-the\-art baseline for crossing minimization, but note that this algorithm has high computational complexity, and cannot be applied to large graphs\.

### 1\.2Understanding high\-dimensional data through projection\.

Paulovich et al\.\[paulovich2025dimensionality\]recently linked dimensionality reduction and graph drawing, showing their close relationship\. Algorithms and interactive systems for exploring high\-dimensional data via projections have long been studied, with early work in the statistics and physics communities\[buja:1996:XGobi,cook1995grand,cutura2018viscoder,Dang2014ScagExplorer,laa2020high,huh2002visualization,Morariu\_2023\]\. Notable examples include the R\-based visualization tool XGobi\[buja:1996:XGobi\]and the Embedding Projector\[smilkov2016embedding\]\.

GraphDice\[GraphDice2010\]is a relevant tool for exploring and analyzing graph data\. It supports exploration of multivariate social network graphs through animated transitions between nearby views\. However, it operates on observed actor and edge attributes rather than on geometric projections of a high\-dimensional graph embedding\. EvoGraphDice\[EvoGraphDice2012\]extends this line of work to general multidimensional data, using interactive evolutionary search to propose specific views based on user feedback\. It is therefore relevant as a visual analytics approach to guided viewpoint discovery, but it is not designed for graph layouts or for optimizing graph\-drawing metrics\.

As far as we are aware, there is little prior work focused on understanding high\-dimensional graph embeddings \(beyond 2/3D\) via optimal projection\. Gajer et al\.\[Gajer\_Goodrich\_Kobourov\_2000\]demonstrated that laying out a graph in a high\-dimensional space and randomly projecting it back to 2D can produce interesting drawings, illustrated using a Möbius strip example; however, their work does not aim to identify optimal views\. Also related is work by Gaertler and Krug, who used animations between different viewpoints of a spectral graph embedding\[gaertler05\]\.

Li et al\.\[li2020visualizing\]investigated methods for visualizing multidimensional data arising from neural networks\. Although they examined nonlinear dimensionality reduction techniques such as t\-SNE and UMAP, linear projections were favored due to their predictable response to changes in the original data\. Etemadpour et al\.\[Etemadpour2015\]conducted a perception\-based user study on multidimensional projections, showing that projection performance depends on tasks and data characteristics\. The study shows that no single projection method yields universally optimal layouts, which supports the argument for generating and evaluating multiple views of high\-dimensional graph layouts\.

Martins et al\.\[Martins2012MDProjection\]introduce multidimensional projection methods for social network visualization, where node placement is guided by node similarity to produce meaningful 2D layouts without heavy force\-based optimization\. The work demonstrates that high\-dimensional embeddings and projection\-driven views can reveal structural patterns beyond conventional graph drawing, reinforcing the motivation for multi\-view exploration of high\-dimensional graph layouts\.

It has been shown that 2D viewpoint drawings acquired from 3D drawings can be of higher quality than 2D drawings produced with the same algorithm\[Wageningen2025\]\. This result is achieved through efficient optimization algorithms, including gradient descent and evolutionary\-inspired metaheuristics, that automatically find high\-quality viewpoints for 3D straight\-line graph drawings, replacing slow brute\-force sampling methods that require evaluating thousands of viewpoints\. Although this viewpoint optimization is similar to the optimal projection found in\\DataFly, we aim to find good viewpoints for a K\-D graph embedding instead\. Recent findings\[joos2025show\]show that, after creating such a 3D drawing, users prefer to view it from angles that yield 2D projections with high values for several 2D quality metrics\. Similar findings were shown for Dimensionality Reduction \(DR\) plots\[castelein2023based\]\. We primarily consider linear projections, though a rich set of non\-linear DR techniques exist; see recent surveys\[DBLP:journals/tvcg/NonatoA19,DBLP:journals/tvcg/EspadotoMKHT21\]\.

## 2Understanding Graphs through High\-Dimensional Layouts

Here, we give a high level overview of our methodology\. We use standard graph definitions:G=\(V,E\)G=\(V,E\), whereVVdenotes the set of vertices andEEdenotes the set of edges\. We posit that embedding graphs in a high\-dimensional space enables a more faithful representation of their intrinsic structure, such as local neighborhoods\. To this end, we first embed the graph inKK\-dimensional space \(e\.g\.,KK= 10\), using an off\-the\-shelf algorithm\. What remains is to investigate potentially interesting 2D viewpoints of this embedding for visualization\.

[Figure˜1](https://arxiv.org/html/2606.31119#S2.F1)illustrates how different projections of the same 10D embedding can reveal qualitatively different structures\. However, it is not clear that such viewpoints offer any real advantages for real\-world data\. We posit the following research questions:

RQ1\. Do high\-dimensional graph embeddings, when combined with carefully designed projection methods, offer advantages over direct 2D layouts in optimizing specific aesthetic metrics?

RQ2\. Can exploring systematic projections of high\-dimensional viewpoints help users understand graph structure?

![Refer to caption](https://arxiv.org/html/2606.31119v1/x2.png)Figure 1:Several projections of an 10D spectral embedding of a subdivided dodecahedron graph\. The leftmost projection looks like a typical embedding, but other viewpoints reveal the structure twists and wraps on itself in the ambient space\.We addressRQ1through the optimal viewpoint formulations developed in this section and the quantitative evaluation in[Section˜3](https://arxiv.org/html/2606.31119#S3)\. We addressRQ2later through qualitative examples \([Section˜3\.3](https://arxiv.org/html/2606.31119#S3.SS3)\) and by introducing an interactive tool,\\DataFly, evaluated through a usability study \([Section˜5](https://arxiv.org/html/2606.31119#S5)\)\.

In this section, we propose carefully designed 2D projections of high\-dimensional layouts, referred to asviewpoints, to addressRQ1\. Our goal is to identify viewpoints that are most likely to provide informative insights into the graph structure\.

### 2\.1High\-dimensional layouts

Many graph layout algorithms generalize naturally to high dimensions\. In our work we consider four approaches: a stress model\[Gansner\_Koren\_North\_2005b\], a force\-directed algorithm\[Hu\_2004\], the spectral embedding algorithm\[Hall\_1970\_eigenplacement\], and PivotMDS\[Brandes\_Pich\_2006\]\. However, the pipeline we describe works for any embedding algorithm in principle\.

For the stress model, we useneato\[Gansner\_Koren\_North\_2005b\]from the Graphviz library\[graphviz\]\. Stress\-based optimization is one of the standard 2D graph embedding methods and is available in most software platforms, ranging from Graphviz and NetworkX\[networkx\], to OGDF\[Chimani2014OGDF\]and yFiles\[Wiese2004\_yfiles\]\. Stress\-based layouts are often included as a baseline when evaluating new methods, and the implementation in Graphviz is a trusted standard\. The notion of stress \([Eq\.˜3](https://arxiv.org/html/2606.31119#S2.E3)\) is based on Euclidean distances and therefore well\-defined in high dimensions\. Human\-subject studies\[Chimani\_2014\_stress,Mooney25\]have shown that lower stress is an important predictor of aesthetic preference in graph visualization\.

Perhaps the only graph embedding methods as popular as stress\-based models are force\-directed methods\[kobourov\_2013\]\. These methods typically minimize a spring\-electrical energy \([Eq\.˜5](https://arxiv.org/html/2606.31119#S2.E5)\) that captures attractive forces between pairs of nodes connected by edges, and repulsive forces between all pairs of nodes\. As a representative of force\-directed algorithms, we use\\sfdp\[Hu\_2004\]from the Graphviz library\[graphviz\]\. This is an efficient implementation frequently used as a baseline for evaluation\.

We evaluate two additional algorithms as they naturally lay out graphs in up to\|V\|\|V\|dimensions\. The spectral algorithm\[Hall\_1970\_eigenplacement\]is based on taking the topKK\-eigenvectors of the Laplacian corresponding to the lowestKK\-eigenvalues\. We use the implementation in the NetworkX library\[networkx\]\. We also include the PivotMDS algorithm\[Brandes\_Pich\_2006\], a scalable version of the classical MDS algorithm\[Torgerson\_MDS\_1952\]\. It works by pickingKKpivot nodes, forming the squared and double\-centered distance matrix to the pivots of dimension\|V\|×K\|V\|\\times K, and performing Principal Component Analysis \(PCA\) on this matrix\. We re\-implemented this based on the description in\[Brandes\_Pich\_2006,hu\_2012\_large\_networks\]\.

Given the wide popularity of the stress metric, we present most of the results for projections of high\-dimensional layouts produced byneato, alongside corresponding results for\\sfdp\. Additional results are given in the Appendix\. We also limit our evaluation to up to 10D embeddings, as GraphViz officially supports graph layouts in at most 10 dimensions\.

Although higher dimensions provide more freedom for the layout to preserve connectivity, the final display of information to human readers is limited to the 2 dimensions of the computer screen\. We must therefore rely on informative 2D projections to interpret these high\-dimensional layouts\.

### 2\.2Optimal Viewpoints

We define aviewpointas a specific 2D projection of aKK\-D layout of a graph\. While there are infinitely many possible viewpoints, some of them are more interesting than others\.

Anoptimal viewpointis defined as the projection fromKK\-D to 2D that minimizes a specific metricℒ\\mathcal\{L\}\. For brevity, we refer to locally optimized viewpoints as optimal viewpoints\. We collectively call these viewpoint\-optimizing algorithms\\DataFly\.

LetX∈ℝ\|V\|×KX\\in\\mathbb\{R\}^\{\|V\|\\times K\}denote a high\-dimensional embedding matrix, where each row represents theKK\-D embedding of a node, obtained from an algorithm\. The goal is then to learn a linear projectionP∈ℝK×2P\\in\\mathbb\{R\}^\{K\\times 2\}that mapsXXto a two\-dimensional space while minimizing a loss of the projected layout\. Formally, we solve

minP∈ℝK×2⁡ℒ​\(X​P\)\\min\_\{P\\in\\mathbb\{R\}^\{K\\times 2\}\}\\ \\mathcal\{L\}\(XP\)\(1\)
whereℒ\\mathcal\{L\}is the graph metric, andX​P∈ℝ\|V\|×2XP\\in\\mathbb\{R\}^\{\|V\|\\times 2\}gives the 2D coordinates resulting from the projection\. Note that this optimization involves only2​K2Kvariables \(e\.g\.,K=10K=10\), which is far fewer than the2​\|V\|2\|V\|variables that algorithms for traditional layouts must optimize over, making the problem easier and less computationally expensive to solve\.

To find the optimal projectionPPin \([Eq\.˜1](https://arxiv.org/html/2606.31119#S2.E1)\), we use an iterative gradient descent algorithm, updated as:

Pt\+1=Pt−η​∇Ptℒ​\(X​Pt\),P\_\{t\+1\}=P\_\{t\}\-\\eta\\nabla\_\{P\_\{t\}\}\\mathcal\{L\}\(XP\_\{t\}\),\(2\)whereη\\etais the step size\.

### 2\.3Metrics

Graph layout quality is inherently multi\-objective, and no single metric consistently captures aesthetic quality across different graphs and tasks\. Different metrics instead reflect different, and sometimes competing, objectives—for example, stress emphasizes structural faithfulness, whereas edge crossings emphasize readability\. Therefore, we explore a diverse set of complementary metrics to evaluate viewpoint quality\. We denotexi∈ℝdx\_\{i\}\\in\\mathbb\{R\}^\{d\}the coordinates of nodeiiindddimensions\.

Stress:This metric measures the stress of the layout,

stress=∑i,j∈V,i≠jwi​j​\(α​‖𝐱i−𝐱j‖−di​j\)2,\\text\{stress\}=\\sum\_\{i,j\\in V,i\\neq j\}w\_\{ij\}\\left\(\\alpha\\\|\\mathbf\{x\}\_\{i\}\-\\mathbf\{x\}\_\{j\}\\\|\-d\_\{ij\}\\right\)^\{2\},\(3\)
wheredi​jd\_\{ij\}is the desired distance betweeniiandjj, andwi​jw\_\{ij\}is set to1/di​j21/d\_\{ij\}^\{2\}\. Here, for a given layout, we apply a scaling factor ofα\\alphato normalize the stress \(e\.g\.,\[wang\_2023\_smartgd,smelser2026scale\]\):

α=∑i,j∈V,i≠jwi​j​di​j​‖𝐱i−𝐱j‖∑i,j∈V,i≠jwi​j​‖𝐱i−𝐱j‖2\.\\alpha=\\frac\{\\sum\_\{i,j\\in V,i\\neq j\}w\_\{ij\}d\_\{ij\}\\\|\\mathbf\{x\}\_\{i\}\-\\mathbf\{x\}\_\{j\}\\\|\}\{\\sum\_\{i,j\\in V,i\\neq j\}w\_\{ij\}\\\|\\mathbf\{x\}\_\{i\}\-\\mathbf\{x\}\_\{j\}\\\|^\{2\}\}\.\(4\)
Spring\-Electrical Energy:This metric treats edges as springs and nodes as electrically repelling particles and measures the energy of the system:

se\_eng=∑\(i,j\)∈E‖𝐱i−𝐱j‖33​Ks−∑i≠jKr2​ln⁡\(‖𝐱i−𝐱j‖\)\.\\text\{se\\\_eng\}=\\sum\_\{\(i,j\)\\in E\}\\frac\{\\\|\\mathbf\{x\}\_\{i\}\-\\mathbf\{x\}\_\{j\}\\\|^\{3\}\}\{3K\_\{s\}\}\-\\sum\_\{i\\neq j\}K\_\{r\}^\{2\}\\ln\(\\\|\\mathbf\{x\}\_\{i\}\-\\mathbf\{x\}\_\{j\}\\\|\)\.\(5\)
whereKsK\_\{s\}andKrK\_\{r\}are spring and repulsion constants\. We set them to 1\. To make this metric scale\-invariant, we use a scaling factorttthat minimizes \([Eq\.˜5](https://arxiv.org/html/2606.31119#S2.E5)\)\. We derive an analytical solution \(proof is given in the Appendix\):t=\(\|V\|​\(\|V\|−1\)∑\(i,j\)∈E‖𝐱i−𝐱j‖3\)1/3t=\\left\(\\frac\{\|V\|\(\|V\|\-1\)\}\{\\sum\_\{\(i,j\)\\in E\}\\\|\\mathbf\{x\}\_\{i\}\-\\mathbf\{x\}\_\{j\}\\\|^\{3\}\}\\right\)^\{1/3\}\.

t\-SNE Score:This metric measures Kullback\-Leibler divergence of node distances between the Euclidean layout space and the graph space using t\-distributed stochastic neighbor embedding \(t\-SNE\[Maaten\_2008\_tSNE\]\)\. It evaluates how well the graph neighborhoods are preserved in the layout\.

tsne=∑i,ji≠jpi​j​log⁡pi​jqi​j,\\text\{tsne\}=\\sum\_\{\\begin\{subarray\}\{c\}i,j\\\\ i\\neq j\\end\{subarray\}\}p\_\{ij\}\\log\\frac\{p\_\{ij\}\}\{q\_\{ij\}\},\(6\)where

pi​j=pj\|i\+pi\|j2​N,pj\|i=exp⁡\(−di​j22​σi2\)∑kk≠iexp⁡\(−di​k22​σi2\),qi​j=\(1\+‖𝐱i−𝐱j‖2\)−1∑k,lk≠l\(1\+‖𝐱k−𝐱l‖2\)−1\.p\_\{ij\}=\\frac\{p\_\{j\|i\}\+p\_\{i\|j\}\}\{2N\},p\_\{j\|i\}=\\frac\{\\exp\(\-\\frac\{d\_\{ij\}^\{2\}\}\{2\\sigma\_\{i\}^\{2\}\}\)\}\{\\sum\_\{\\begin\{subarray\}\{c\}k\\\\ k\\neq i\\end\{subarray\}\}\\exp\(\-\\frac\{d\_\{ik\}^\{2\}\}\{2\\sigma\_\{i\}^\{2\}\}\)\},q\_\{ij\}=\\frac\{\(1\+\|\|\\mathbf\{x\}\_\{i\}\-\\mathbf\{x\}\_\{j\}\|\|^\{2\}\)^\{\-1\}\}\{\\sum\_\{\\begin\{subarray\}\{c\}k,l\\\\ k\\neq l\\end\{subarray\}\}\(1\+\|\|\\mathbf\{x\}\_\{k\}\-\\mathbf\{x\}\_\{l\}\|\|^\{2\}\)^\{\-1\}\}\.
The t\-SNE metric is not scale\-invariant\. When computing the optimal 2D projection, to ensure good initialization and a differentiable loss function, we normalize the coordinates using \([Eq\.˜4](https://arxiv.org/html/2606.31119#S2.E4)\)\. However, to ensure a fair comparison across methods, we rescale the coordinates for each layout using a 1\-D minimization step before computing the reported metric\.

Edge crossings: This metric counts the number of edge crossings in the layout, indicating visual clutter\. Fewer crossings suggest a cleaner, more readable layout\.

xing=∑\(i,j\),\(k,l\)∈E,\(i,j\)≠\(k,l\)𝕀​\[edge​\(i,j\)​crosses​\(k,l\)\]\.\\text\{xing\}=\\sum\_\{\(i,j\),\(k,l\)\\in E,\(i,j\)\\neq\(k,l\)\}\\mathbb\{I\}\\left\[\\text\{edge \}\(i,j\)\\text\{ crosses \}\(k,l\)\\right\]\.\(7\)To apply gradient descent for minimizing edge crossing, a differentiable surrogate function is required\.SGD2\\text\{SGD\}^\{2\}\[sgd2\]learns a differentiable proxy for edge crossings by training a neural network to predict whether two edges intersect, based on their node coordinates\. We instead propose a simpler approach\. Lett,ut,ube the intersection parameters in[Fig\.˜2](https://arxiv.org/html/2606.31119#S2.F2)\(left\)\. Geometrically, whent∈\[0,1\]t\\in\[0,1\], this means that the second edge, or its extension, hits the first edge\. The parameteruuis defined symmetrically tott\. Hence, if the two edges cross, we must have that botht,u∈\[0,1\]t,u\\in\[0,1\]\. But instead of using such a hard, non\-differentiable condition to determine whether edges intersect, we use sigmoid\-based soft masks:

Mt=σ​\(t\)​\[1−σ​\(t−1\)\]Mpeak,Mu=σ​\(u\)​\[1−σ​\(u−1\)\]Mpeak,M\_\{t\}=\\frac\{\\sigma\(t\)\\,\[1\-\\sigma\(t\-1\)\]\}\{M\_\{\\text\{peak\}\}\},\\qquad M\_\{u\}=\\frac\{\\sigma\(u\)\\,\[1\-\\sigma\(u\-1\)\]\}\{M\_\{\\text\{peak\}\}\},whereσ​\(t\)=11\+e−T⋅t\\sigma\(t\)=\\frac\{1\}\{1\+e^\{\-T\\cdot t\}\}is the sigmoid function, with temperatureTTcontrolling the steepness, andMpeakM\_\{\\text\{peak\}\}normalizing the peak value to 1\.

The differentiable probability of an edge crossing is thenP​\(crossing\)=Mt​MuP\(\\text\{crossing\}\)=M\_\{t\}M\_\{u\}\. It approaches 1 when the edges intersect \(t,u∈\[0,1\]t,u\\in\[0,1\]\), but quickly and smoothly decreases toward 0 otherwise\. We call this loss functionSigmoidX\. It is a continuous, differentiable proxy for the binary “edges intersect” test, enabling gradient\-based minimization of edge crossings\. We illustrate this function in[Fig\.˜2](https://arxiv.org/html/2606.31119#S2.F2)\. Additional details are given in the Appendix[Appendix˜A1](https://arxiv.org/html/2606.31119#A1)\. SigmoidX is used to find the optimal projection, but we report the actual edge crossings\.

![Refer to caption](https://arxiv.org/html/2606.31119v1/images/optimal_projection/xing_params.png)

![Refer to caption](https://arxiv.org/html/2606.31119v1/images/optimal_projection/edge_loss_surrogate.png)

Figure 2:Left: the edge\-crossing parameters\. Right: contour of the SigmoidX surrogate function for edge crossings\.Edge Length Variation:This metric measures the standard deviation of all edge lengths\. Letℓi​j=‖𝐱i−𝐱j‖\\ell\_\{ij\}=\\\|\\mathbf\{x\}\_\{i\}\-\\mathbf\{x\}\_\{j\}\\\|be the length of edge\(i,j\)∈E\(i,j\)\\in E, thenedgelen\_var=std​\(ℓi​j\)/mean​\(ℓi​j\)\\text\{edgelen\\\_var\}=\\mathrm\{std\}\(\\ell\_\{ij\}\)/\\mathrm\{mean\}\(\\ell\_\{ij\}\)\. It captures uniformity in edge representation, affecting aesthetic clarity and interpretability\.

Angular Resolution:This metric measures the minimum angle between incident edges at each node — higher resolution improves visual distinction and reduces ambiguity\. Typically, it is defined asπ−minv∈V⁡min\(v,u\),\(v,w\)∈Eu≠w⁡arccos⁡\(𝐱u−𝐱v,𝐱w−𝐱v\)\.\\pi\-\\min\_\{v\\in V\}\\min\_\{\\begin\{subarray\}\{c\}\(v,u\),\(v,w\)\\in E\\\\ u\\neq w\\end\{subarray\}\}\\arccos\{\(\\mathbf\{x\}\_\{u\}\-\\mathbf\{x\}\_\{v\},\\mathbf\{x\}\_\{w\}\-\\mathbf\{x\}\_\{v\}\)\}\.However, this measure only focuses on the worst case, which has limited expressivity\. In this paper, we aim to improve the angular resolution across all nodes; hence, we adopt an average measure of how evenly the neighbors of each node are distributed around it\. For a nodevvwith degreedv≥2d\_\{v\}\\geq 2, the ideal angular spacing isθv∗=2​πdv\\theta\_\{v\}^\{\*\}=\\tfrac\{2\\pi\}\{d\_\{v\}\}\. Letθ^vmin\\hat\{\\theta\}\_\{v\}^\{\\min\}be the smallest actual angle between consecutive neighbors ofvv\. We define angular resolution as

ang\_res=1\|V′\|​∑v∈V′\(θv∗−θ^vmin\)2,\\text\{ang\\\_res\}=\\sqrt\{\\frac\{1\}\{\|V^\{\\prime\}\|\}\\sum\_\{v\\in V^\{\\prime\}\}\\left\(\\theta\_\{v\}^\{\*\}\-\\hat\{\\theta\}\_\{v\}^\{\\min\}\\right\)^\{2\}\},\(8\)whereV′=\{v∣dv≥2\}V^\{\\prime\}=\\\{v\\mid d\_\{v\}\\geq 2\\\}is the set of nodes with degree over 1\.

## 3Experimental Evaluation

To quantify the potential benefits of\\DataFly, we compare the metrics of baseline algorithms with those of the\\DataFlyoptimal viewpoints\. Hereafter, we useneato\(KK\) to denote the outcome of laying out a graph in K\-D; the same notation applies to other algorithms\. By default,neatorefers toneato\(2\)\. We denote DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}\(K\) as the result of the\\DataFlyoptimal viewpoint on theneatoK\-D layout for a specific metric, and write DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}for DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}\(10\)\. Similarly, DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}denotes the optimal projection on the\\sfdp10\-D layout for a specific metric\.

### 3\.1Experimental Setup

Optimization procedure\. To find the optimal projectionPPin \([Eq\.˜1](https://arxiv.org/html/2606.31119#S2.E1)\), we initializePPusing the first two principal components of the K\-dimensional layoutXX\. At each iterationtt, we perform a gradient descent update \([Eq\.˜2](https://arxiv.org/html/2606.31119#S2.E2)\) with a learning rate ofη=0\.1\\eta=0\.1, using the Adam\[AdamW\]optimizer\. The algorithm tracks the best projection matrixPPand the corresponding projected coordinatesX​PXPthat yield the lowest metric value observed during training\. We run up to 200 epochs, which we found sufficient for convergence in most cases\.

Across multiple runs, we found that DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}and DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}produced stable results, likely due to the deterministic initialization ofPP\. The GraphViz implementations ofneatoand\\sfdpare known to be stable, and provide consistent results\.

Benchmark Datasets\. We evaluate using two benchmark datasets\. SuiteSparse14 consists of 14 test graphs chosen from the SuiteSparse matrix collection\[Davis\_Hu\_ufl\_2009\]\. They are selected to include graphs of different types \(e\.g\., originating from structural, power network, and scientific problems\)\. The two exceptions are “mobius”, which is generated from a5×505\\times 50mesh graph of a Möbius strip\. This graph was used as an example in Gajer et al\.\[Gajer\_Goodrich\_Kobourov\_2000\]to motivate the need for high\-dimensional layouts of graphs\. We also include the2102^\{10\}\-dimensional hypercube “hypercube10”, since it can only be realized without distortion in 10D\. The number of nodes and edges is shown in the first column of[Table˜1](https://arxiv.org/html/2606.31119#S3.T1), under the name of each graph\. For disconnected graphs, the largest component is taken\.

The second benchmark is the Rome graph collection\[rome\_graphs\], comprising 11,531 relatively sparse graphs with 10–100 vertices and 9–158 edges\. We include this dataset to enable comparison with metric\-optimizing baseline algorithmsSGD2\\text\{SGD\}^\{2\}\[sgd2\]and SmartGD\[wang\_2023\_smartgd\], which were originally designed for smaller graphs and therefore require a small\-graph benchmark\. To present results here, we randomly selected 20 graphs from the benchmark for illustration purposes, which we refer to as Rome20\.

Aggregating performance metrics\. To summarize the comparison between\\DataFlyand other algorithms over many graphs, we report the SPC metric\[deepgd\], defined as the average relative difference between the metrics of\\DataFlyand the baseline, over the test graphs\.*Smaller values are better*\. For example, when comparing algorithm A with the baseline algorithm B, a SPC=−0\.05=\-0\.05indicates that A is on average 5% better than B under that metric\. Specifically,

SPC=1m​∑i=1mAi−Bimax⁡\(\|Ai\|,\|Bi\|\),\\text\{SPC\}=\\frac\{1\}\{m\}\\sum\_\{i=1\}^\{m\}\\frac\{A\_\{i\}\-B\_\{i\}\}\{\\max\(\|A\_\{i\}\|,\|B\_\{i\}\|\)\},\(9\)whereAiA\_\{i\}andBiB\_\{i\}denote the metrics for the algorithm of interest and the baseline algorithm, for test graphii, andmmis the number of test graphs\.

### 3\.2Comparing\\DataFlywith Baseline Algorithms

Traditional graph layout algorithms, such as force\-directed methods and stress majorization\[Eades\_1984,Fruchterman\_Reingold\_1991,neato\], optimize functions directly in low\-dimensional spaces \(e\.g\., 2D or 3D\), where they are more likely to get trapped in local minima\. In contrast, optimizing the same problem in higher dimensions offers more degrees of freedom, helping escape local minima that often become saddle points in higher dimensions, which still admit descent directions\[goodfellow2016deep\]\(Section 8\.2\.3\)\. Given this, a natural question arises: can\\DataFlyachieve lower stress thanneato, lower spring\-electrical energy than\\sfdp, and improvements across other layout quality metrics?

Table 1:Comparison of metrics on SuiteSparse14 for two method pairs:neato\(2D\) vs\. DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}and\\sfdp\(2D\) vs\. DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}\.*Lower is better for all metrics*\. The best value within each method pair and graph/metric combination is highlighted in bold\. The number of nodes/edges is shown in parentheses under the graph names\. SPC values marked with \* are statistically significant \(Appendix[Fig\.˜A1](https://arxiv.org/html/2606.31119#A5.F1)\)\.neatovs\. DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}\\sfdpvs\. DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}Graphmthdang\_r↓\\downarrowedglen↓\\downarrowspr\_el↓\\downarrowstress↓\\downarrowt\-SNE↓\\downarrowxing↓\\downarrowmthdang\_r↓\\downarrowedglen↓\\downarrowspr\_el↓\\downarrowstress↓\\downarrowt\-SNE↓\\downarrowxing↓\\downarrow1138\_bus\(1138,1458\)neato1\.1730\.296\-3\.9880\.0692\.3651406\\sfdp1\.0460\.550\-4\.3690\.0921\.415511DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}1\.1540\.409\-4\.2120\.0762\.0851199DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}1\.1050\.569\-4\.3220\.0981\.584756bcspwr07\(1612,2106\)neato1\.1600\.291\-4\.4650\.0462\.1491572\\sfdp1\.0330\.672\-4\.9980\.0881\.365678DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}1\.0960\.402\-4\.6640\.0622\.0241570DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}1\.0300\.568\-4\.9370\.0841\.526814email\(1133,5451\)neato1\.1910\.515\-2\.1510\.1331\.369544390\\sfdp1\.2070\.546\-2\.3060\.1421\.308425417DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}0\.9960\.477\-2\.1140\.1631\.336517851DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}1\.0970\.529\-2\.2690\.1501\.309399532Erdos991\(446,1413\)neato1\.2770\.439\-2\.1700\.1091\.45135664\\sfdp1\.2600\.502\-2\.3130\.1421\.37832632DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}1\.1320\.409\-2\.1570\.1291\.39829064DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}1\.1830\.465\-2\.2800\.1311\.33427855EX1\(560,4368\)neato0\.3900\.339\-1\.4580\.1671\.153487248\\sfdp0\.3900\.205\-1\.4930\.1701\.167492441DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}0\.3740\.201\-1\.4770\.1761\.123340132DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}0\.3670\.183\-1\.4860\.1751\.089364237football\(115,613\)neato0\.5410\.424\-1\.1670\.1280\.5396843\\sfdp0\.5390\.519\-1\.1960\.1380\.5085622DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}0\.5290\.399\-1\.1700\.1320\.5185183DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}0\.5260\.438\-1\.1810\.1350\.5075334hypercube10\(1024,5120\)neato0\.5950\.341\-1\.9350\.1982\.302478232\\sfdp0\.4860\.081\-1\.9800\.2022\.343518024DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}0\.3230\.007\-1\.9720\.2061\.987247989DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}0\.3560\.040\-1\.9700\.2061\.991264487Journals\(124,5972\)neato0\.0770\.4340\.2100\.1670\.0793747293\\sfdp0\.0770\.4460\.2100\.1700\.0783720645DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}0\.0750\.4630\.2430\.1820\.0773156318DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}0\.0740\.4620\.2370\.1820\.0773741769mobius\(250,450\)neato0\.6390\.187\-3\.5350\.0290\.75972\\sfdp0\.8190\.524\-3\.6070\.0380\.81237DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}0\.4640\.089\-3\.6030\.0340\.73537DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}0\.6500\.193\-3\.5920\.0380\.73014qh882\(1764,3354\)neato1\.2770\.421\-4\.1770\.0562\.0657590\\sfdp1\.3110\.874\-4\.4840\.0831\.3994337DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}1\.2370\.323\-4\.3690\.0721\.7305361DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}1\.3560\.752\-4\.4420\.0871\.5155283Si2\(769,8516\)neato0\.3170\.536\-1\.4210\.1590\.9071745046\\sfdp0\.3260\.569\-1\.5680\.1410\.8251242149DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}0\.3080\.384\-1\.5550\.1430\.8281150696DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}0\.3070\.393\-1\.5610\.1460\.8301229816spiral\(665,7657\)neato0\.5330\.467\-2\.4640\.0441\.443853028\\sfdp0\.3490\.685\-2\.8570\.0761\.424725528DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}0\.2830\.440\-2\.6530\.0621\.284638663DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}0\.3400\.618\-2\.8150\.0791\.193689046Trefethen\(700,5977\)neato0\.3530\.616\-1\.4810\.1771\.209667259\\sfdp0\.3520\.519\-1\.5120\.1801\.239696237DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}0\.3460\.341\-1\.4980\.1811\.103457958DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}0\.3480\.348\-1\.5030\.1791\.132519729USpower\(4941,6594\)neato1\.2460\.372\-4\.8290\.0583\.10012364\\sfdp1\.0660\.795\-5\.6330\.0981\.7123614DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}1\.1560\.438\-5\.1770\.0732\.78212273DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}1\.1290\.773\-5\.5790\.1012\.0445207SPC↓\\downarrow\-0\.128∗\-0\.169∗\-0\.0190\.122∗\-0\.0744∗\-0\.233∗\-0\.050∗\-0\.178∗0\.017∗0\.015\-0\.011\-0\.068![Refer to caption](https://arxiv.org/html/2606.31119v1/x3.png)

![Refer to caption](https://arxiv.org/html/2606.31119v1/x4.png)

![Refer to caption](https://arxiv.org/html/2606.31119v1/x5.png)

Figure 3:Example curves showing how metrics change whenneatoand\\sfdpare embedded in higher dimensions: solid lines show values in the ambient dimension, and dotted lines show values after optimal projection\. Stress improves markedly with added dimensions, but this also makes low\-stress projections harder to obtain\. Spring\-electrical energy gains little from extra dimensions\. For the t\-SNE score, higher dimensions often increase the metric, while projecting down from them typically outperforms direct 2D optimization\.Comparison withneato\.Results are summarized in[Table˜1](https://arxiv.org/html/2606.31119#S3.T1)\. We first note that, as embedding dimensions increase, it is intuitively much easier for the Euclidean distance in the layout to approximate the graph\-theoretic distance, resulting in lower stress \([Eq\.˜3](https://arxiv.org/html/2606.31119#S2.E3)\)\. As shown in[Fig\.˜3](https://arxiv.org/html/2606.31119#S3.F3)and[Table˜A3](https://arxiv.org/html/2606.31119#A3.T3)in the Appendix, the stress decreases as the ambient dimension increases\. Stress in 10\-D is 84\.4% lower on average than in 2D\.

However, does this lower stress in high dimensions result in\\DataFlyachieving lower stress than a 2D optimizedneato? As shown in[Table˜1](https://arxiv.org/html/2606.31119#S3.T1)\(left\), this can be true in some cases, e\.g\., for “Si2”, DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}achieved a stress of 0\.143 vs 0\.159 forneato\. Overall, the SPC aggregate metric shows that stress for DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}is 12\.2% higher \(worse\) than forneato\. This suggests that whileneatocan occasionally get trapped in a local optimum in 2D, this is infrequent —neatogenerally achieves lower stress in 2D\.

When considering other metrics, the advantage of\\DataFlybecomes clearer\. DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}achieves 12\.8% better angular resolution, 16\.9% lower edge\-length variance, 7\.44% better t\-SNE score, and 23\.3% fewer edge crossings\. Its performance on spring\-electrical energy is on par with directneato2D layouts\.

[Fig\.˜A1](https://arxiv.org/html/2606.31119#A5.F1)\(top\) in the Appendix shows paired\-bootstrap \(n=5000n=5000\) 95% confidence intervals \(CIs\) for the SPC metric across evaluation measures\. A CI entirely below 0 indicates DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}significantly outperformsneato\. Under this view, DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}clearly outperforms in angular resolution, edge length variance, t\-SNE score, and edge crossings\. SPC values for these metrics in[Table˜1](https://arxiv.org/html/2606.31119#S3.T1)are marked with \*, and the same convention is used for subsequent results\.

Comparison with\\sfdp\.The spring\-electrical energy, which\\sfdpoptimizes, decreases as the layout dimension increases, as shown in[Fig\.˜3](https://arxiv.org/html/2606.31119#S3.F3)and[Table˜A5](https://arxiv.org/html/2606.31119#A4.T5)in the Appendix\. However, the reduction is not as drastic as we saw with stress andneato\([Table˜A3](https://arxiv.org/html/2606.31119#A3.T3)\)\. Overall, spring\-electrical energy is 10\.6% lower in 10\-D than in 2D\. In particular, when the graphs originate from 2D applications \(e\.g\., 1138\_bus and USpowerGrid\), higher dimensions do not help at all, while for other graphs \(e\.g\., email, EX1\), higher dimensions improve spring\-electrical energy, though not significantly\. This seems to indicate that an optimal layout for this loss function can be realized well in a lower dimension\. One way to assess whether a high\-dimensional embedding can “live” comfortably in a lower\-dimensional subspace is to examine the variance explained by PCA\[hotelling1933\_PCA\]when projecting to that lower dimension\. The results in the Appendix \([Table˜A6](https://arxiv.org/html/2606.31119#A4.T6)\) support this: at the same target dimension, the PCA\-explained variance for\\sfdpis consistently higher than forneato\. For example, projecting 10D embeddings to 2D via PCA retains 75\.4% of the original variance for\\sfdp, compared with 70\.2% forneato\.

The optimal projection DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}achieves similar \(within 2%\) spring\-electrical energy, stress, and t\-SNE score as\\sfdp; see[Table˜1](https://arxiv.org/html/2606.31119#S3.T1)\(right\)\.

However, DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}performs better on angular resolution, edge length variance, and edge crossings\.

![Refer to caption](https://arxiv.org/html/2606.31119v1/x6.png)Figure 4:Optimal projections of the 10Dneatolayout\. Numbers represent \(neato, DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}\) metrics\.Bold= better\. Edges colored by length:red= short,blue= long\.
### 3\.3Visualizations with\\DataFly

It is informative to compare the optimal projection viewpoints identified by\\DataFlywith those obtained from the corresponding baseline algorithms\.[Fig\.˜4](https://arxiv.org/html/2606.31119#S3.F4)presents visualizations from DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}alongside those fromneato\.\\DataFlysometimes reveals compelling graph structures that differ markedly from those produced byneato\. For example, for the “Si2” and “EX1” graphs, the\\DataFlyvisualization that optimizes edge crossings highlights a lattice structure that is less apparent in theneatolayout\. Similarly, for “mobius”, the\\DataFlyviewpoint that minimizes edge crossings is much smoother and has half as many edge crossings asneato\(37 vs\. 72\)\. In “Trefethen\_700”,\\DataFlyproduces a clearer, more organized layout that reveals linear and rectangular patterns, while theneatolayout appears more entangled, with increased edge crossings and visual clutter\. Additional visual comparisons betweenneato,\\sfdp, and\\DataFlyare given in the Appendix \([Tables˜A1](https://arxiv.org/html/2606.31119#A3.T1)and[A2](https://arxiv.org/html/2606.31119#A3.T2)and[Table˜A4](https://arxiv.org/html/2606.31119#A4.T4)\), which further confirm the effectiveness of\\DataFlyin revealing graph structures that may be obscured in direct 2D layouts\. These findings provide strong support forRQ1\.

### 3\.4Comparing\\DataFlywith Other Algorithms

We observe that embedding graphs into high dimensions and then finding the optimal 2D projection can produce layouts with substantially better metric scores than those generated by baseline 2D algorithms \(e\.g\.,neatoor\\sfdp\)\. To understand how effective this approach is compared with directly optimizing the metrics, we compare\\DataFlywithSGD2\\text\{SGD\}^\{2\}\[sgd2\]and SmartGD\[wang\_2023\_smartgd\]in minimizing edge crossings, and withSGD2\\text\{SGD\}^\{2\}on angular resolution and edge length variance\. We were unable to obtain comparable results forSGD2\\text\{SGD\}^\{2\}and SmartGD across other metrics from the authors of these algorithms\.

SGD2\\text\{SGD\}^\{2\}is designed to optimize several graph aesthetic metrics directly using gradient descent on relatively small graphs\. SmartGD was also trained on the Rome collection of small graphs to optimize specific metrics using a generative adversarial network\[rgan\]\. Therefore, we first used the Rome20 dataset for comparison\.

The results are given in[Table˜2](https://arxiv.org/html/2606.31119#S3.T2), with CIs analyzed in the Appendix[Fig\.˜A2](https://arxiv.org/html/2606.31119#A5.F2)\. For minimizing edge crossings, the last row of the table shows that DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}achieves 22\.1% fewer crossings on average thanSGD2\\text\{SGD\}^\{2\}, while SmartGD is 4\.4% better thanSGD2\\text\{SGD\}^\{2\}\. In terms of CPU time, DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}took only 10 seconds, SmartGD takes 1 second, while theSGD2\\text\{SGD\}^\{2\}algorithm took over 1280 seconds\. This also indicates that the gradient for crossing minimization inSGD2\\text\{SGD\}^\{2\}, which involves a neural network model that approximates the crossings, requires significant computational effort\. The best performing algorithm, Vertex Movement \(VM\)\[radermacher2019geometric\], takes 760 seconds and is 60\.3% better thanSGD2\\text\{SGD\}^\{2\}\. While VM achieves low crossing numbers, it produces drawings that do not optimize other aesthetic criteria; see the example drawing of grafo7023 in[Fig\.˜A4](https://arxiv.org/html/2606.31119#A5.F4), versus the corresponding drawing produced by\\DataFlyin[Table˜A2](https://arxiv.org/html/2606.31119#A3.T2)\(row 5, last column\)\.

In terms of angular resolution, DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}is 62% better thanSGD2\\text\{SGD\}^\{2\}and again much faster, requiring only a few seconds vs 218 seconds forSGD2\\text\{SGD\}^\{2\}\. When looking at edge length variance,SGD2\\text\{SGD\}^\{2\}reduced it to almost zero and performed better than\\DataFly\. This is not a surprise, since even though edge length variance should be very low in high dimensions, linear projection is unlikely to preserve it due to the different orientations of edges in high dimensions\. However,SGD2\\text\{SGD\}^\{2\}required 148 seconds, while DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}completed in 10 seconds\.

Table 2:Comparison on Rome20 graphs\. We usedSGD2\\text\{SGD\}^\{2\}as the benchmark\. \*=statistically significant \(Appendix[Fig\.˜A2](https://arxiv.org/html/2606.31119#A5.F2)\)\.xing↓\\downarrowang\_res↓\\downarrowedge\_len↓\\downarrowgrafo idDFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}SGD2\\text\{SGD\}^\{2\}SmartGDVMDFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}SGD2\\text\{SGD\}^\{2\}DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}SGD2\\text\{SGD\}^\{2\}2815\.3528210\.7002\.1580\.0970\.01210379\.95828082400\.7731\.8810\.2820\.3889033\.80545449300\.7781\.6810\.2790\.30611674\.3335610\.6151\.9970\.1220\.000847\.22416500\.6221\.8080\.0510\.000762\.2735310\.5681\.9800\.1570\.00011412\.3210100\.6962\.1710\.1160\.0001583\.6612181590\.7042\.0850\.2450\.2063587\.3678730\.8681\.8480\.1730\.0138296\.86665079420\.7711\.7980\.2400\.30211543\.3407400\.7532\.1140\.1740\.0217023\.458161220\.7182\.0340\.1940\.0959592\.7213241830\.6922\.1020\.2040\.1743138\.60311842200\.8892\.1610\.2260\.2942955\.386101650\.7541\.9680\.2170\.03910480\.95598163300\.8351\.9340\.2960\.3072747\.1600000\.6352\.0380\.0460\.0005780\.48253033130\.8861\.7030\.2030\.10211311\.40412650\.6962\.0970\.2160\.14310451\.959664116500\.8681\.8050\.2780\.337SPC\-0\.221∗0\-0\.044\-0\.603∗\-0\.620∗00\.456∗0
### 3\.5Additional comparison on edge crossing minimization

As seen in[Table˜1](https://arxiv.org/html/2606.31119#S3.T1), edge\-crossings for both DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}and\\sfdpare generally lower than forneatoand DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}\. We formally compare these four algorithms together with SmartGD andSGD2\\text\{SGD\}^\{2\}on crossing minimization of the SuiteSparse14 dataset in[Table˜3](https://arxiv.org/html/2606.31119#S3.T3)\. Indeed, on average,\\sfdphas 25\.9% fewer crossings thanneato, while DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}has 33\.9% fewer crossings\. This is surprising given that in the graph drawing community, minimizing stress \(whichneatodoes\) and minimizing crossings were known to be highly associated with user preference\[Purchase\_1997,Tim\-user\-study\]\.

We observe that SmartGD andSGD2\\text\{SGD\}^\{2\}have substantially higher edge crossings \(70% and 54\.5% worse thanneato\), while the two\\DataFlyalgorithms achieve improvements of 23\.3% and 35% overneato\. This confirms that neither SmartGD norSGD2\\text\{SGD\}^\{2\}works well for large graphs\. Comparing the runtime on these large graphs, DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}and DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}take 10,470 and 9,800 seconds, respectively\.SGD2\\text\{SGD\}^\{2\}is the slowest, taking 16,500 seconds\. DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}and DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}are only moderately faster because, although their optimization involves only 20 variables, compared with2​\|V\|2\|V\|variables forSGD2\\text\{SGD\}^\{2\}, that advantage is offset by the high cost of computing the edge\-crossings loss required for these algorithms\. Inference with SmartGD is fast on these large graphs, taking only 315 seconds, reflecting its neural network–based, non\-iterative nature; however, training the model demands substantial data and computational resources\.

Table 3:Comparing edge\-crossings for various algorithms on SuiteSparse14\. \*=statistically significant \(Appendix[Fig\.˜A3](https://arxiv.org/html/2606.31119#A5.F3)\)neatoDFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}\\sfdpDFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}SmartGDSGD2\\text\{SGD\}^\{2\}1138\_bus140611995117561206115431bcspwr07157215706788142217647795email54439051785142541739953228528171234370Erdos9913566429064326322785513841147165EX14872483401324924413642371862924741591football684351835622533493535578hypercube1047823224798951802426448728211321111129Journals374729331563183720645374176936070802772694mobius7237371410704544qh882759053614337528347892203344Si2174504611506961242149122981641816063347111spiral85302863866372552868904623388552006179Trefethen66725945795869623751972918786131723377USpower1236412273361452072012921254507SPC0\-0\.233∗\-0\.259∗\-0\.339∗0\.700∗0\.545∗Table 4:SPC for edge crossings relative toneatoover all Rome Graphs\. \*=statistically significant \(Appendix[Fig\.˜A3](https://arxiv.org/html/2606.31119#A5.F3)\)neatoDFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}\\sfdpDFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}SmartGDSGD2\\text\{SGD\}^\{2\}SPC0\-0\.268∗\-0\.0338∗\-0\.215∗\-0\.0393∗\-0\.0477∗We also ran the edge\-crossing minimization algorithmVertex Movement\(VM\)\[radermacher2019geometric\]on these large graphs, but it did not finish within 24 hours for 9 of the 14 instances, confirming its scalability limitations\.

Finally, for a comprehensive evaluation of edge\-crossing minimization on small graphs, we compared all the algorithms on the complete Rome graph collection in[Table˜4](https://arxiv.org/html/2606.31119#S3.T4)\. On this full set of 11,531 graphs, DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}and DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}have 26\.8% and 21\.5% lower edge crossings thanneato, respectively\. Both substantially outperformSGD2\\text\{SGD\}^\{2\}and SmartGD, although the latter two do surpassneato\. Regarding the runtime, DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}and DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}each take about 3300 seconds for all graphs\. The inference of SmartGD is also very fast on these small graphs, taking just 615 seconds\. The geometric heuristic VM is slower, taking 122,000 seconds; furthermore,*it failed on 12% of the graphs due to floating\-point errors\.*Therefore, we do not include VM in the table\.SGD2\\text\{SGD\}^\{2\}is the slowest, requiring 558,000 seconds, almost 170 times longer than DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}and DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}\.

### 3\.6Findings forRQ1

When comparing\\DataFlywith the two baseline algorithmsneatoand\\sfdpacross eight different evaluation metrics,\\DataFlyoutperformed them on six metrics\. When comparing it with the two state\-of\-the\-art algorithms SmartGD andSGD2\\text\{SGD\}^\{2\}, which are designed to optimize specific metrics, we found that\\DataFlyis highly effective at minimizing edge crossings and improving angular resolution\. While VM achieves the absolute lowest number of edge crossings, it is designed solely for that objective and has very high computational complexity\. VM does not produce an aesthetic layout and is not suitable for large graphs\. These observations support an affirmative answer toRQ1\.

## 4\\DataFlyExploration Tool

The\\DataFlyapplication is a web\-based platform for interactively exploring high\-dimensional graph embeddings through different viewpoints\. Having established that these viewpoints are quantitatively strong \([Section˜3](https://arxiv.org/html/2606.31119#S3)\) and qualitatively informative \([Fig\.˜4](https://arxiv.org/html/2606.31119#S3.F4)\), we now describe the system that exposes them to users\. Through interactive 2D visualizations,\\DataFlyreveals patterns and structures that may be hidden in a static layout by allowing users to navigate among related projections\. In addition to optimal projections,\\DataFlysupports PCA projections, which are particularly well\-suited for interactive exploration\. We describe the PCA projections next, then the system’s main functionality, and finally a usability study\.

### 4\.1PCA viewpoints and interactive rotation

Given theKK\-dimensional node\-embeddings matrixX∈ℝ\|V\|×KX\\in\\mathbb\{R\}^\{\|V\|\\times K\}, we compute its principal components through Principal Component Analysis \(PCA\)\[hotelling1933\_PCA\]\. PCA yieldsKKorthogonal directions in the embedding space, ranked by decreasing variance of the projected points, that is, the spread of nodes in the projection\. The first principal component captures the most variance, the second the next most, and so on\. Any pair of principal components can be used as the axes of a 2D projection, yielding up to\(K2\)=K​\(K−1\)/2\\binom\{K\}\{2\}=K\(K\-1\)/2possible 2D viewpoints\. In the tool, these viewpoints provide a structured set of interpretable linear projections that users can browse interactively\.

To support informed exploration, we compute quality metrics such as stress and edge crossings for each candidate viewpoint and display them as a line chart at the bottom of the UI \([Fig\.˜5](https://arxiv.org/html/2606.31119#S4.F5)\)\. The user can then navigate to the view that optimizes a given metric by clicking either the corresponding point on the line chart or its thumbnail\. Since computing these metrics for every possible viewpoint is expensive, we first rank all principal\-component pairs by their explained variance and retain only the top 50\. Beyond the practical limitation that the interface can display only a limited number of viewpoints along thexx\-axis, this filtering is motivated by the observation that low\-variance projections tend to cluster nodes tightly, leading to visual occlusion\. Empirically, the most insightful views typically correspond to component pairs that capture relatively high variance \(i\.e\., greater spatial spread\)\.

### 4\.2User Interface and Functionality

[Fig\.˜5](https://arxiv.org/html/2606.31119#S4.F5)shows the\\DataFlyvisualization and interactive exploration system, with components labeled in the figure \(e\.g\.,A\)\. Users begin by uploading a file or selecting a built\-in demo datasetA\. The system processes the data and renders the visualizationI\. Users can configure the layout algorithm \(neato,\\sfdp,PivotMDS, orspectral\), the initial embedding dimension \(2–10\), and the projection method \(PCA or optimal projection\) viaB\.

By default, edges are colored according to a breadth\-first traversal of the graph\. The coloring is intended as a stable visual anchor \(rather than a semantic encoding of graph properties\) that helps preserve the user’s mental map as the visualization “flies” from one viewpoint to another\. Nodes can also be highlighted to further support mental map preservation\. Mouse\-based zooming, panning are supported and instructed in a translucent information panelH\.

![Refer to caption](https://arxiv.org/html/2606.31119v1/images/system/system_interface.png)Figure 5:\\DataFlyinterface and functionality overview\. Main: Rotated PCA view of Harry Potter co\-occurrence graph\. Inset: Optimal Projection view of hypercube 8D\.Interaction modes\.Three modes are provided\. The first is the*optimal\-projection*mode: when the “Projection Algorithm” inBis set to “optimal projection,”\\DataFlypresents the layout that optimizes stress as the main displayJ\(depicted inset in the figure for demonstration\), along with six thumbnailsKcorresponding to optimal projections for stress, edge crossings, t\-SNE, spring\-electrical energy, angular resolution, and edge\-length variation, respectively\. The value achieved by each of the six layouts for the currently selected metric is displayed above the thumbnailsKin the optimal projection interfaceJ\.

The PCA mode displays thumbnailsGas previews of 2D projections from PCA component pairs sorted by the selected metric \(default: highest to lowest variance\)\. The toggle list inFsupports re\-sorting by a different metric\. Users can browse thumbnails to spot patterns and click to transition to the selected viewpoint\. The “Projection” sliders inDlet users manually select PC combinations, triggering a transition animation\. From the panelE, users can view projection details, bookmark rotated projections, and quickly switch to saved views\.

Thirdly, the system also provides an automatic cinematic mode, allowing users to initiate transition animations through all viewpoints by clicking the “DataFly” buttonC, and to pause or resume at any time by pressing the space key\.

Interactive rotation\.Users can drag the view to perform a local rotation within a 3D exploration subspace\. For PCA projections, this subspace is spanned by the two principal components of the current viewpoint and the remaining component with the highest variance; in optimal projection mode, it is formed by the optimized projection plane and an additional orthogonal direction\. In both cases, the resulting viewpoint remains a linear projection, preserving interpretability while letting users perturb a view, extract structure from motion, mitigate node occlusion, and inspect nearby projections without losing continuity\.

Together, these functionalities and interface elements are intended to support three complementary tasks: overview, selection, and refinement\. ThumbnailsGandKprovide an overview of all viewpoints at once, and users can switch easily to a projection by clicking on a thumbnail; the line chart sectionFsupports quantitative comparison by quality metrics and selection of views that optimize a chosen criterion; and direct interactions \(e\.g\., zooming, panning, interactive rotation\) support local refinement near a promising view\. Animated transitions and a consistent edge\-coloring scheme, together with node highlighting, help preserve correspondence between successive projections, so users can compare related views as transformations of the same embedding rather than as unrelated layouts\.

Scalability\.\\DataFlyleverages GPU\-accelerated rendering and scalable layout algorithms such assfdpand PivotMDS\. Scalability results are provided in[Appendix˜A7](https://arxiv.org/html/2606.31119#A7)\.

## 5Usability Study and Results

To evaluate the practical utility of\\DataFlyin helping users explore high\-dimensional graph layouts \(RQ2\), we conducted a qualitative usability study with four participants via remote video conferencing using a screen\-sharing and think\-aloud protocol\. Each 40–50 minute session comprised four phases: a pre\-interview on the participant’s expertise, a tutorial video, hands\-on exploration of a few datasets, and a closing Likert\-scale questionnaire\. During exploration, participants used the live system to perform open\-ended exploration and tasks such as identifying nodes of interest, tracing neighbors, switching between projections, and comparing viewpoints across metrics, while verbalizing their observations\.

Participants\.We recruited four participants: two visualization experts \(E1, E2\), both PhD students who regularly work with graph data using tools such as NetworkX, Gephi, and D3\. The other two participants were non\-experts \(N1, N2\) with computer science backgrounds but no prior experience with graph embeddings or dimensionality reduction\. This setup enabled us to evaluate\\DataFly’s accessibility to a broader audience\. Datasets were selected by the participants to match their backgrounds: experts explored subsets of a VIS co\-authorship network, a hypercube graph, a Game of Thrones character network, and a Last\.fm music similarity network, while non\-experts explored the Last\.fm and Game of Thrones networks due to greater domain familiarity\. None of the participants had prior exposure to\\DataFly\.

Expert feedback\.Both experts found the tool visually appealing and intuitive\. E1 searched for recognizable node names and topological features, appreciating the ability to highlight and track nodes across projections, noting that*“in one layout a node looked like a neighbor, but in the other layout didn’t\.”*They particularly valued the interactive rotation for resolving occlusion, finding that it revealed bridge connections between clusters that were hidden in static views\. E1 also observed a duality between exploratory and communicative uses, noting that PCA viewpoints supported open\-ended exploration while optimal projections could serve presentation:*“I like the duality of being able to explore, but also look at a layout that maybe is nice to present in a communication setting\.”*On the hypercube dataset, they noted that higher\-dimensional embeddings produced markedly different projections, whereas the traditional 2D layout made all views appear noisy and similar —*“in 2D, they all kind of look the same\. It looks noisy\.”*

E2 began by comparing layout algorithms and exploring the animated transitions\. They found the transitions helpful for building intuition about the high\-dimensional structure, though initially confused by the apparent rotation direction of the animations\. They praised the metric chart for validating projection quality, finding it helpful to*“compare the values for the others as well\.”*They also found the edge coloring scheme useful as a spatial anchor — when colors became*“all kind of messed up”*in a new projection, it prompted them to rotate the view to*“separate them back out again\.”*On the music dataset, they remarked that thumbnails were effective for previewing and skipping uninformative projections\.

Each expert proposed improvements: E1 suggested ego\-network highlighting and edge tracking through transitions, while E2 recommended supporting additional graph formats\.

Non\-expert feedback\.N1 started with the Last\.fm network, independently noting its small\-world structure\. They went on to identify coherent genre clusters, pointing out a tight grouping of nu\-metal and alternative metal bands, and used the rotation feature to position the cluster as visually salient\.

N2 began with the Game of Thrones network, and made immediate comparisons with other 3D viewpoint editors, noting that the navigation controls of\\DataFlyfelt both natural and intuitive\. Using the projection thumbnails to see different viewpoints, they noted that the separation between the Westeros and Essos storylines was visually salient across nearly all projections\. They went on to explore the optimal projection and found the stress\-optimized projection particularly informative\. They also used the bookmark feature to save interesting viewpoints\. Later in the session, they discovered the shift\-click highlight feature on their own and found it valuable\.

Both non\-expert participants found similar insights in the Game of Thrones graph: both expressed surprise at the large number of connections of Robert Baratheon, and the unexpectedly small number of connections for many of the Stark family characters\. Each participant used rotation and alternate projections to “convince” themselves of these observations, with N2 noting that they would not have believed this without the visualization\. Both rated the tool’s visual appeal highly, with N1 noting that it felt “natural” and N2 praising its intuitive panning and rotation\.

Likert ratings\.Table[5](https://arxiv.org/html/2606.31119#S5.T5)summarizes post\-session ratings \(1–5\) across five aspects:Intuitive\(how intuitive the tool is to use?\),Would Use\(how likely would you use the tool for work or personal interest\),Appeal\(overall visual appeal of the tool\),Suitable\(how suitable the tool is for answering questions about the data\), andInteraction\(overall effectiveness of the tool’s user interactions and instructions\)\.

The tool received consistently high marks for visual appeal \(median 4\.5\) and user interaction quality \(median 4\.5\)\. Intuitiveness ratings were high among experts and N1 \(all rated 4\) but lower for N2 \(rated 2\), who struggled with interpreting the axes and viewpoints\. Suitability scores were consistently high \(median 4\)\. Likelihood of use for work or personal interest varied \(range 2–4\)\.

Table 5:Likert\-scale ratings \(1 = poor, 5 = excellent\) from all four participants\.ParticipantIntuitiveWould UseAppealSuitableInteractionExpert 1 \(E1\)44445Expert 2 \(E2\)43544Non\-expert 1 \(N1\)44454Non\-expert 2 \(N2\)22545Median43\.54\.544\.5Results and findings summary\.The usability study indicates that\\DataFlyhelps users relate high\-dimensional embeddings to recognizable structural patterns\. Interactive rotation was repeatedly cited as essential for resolving occlusion and revealing otherwise hidden connections\. Participants also valued the duality between exploratory and communicative uses: while PCA viewpoints supported open\-ended exploration, the optimal projection mode was favored for producing aesthetic, presentation\-ready layouts\. Quantitatively, participants gave consistently high ratings for visual appeal, interaction quality, and task suitability\. Based on participant feedback, we have iteratively improved the tool and outlined further updates in the Appendix \([Appendix˜A8](https://arxiv.org/html/2606.31119#A8)\)\.

Discussion on RQ2\.Taken together, the qualitative examples presented earlier \(e\.g\.,[Fig\.˜4](https://arxiv.org/html/2606.31119#S3.F4)\), which demonstrate\\DataFly’s ability to reveal graph structures not visible in direct 2D layouts, along with the findings of this usability study, provide qualitative evidence supporting an affirmative answer toRQ2\.

## 6Discussions and Limitations

While\\DataFlycurrently supports viewpoints via linear projection, either optimized for one of the six metrics or allowing users to explore through the sequence of PCA projections, one might also consider non\-linear projection techniques such as Multidimensional Scaling\[Kruskal\_MDS\_1964\], t\-SNE\[Maaten\_2008\_tSNE\], or UMAP\[mcinnes2018\_umap\]\. These methods can also produce a 2D representation of a high\-dimensional graph layout, but they are less suitable for the interactive exploration scenario considered here\. First, they produce only a single projection for a fixed parameter setting, so exploring alternative views requires rerunning the method, potentially yielding layouts that differ significantly\. Second, due to non\-linearity, there is no meaningful relationship between two different embeddings\. In this sense, non\-linear projections are an abstract representation of high\-dimensional data\.\\DataFlyallows users to explore, in a principled way, several viewpoints of a high\-dimensional graph embedding using interpretable linear projection techniques\.

Due to Graphviz limitations, our experiments were restricted to layouts up to 10 dimensions; future work will investigate whether higher\-dimensional layouts provide additional benefits\.

## 7Conclusions

This paper proposes an approach to graph visualization that embeds graphs in high\-dimensional space and systematically searches for informative 2D viewpoints, moving beyond the conventional paradigm of producing a single static 2D layout\.

Our experimental results show that optimal projections derived from high\-dimensional embeddings can produce superior visualizations, with metrics \(e\.g\., edge crossings\) outperforming 2D layout algorithms such asSGD2\\text\{SGD\}^\{2\}and SmartGD which are explicitly designed to minimize them\.\\DataFlyalso reveals structural patterns not apparent in widely used 2D layouts such asneatoand\\sfdp\. We further introduce SigmoidX, an effective differentiable loss function for edge crossings that is simpler to implement than the neural network–based surrogate used inSGD2\\text\{SGD\}^\{2\}, while achieving significantly fewer crossings than bothSGD2\\text\{SGD\}^\{2\}and SmartGD in our experiments\.

To make these high\-dimensional embeddings accessible to users, we propose an interactive tool\\DataFlyfor exploring high\-dimensional graph embeddings and discovering insights\. The system enables users to find projections that optimize specific metrics or explore a set of PCA\-based viewpoints, accompanied by a chart displaying quality metrics such as stress, t\-SNE score, and edge crossings\. Users can interactively explore the high\-dimensional layout landscape by “flying” from one viewpoint to another\. Our usability study with experts and non\-experts provides encouraging evidence that\\DataFlyhelps users discover structural patterns that remain hidden in conventional 2D layouts\.

## Acknowledgments

The authors thank Pan Xu for her input during the initial exploration of this project\.

## References

## Appendix

## Appendix A1Differentiable surrogate function for edge crossing

We define a differentiable approximation for detecting whether two edges intersect as follows\.

Let the two edges be

e1:p\+t​r,e2:q\+u​s,t,u∈\[0,1\],e\_\{1\}:p\+tr,\\quad e\_\{2\}:q\+us,\\quad t,u\\in\[0,1\],wherep,q∈ℝ2p,q\\in\\mathbb\{R\}^\{2\}are the start points, andr,sr,sare the direction vectors\. The endpoints of these two edges arep\+rp\+randq\+sq\+s\.

The intersection parameters are computed as

t=\(q−p\)×sr×s,u=\(q−p\)×rr×s,t=\\frac\{\(q\-p\)\\times s\}\{r\\times s\},\\qquad u=\\frac\{\(q\-p\)\\times r\}\{r\\times s\},where “×\\times” denotes the 2D cross product\. Geometrically, whent∈\[0,1\]t\\in\[0,1\], it means thate2e\_\{2\}, or its extension, hitse1e\_\{1\}\. Similarly,u∈\[0,1\]u\\in\[0,1\]means thate1e\_\{1\}, or its extension, hitse2e\_\{2\}\. Hence, if the two edges cross, we must havet,u∈\[0,1\]t,u\\in\[0,1\]\.

Instead of using a hard condition\(t,u∈\[0,1\]\)\(t,u\\in\[0,1\]\)to decide whether the edges intersect, which is not differentiable, we use sigmoid\-based soft masks:

Mt=σT​\(t\)​\[1−σT​\(t−1\)\]Mpeak,Mu=σT​\(u\)​\[1−σT​\(u−1\)\]Mpeak,M\_\{t\}=\\frac\{\\sigma\_\{T\}\(t\)\\,\[1\-\\sigma\_\{T\}\(t\-1\)\]\}\{M\_\{\\text\{peak\}\}\},\\qquad M\_\{u\}=\\frac\{\\sigma\_\{T\}\(u\)\\,\[1\-\\sigma\_\{T\}\(u\-1\)\]\}\{M\_\{\\text\{peak\}\}\},whereσT​\(u\)=1/\(1\+e−T​u\)\\sigma\_\{T\}\(u\)=1/\(1\+e^\{\-Tu\}\)is the sigmoid function with temperatureTT\(we useT=10T=10\), andMpeak=σT​\(0\.5\)​\[1−σT​\(−0\.5\)\]M\_\{\\text\{peak\}\}=\\sigma\_\{T\}\(0\.5\)\\,\[1\-\\sigma\_\{T\}\(\-0\.5\)\]normalizes the peak value\.

The differentiable edge\-crossing probability is

P​\(crossing\)=Mt​Mu,P\(\\text\{crossing\}\)=M\_\{t\}M\_\{u\},which approaches 1 when the edges intersect \(t,u∈\[0,1\]t,u\\in\[0,1\]\) and smoothly decreases toward 0 otherwise\. We call this loss functionSigmoidX\. It is a continuous, differentiable proxy for the binary “edges intersect” test, enabling gradient\-based optimization to minimize edge crossings\. We also experimented with other surrogate functions, including a pyramid function over the unit square, but found thatSigmoidXperforms better\.

## Appendix A2Scale\-invariant spring\-electrical energy

This metric treats edges as springs and nodes as electrically repelling particles and measures the energy of the system:

se\_eng​\(x\)=∑\(i,j\)∈E‖𝐱i−𝐱j‖33​Ks−∑i≠jKr2​ln⁡\(‖𝐱i−𝐱j‖\)\.\\text\{se\\\_eng\}\(x\)=\\sum\_\{\(i,j\)\\in E\}\\frac\{\\\|\\mathbf\{x\}\_\{i\}\-\\mathbf\{x\}\_\{j\}\\\|^\{3\}\}\{3K\_\{s\}\}\-\\sum\_\{i\\neq j\}K\_\{r\}^\{2\}\\ln\(\\\|\\mathbf\{x\}\_\{i\}\-\\mathbf\{x\}\_\{j\}\\\|\)\.whereKsK\_\{s\}andKrK\_\{r\}are spring and repulsion constants\. We set them to 1\. To make this metric scale\-invariant, we can find a scaling factorttsuch thatse\_eng​\(t​x\)\\text\{se\\\_eng\}\(tx\)is minimized\. If we denote the first term in the energy asA/3A/3\(for attractive spring energy\) and the second asRR\(repulsive electrical energy\), then we want to minimize

se\_eng​\(t​x\)=A3​t3−R−ln⁡\(t\)∗M\\text\{se\\\_eng\}\(tx\)=\\frac\{A\}\{3\}t^\{3\}\-R\-\\ln\(t\)\*M
whereM=\|V\|​\(\|V\|−1\)M=\|V\|\(\|V\|\-1\)\. Taking the derivative and setting it to zero, the optimum is achieved whent=\(MA\)1/3t=\(\\frac\{M\}\{A\}\)^\{1/3\}\. The final energy is

se\_eng​\(x\)\\displaystyle\\text\{se\\\_eng\}\(x\)=M3\+M3​ln⁡AM−R\\displaystyle=\\frac\{M\}\{3\}\+\\frac\{M\}\{3\}\\ln\\frac\{A\}\{M\}\-R=\|V\|​\(\|V\|−1\)3\+\|V\|​\(\|V\|−1\)3​ln⁡∑\(i,j\)∈E‖𝐱i−𝐱j‖3\|V\|​\(\|V\|−1\)\\displaystyle=\\frac\{\|V\|\(\|V\|\-1\)\}\{3\}\+\\frac\{\|V\|\(\|V\|\-1\)\}\{3\}\\ln\\frac\{\\sum\_\{\(i,j\)\\in E\}\\\|\\mathbf\{x\}\_\{i\}\-\\mathbf\{x\}\_\{j\}\\\|^\{3\}\}\{\|V\|\(\|V\|\-1\)\}−∑i≠jln⁡\(‖𝐱i−𝐱j‖\)\\displaystyle\\ \\ \\ \\ \-\\sum\_\{i\\neq j\}\\ln\(\\\|\\mathbf\{x\}\_\{i\}\-\\mathbf\{x\}\_\{j\}\\\|\)

## Appendix A3Additional results forneato

Tables[A1](https://arxiv.org/html/2606.31119#A3.T1)\-[A2](https://arxiv.org/html/2606.31119#A3.T2)show visualizations of graphs in the Rome20 dataset, comparingneatowith\\DataFly’s optimal projection DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}across various metrics\. The corresponding metric values forneatoand DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}are shown below each visualization\. Because DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}is a projection of a high\-dimensionalneatolayout, its drawings remain visually similar to those ofneato, yet apart from stress and spring\-electrical loss, DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}typically achieves better metric values\. For example, for grafo11543, DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}find a planar layout, whereasneatohas four edge crossings\.

Table[A3](https://arxiv.org/html/2606.31119#A3.T3)presents the stress achieved byneatowhen optimizing directly in various dimensions from 2D to 10D\. As expected, stress is significantly lower in 10D for all graphs\.

Table A1:Optimal projections of the 10D Neato layout of the Rome20 dataset \(part 1/2\)\. Numbers show Neato vs\. DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}metrics\.Bold= better\. Edges colored by length:red= short,blue= long\.neatoang\_res↓\\downarrowedge\_len↓\\downarrowspr\_elec↓\\downarrowstress↓\\downarrowtsne↓\\downarrowxing↓\\downarrowgrafo10379\.95![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x7.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x8.png)1\.036,0\.773![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x9.png)0\.214,0\.282![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x10.png)\-2\.139,\-2\.16![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x11.png)0\.085,0\.087![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x12.png)1\.335,1\.28![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x13.png)99,82grafo10451\.95![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x14.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x15.png)1\.046,0\.868![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x16.png)0\.212,0\.278![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x17.png)\-2\.032,\-2\.063![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x18.png)0\.103,0\.1![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x19.png)1\.444,1\.307![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x20.png)117,96grafo10480\.95![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x21.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x22.png)0\.893,0\.835![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x23.png)0\.245,0\.296![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x24.png)\-2\.191,\-2\.244![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x25.png)0\.091,0\.088![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x26.png)1\.381,1\.288![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x27.png)70,59grafo11311\.40![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x28.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x29.png)0\.87,0\.696![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x30.png)0\.196,0\.216![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x31.png)\-1\.726,\-1\.727![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x32.png)0\.066,0\.073![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x33.png)0\.784,0\.716![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x34.png)6,4grafo11412\.32![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x35.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x36.png)0\.842,0\.696![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x37.png)0\.101,0\.116![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x38.png)\-1\.979,\-2\.035![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x39.png)0\.032,0\.033![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x40.png)0\.569,0\.512![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x41.png)3,1grafo11543\.34![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x42.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x43.png)0\.872,0\.753![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x44.png)0\.092,0\.174![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x45.png)\-1\.829,\-1\.839![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x46.png)0\.036,0\.042![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x47.png)0\.625,0\.6![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x48.png)4,0grafo11674\.33![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x49.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x50.png)0\.924,0\.615![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x51.png)0\.121,0\.122![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x52.png)\-1\.611,\-1\.713![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x53.png)0\.074,0\.061![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x54.png)0\.893,0\.66![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x55.png)11,3grafo1583\.66![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x56.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x57.png)0\.815,0\.704![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x58.png)0\.186,0\.245![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x59.png)\-2\.253,\-2\.252![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x60.png)0\.06,0\.066![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x61.png)0\.964,0\.927![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x62.png)13,12grafo2747\.16![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x63.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x64.png)0\.845,0\.635![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x65.png)0\.07,0\.046![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x66.png)\-1\.438,\-1\.472![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x67.png)0\.01,0\.01![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x68.png)0\.204,0\.184![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x69.png)0,0grafo2815\.35![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x70.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x71.png)0\.934,0\.7![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x72.png)0\.108,0\.097![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x73.png)\-1\.872,\-1\.896![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x74.png)0\.032,0\.033![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x75.png)0\.603,0\.582![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x76.png)5,2Table A2:Optimal projections of the 10D Neato layout of the Rome20 dataset \(part 2/2\)\. Numbers show Neato vs\. DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}metrics\.Bold= better\. Edges colored by length:red= short,blue= long\.neatoang\_res↓\\downarrowedge\_len↓\\downarrowspr\_elec↓\\downarrowstress↓\\downarrowtsne↓\\downarrowxing↓\\downarrowgrafo2955\.38![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x77.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x78.png)0\.96,0\.754![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x79.png)0\.153,0\.217![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x80.png)\-1\.692,\-1\.7![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x81.png)0\.063,0\.07![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x82.png)0\.78,0\.708![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x83.png)9,6grafo3138\.60![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x84.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x85.png)1\.137,0\.889![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x86.png)0\.244,0\.226![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x87.png)\-1\.809,\-1\.791![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x88.png)0\.089,0\.095![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x89.png)1\.054,1\.0![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x90.png)42,31grafo3587\.36![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x91.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x92.png)1\.186,0\.868![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x93.png)0\.176,0\.173![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x94.png)\-1\.748,\-1\.755![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x95.png)0\.041,0\.046![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x96.png)0\.621,0\.669![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x97.png)6,7grafo5780\.48![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x98.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x99.png)1\.143,0\.886![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x100.png)0\.221,0\.203![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x101.png)\-1\.657,\-1\.672![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x102.png)0\.086,0\.089![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x103.png)0\.972,0\.892![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x104.png)41,25grafo7023\.45![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x105.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x106.png)0\.913,0\.718![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x107.png)0\.156,0\.194![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x108.png)\-1\.836,\-1\.84![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x109.png)0\.063,0\.069![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x110.png)0\.886,0\.816![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x111.png)15,8grafo762\.27![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x112.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x113.png)0\.807,0\.568![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x114.png)0\.123,0\.157![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x115.png)\-1\.544,\-1\.567![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x116.png)0\.052,0\.057![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x117.png)0\.579,0\.554![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x118.png)3,3grafo8296\.86![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x119.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x120.png)0\.964,0\.771![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x121.png)0\.214,0\.24![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x122.png)\-2\.071,\-2\.079![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x123.png)0\.092,0\.102![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x124.png)1\.361,1\.257![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x125.png)81,66grafo847\.22![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x126.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x127.png)1\.136,0\.622![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x128.png)0\.121,0\.051![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x129.png)\-1\.257,\-1\.257![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x130.png)0\.051,0\.056![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x131.png)0\.421,0\.373![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x132.png)4,4grafo9033\.80![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x133.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x134.png)1\.025,0\.778![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x135.png)0\.219,0\.279![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x136.png)\-2\.066,\-2\.079![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x137.png)0\.083,0\.09![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x138.png)1\.235,1\.138![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x139.png)61,54grafo9592\.72![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x140.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x141.png)0\.859,0\.692![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x142.png)0\.142,0\.204![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x143.png)\-2\.584,\-2\.66![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x144.png)0\.037,0\.044![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x145.png)0\.913,0\.845![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x146.png)16,13Table A3:Stress for Neato layouts at different dimensions, vs that for the Optimal projection to 2D, and for the best PCA projection to 2D\.neato at KOptProj \(from K\-D to 2D\)PCA \(from K\-D to 2D\)23571035710357101138\_bus0\.0690\.0300\.0130\.0080\.0060\.0740\.0830\.0800\.0760\.0750\.0870\.1010\.113bcspwr070\.0460\.0220\.0100\.0070\.0060\.0570\.0620\.0630\.0620\.0580\.0680\.0740\.077email0\.1330\.0850\.0470\.0330\.0250\.1450\.1570\.1560\.1630\.1450\.1590\.1650\.171Erdos9910\.1090\.0630\.0330\.0230\.0170\.1240\.1310\.1310\.1290\.1250\.1430\.1500\.161EX10\.1670\.1030\.0500\.0350\.0330\.1690\.1730\.1760\.1760\.1690\.1770\.1770\.177football0\.1280\.0770\.0430\.0360\.0320\.1320\.1330\.1330\.1320\.1320\.1340\.1360\.139hypercube100\.1980\.1350\.0810\.0580\.0410\.1990\.2020\.2040\.2060\.1990\.2020\.2050\.208Journals0\.1670\.1140\.0710\.0550\.0430\.1760\.1800\.1850\.1820\.1780\.1820\.1940\.198mobius0\.0290\.0170\.0160\.0160\.0160\.0350\.0360\.0360\.0340\.0420\.0430\.0430\.043qh8820\.0560\.0280\.0140\.0100\.0090\.0690\.0710\.0740\.0720\.0720\.0800\.0870\.089Si20\.1590\.0880\.0500\.0360\.0270\.1390\.1420\.1430\.1430\.1390\.1420\.1430\.144spiral0\.0440\.0250\.0170\.0140\.0120\.0530\.0640\.0640\.0620\.0540\.0640\.0750\.078Trefethen\_7000\.1770\.1180\.0720\.0550\.0440\.1780\.1800\.1810\.1810\.1800\.1970\.2070\.218USpowerGrid0\.0580\.0280\.0110\.0070\.0050\.0660\.0740\.0770\.0730\.0660\.0770\.0820\.086SPC0\-0\.459\-0\.707\-0\.794\-0\.8440\.0880\.1220\.1270\.1220\.0880\.1360\.1630\.176
## Appendix A4Additional results for\\sfdp

Table[A4](https://arxiv.org/html/2606.31119#A4.T4)presents visualizations produced by\\sfdpon the SuiteSparse14 dataset\. Optimizing different metrics results in markedly different layouts\. For example, optimizing edge crossings \(xing\) produces a layout that differs substantially from the default\\sfdpoutput\. Optimizing the t\-SNE metric yields a visually less complex layout that appears to contain fewer edge crossings; however, closer inspection reveals many crossings hidden at finer scales\.

Table[A5](https://arxiv.org/html/2606.31119#A4.T5)presents the stress achieved by\\sfdpwhen optimizing directly in various dimensions from 2D to 10D\. It is surprising that the spring\-electrical energy obtained by\\sfdp\(10\) is mostly similar to that for\\sfdp\(2\), except for the Journals graph\. Furthermore, the spring\-electrical loss by DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}and DFsfdpPCA\{\}\_\{\\text\{PCA\}\}^\{\\text\{sfdp\}\}are also quite similar to that ofneato\. This seems to indicate that optimal layout for this loss function can be realized well in lower dimension\.

One way to decide whether a high\-dimensional embedding can “live” happily in a lower\-dimensional hyperplane is to look at the variance explained by PCA projections at a lower dimension\. This is confirmed in Table[A6](https://arxiv.org/html/2606.31119#A4.T6), where we looked at variance explained for SuiteSparse14 graphs by PCA projection from 10D of\\sfdpvsneato\. We can see that the variance explained for\\sfdpis indeed higher than forneato, at the same layout dimension\. E\.g\., PCA projection of 10D embeddings to 3D can explain 87\.8% of the original variance, vs 82\.5% forneato\.

Table A4:Optimal projections of the 10D\\sfdplayout\. Numbers show\\sfdpvs\. DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}metrics\.Bold= better\. Edges colored by length:red= short,blue= long\.sfdpang\_resedge\_lenspr\_elecstresstsnexing1138\_bus![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x147.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x148.png)1\.046,1\.105![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x149.png)0\.55,0\.569![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x150.png)\-4\.369,\-4\.322![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x151.png)0\.092,0\.098![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x152.png)1\.415,1\.584![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x153.png)511,756bcspwr07![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x154.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x155.png)1\.033,1\.03![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x156.png)0\.672,0\.568![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x157.png)\-4\.998,\-4\.937![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x158.png)0\.088,0\.084![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x159.png)1\.365,1\.526![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x160.png)678,1047email![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x161.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x162.png)1\.207,1\.097![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x163.png)0\.546,0\.529![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x164.png)\-2\.306,\-2\.269![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x165.png)0\.142,0\.15![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x166.png)1\.308,1\.309![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x167.png)425417,396344Erdos991![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x168.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x169.png)1\.26,1\.183![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x170.png)0\.502,0\.465![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x171.png)\-2\.313,\-2\.28![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x172.png)0\.142,0\.131![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x173.png)1\.378,1\.334![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x174.png)32632,27123EX1![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x175.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x176.png)0\.39,0\.367![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x177.png)0\.205,0\.183![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x178.png)\-1\.493,\-1\.486![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x179.png)0\.17,0\.175![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x180.png)1\.167,1\.089![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x181.png)492441,349871football![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x182.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x183.png)0\.539,0\.526![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x184.png)0\.519,0\.438![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x185.png)\-1\.196,\-1\.181![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x186.png)0\.138,0\.135![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x187.png)0\.508,0\.507![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x188.png)5622,4759hypercube10![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x189.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x190.png)0\.486,0\.356![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x191.png)0\.081,0\.04![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x192.png)\-1\.98,\-1\.97![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x193.png)0\.202,0\.206![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x194.png)2\.343,1\.991![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x195.png)518024,254686Journals![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x196.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x197.png)0\.077,0\.074![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x198.png)0\.446,0\.462![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x199.png)0\.21,0\.237![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x200.png)0\.17,0\.182![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x201.png)0\.078,0\.077![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x202.png)3720645,3241000mobius![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x203.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x204.png)0\.819,0\.65![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x205.png)0\.524,0\.193![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x206.png)\-3\.607,\-3\.592![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x207.png)0\.038,0\.038![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x208.png)0\.812,0\.73![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x209.png)37,21qh882![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x210.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x211.png)1\.311,1\.356![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x212.png)0\.874,0\.752![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x213.png)\-4\.484,\-4\.442![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x214.png)0\.083,0\.087![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x215.png)1\.399,1\.515![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x216.png)4337,5566Si2![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x217.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x218.png)0\.326,0\.307![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x219.png)0\.569,0\.393![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x220.png)\-1\.568,\-1\.561![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x221.png)0\.141,0\.146![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x222.png)0\.825,0\.83![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x223.png)1242149,1180389spiral![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x224.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x225.png)0\.349,0\.34![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x226.png)0\.685,0\.618![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x227.png)\-2\.857,\-2\.815![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x228.png)0\.076,0\.079![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x229.png)1\.424,1\.193![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x230.png)725528,615614Trefethen\_700![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x231.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x232.png)0\.352,0\.348![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x233.png)0\.519,0\.348![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x234.png)\-1\.512,\-1\.503![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x235.png)0\.18,0\.179![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x236.png)1\.239,1\.132![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x237.png)696237,504843USpowerGrid![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x238.png)![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x239.png)1\.066,1\.129![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x240.png)0\.795,0\.773![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x241.png)\-5\.633,\-5\.579![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x242.png)0\.098,0\.101![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x243.png)1\.712,2\.044![[Uncaptioned image]](https://arxiv.org/html/2606.31119v1/x244.png)3614,5207Table A5:Spring\-electrical loss for SFDP layouts at different dimensions, vs that for the Optimal projection to 2D, and for the best PCA projection to 2D\.sfdp at KOptProj \(from K\-D to 2D\)PCA \(from K\-D to 2D\)23571035710357101138\_bus\-4\.369\-4\.425\-4\.396\-4\.362\-4\.337\-4\.417\-4\.352\-4\.374\-4\.322\-4\.415\-4\.336\-4\.339\-4\.264bcspwr07\-4\.998\-4\.977\-4\.970\-4\.950\-4\.852\-4\.989\-4\.992\-4\.984\-4\.937\-4\.984\-4\.984\-4\.976\-4\.918email\-2\.306\-2\.368\-2\.392\-2\.402\-2\.406\-2\.274\-2\.258\-2\.262\-2\.269\-2\.273\-2\.237\-2\.227\-2\.234Erdos991\-2\.313\-2\.377\-2\.369\-2\.392\-2\.389\-2\.291\-2\.260\-2\.283\-2\.280\-2\.287\-2\.227\-2\.258\-2\.235EX1\-1\.493\-1\.606\-1\.678\-1\.700\-1\.701\-1\.487\-1\.486\-1\.486\-1\.486\-1\.476\-1\.459\-1\.469\-1\.458football\-1\.196\-1\.260\-1\.292\-1\.294\-1\.285\-1\.189\-1\.188\-1\.190\-1\.181\-1\.187\-1\.182\-1\.177\-1\.172hypercube10\-1\.980\-2\.084\-2\.160\-2\.192\-2\.203\-1\.966\-1\.975\-1\.973\-1\.970\-1\.961\-1\.936\-1\.909\-1\.951Journals0\.2100\.1050\.0410\.013\-0\.0070\.2250\.2460\.2380\.2370\.2280\.2600\.2720\.282mobius\-3\.607\-3\.606\-3\.591\-3\.590\-3\.523\-3\.605\-3\.603\-3\.605\-3\.592\-3\.596\-3\.594\-3\.594\-3\.588qh882\-4\.484\-4\.493\-4\.473\-4\.457\-4\.446\-4\.474\-4\.459\-4\.447\-4\.442\-4\.472\-4\.454\-4\.436\-4\.429Si2\-1\.568\-1\.628\-1\.666\-1\.679\-1\.686\-1\.566\-1\.562\-1\.562\-1\.561\-1\.566\-1\.562\-1\.561\-1\.560spiral\-2\.857\-2\.865\-2\.828\-2\.805\-2\.819\-2\.839\-2\.799\-2\.791\-2\.815\-2\.833\-2\.786\-2\.753\-2\.761Trefethen\_700\-1\.512\-1\.608\-1\.667\-1\.692\-1\.703\-1\.502\-1\.503\-1\.504\-1\.503\-1\.492\-1\.471\-1\.411\-1\.420USpowerGrid\-5\.633\-5\.619\-5\.593\-5\.564\-5\.532\-5\.629\-5\.614\-5\.595\-5\.579\-5\.626\-5\.609\-5\.576\-5\.573SPC\(sfdp\)↓\\downarrow0\.000\-0\.059\-0\.090\-0\.102\-0\.1060\.0090\.0180\.0150\.0170\.0110\.0280\.0360\.039Table A6:Average variance explained for SuiteSparse14 graphs by PCA projection from 10D of two algorithms, with\\sfdpshowing higher explained variance thanneato\. This suggests that minimum spring\-electrical energy is better realized in lower dimensions than minimum stress energy\.Dimensionneato\\sfdp10\.4710\.49720\.7020\.75430\.8250\.87840\.9010\.94050\.9480\.972
## Appendix A5Statistical Significance of Outperformance of DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}and DFsfdpPCA\{\}\_\{\\text\{PCA\}\}^\{\\text\{sfdp\}\}

Figure[A1](https://arxiv.org/html/2606.31119#A5.F1)\(top\) shows paired\-bootstrap \(n=5000n=5000\) 95% confidence intervals \(CIs\) for the SPC metric across various evaluation measures\. When a CI lies entirely below 0, it indicates that DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}outperformsneatowith statistical significance \(confirmed via a two\-sided bootstrap test\)\. Under this view, DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}clearly outperforms in angular resolution, t\-SNE score, and edge crossings\. Paired\-bootstrap 95% confidence intervals \(CIs\) for the SPC metric across various evaluation measures are given in Figure A2 in the Appendix\. We see that the outperformance of DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}over\\sfdpin angular resolution and edge length variation is statistically significant\.

Additional CIs for the other comparisons are shown in Figures A3 and A4\. Most of the differences are clear, except for DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}and SmartGD in crossings compared toSGD2\\text\{SGD\}^\{2\}\. Here, the SmartGD interval intersects 0, suggesting that there may be no difference in overall performance in crossing minimization between these graphs\. The DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}interval is close, but does not contain 0\. If we compute app\-value for whether the mean SPC between the algorithms is 0, we get0\.03940\.0394\.

![Refer to caption](https://arxiv.org/html/2606.31119v1/x245.png)

![Refer to caption](https://arxiv.org/html/2606.31119v1/x246.png)

Figure A1:Bootstrapped 95% confidence intervals for DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}\(top\) and DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}\(bottom\) improvement over Neato and SFDP, respectively, on SuiteSparse14 graphs\. Negative values favor DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}and DFoptsfdp\{\}^\{\\text\{sfdp\}\}\_\{\\text\{opt\}\}\.![Refer to caption](https://arxiv.org/html/2606.31119v1/x247.png)

![Refer to caption](https://arxiv.org/html/2606.31119v1/x248.png)

Figure A2:\(Top\) Bootstrapped CIs for crossing improvement of algorithms compared toS​G​D2SGD^\{2\}\. \(Bottom\) Bootstrapped CIs for DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}improvement overSGD2\\text\{SGD\}^\{2\}in angular resolution and edge length variation\. Both on the Rome20 graphs\.![Refer to caption](https://arxiv.org/html/2606.31119v1/x249.png)

![Refer to caption](https://arxiv.org/html/2606.31119v1/x250.png)

Figure A3:\(Top\) Bootstrapped CIs for crossing improvement of algorithms compared toneato, on the 14 SuiteSparse graphs in our benchmark\. \(Bottom\) Directly computed CIs across all Rome graphs for each algorithm’s improvement over Neato in edge crossings\.![Refer to caption](https://arxiv.org/html/2606.31119v1/x251.png)Figure A4:Drawing of the rome graph grafo7023 obtained by the Vertex Movement algorithm\. The drawing has a crossing number of 2, but ignores other aesthetic criteria\. See Table[A2](https://arxiv.org/html/2606.31119#A3.T2)row 5, last column, for a drawing of the same graph by DFoptneato\{\}^\{\\text\{neato\}\}\_\{\\text\{opt\}\}, which is aesthetically much more pleasing but with a crossing number of 8\.
## Appendix A6System Details

The system uses a Python/Flask backend with SocketIO for real\-time communication, and a JavaScript/HTML/CSS frontend featuring 3D graph visualization via Three\.js and interactive metric charts with Chart\.js\.

We improve efficiency and flexibility through on\-demand metric computation and lazy loading\. Expensive metrics like t\-SNE are computed and loaded only when requested via asynchronous API calls, allowing users to explore the visualization and basic metrics immediately while more expensive metrics \(e\.g\., t\-SNE, edge crossings\) are used as needed, keeping the system responsive and preventing blocking\. Smart caching and cancellation further enhance usability: users can cancel in\-progress jobs, toggle metric curves via the legend, and reuse cached results to avoid redundant work\.

## Appendix A7Scalability of the system

Built onThree\.js, an OpenGL\-based library,\\DataFlyleverages GPU acceleration to enable smooth, real\-time interaction with large graphs\. On the backend, the server\-side pipeline is also optimized for scalability\. These, combined with scalable layout algorithms, e\.g\.,sfdpor PivotMDS, allow\\DataFlyto quickly visualize the layouts of large graphs\.[Fig\.˜A7](https://arxiv.org/html/2606.31119#A7.F7)demonstrates the system rendering a 169K\-node graph using thesfdplayout from four different viewpoints\.

To quantitatively evaluate the practical scalability of our system, we measured both the total backend runtime and the time spent computing layouts across graphs of increasing size\. We selected graphs from the SuiteSparse matrix collection, ranging from 115 to 170k nodes: football \(115 nodes\), Erdos991 \(446 nodes\), email \(1133 nodes\), USpower \(4941 nodes\), c\-40 \(9941 nodes\), c\-66 \(49989 nodes\) and c\-73 \(169422 nodes\)\. Figure[A5](https://arxiv.org/html/2606.31119#A7.F5)and[A6](https://arxiv.org/html/2606.31119#A7.F6)show that layout computation dominates the overall runtime, especially for larger graphs\. The gap between the total backend time and the layout\-only time remains relatively modest compared with the layout cost itself, indicating that the main computational bottleneck lies in the layout generation procedure rather than in backend overhead\. This suggests that the system is scalable to large graphs\.

![Refer to caption](https://arxiv.org/html/2606.31119v1/x252.png)Figure A5:Computation time comparison between Pivot MDS only and Pivot MDS \+ system processing![Refer to caption](https://arxiv.org/html/2606.31119v1/x253.png)Figure A6:Computation time comparison between sfdp only and sfdp \+ system processing![Refer to caption](https://arxiv.org/html/2606.31119v1/images/system/c-73_vp1.png)![Refer to caption](https://arxiv.org/html/2606.31119v1/images/system/c-73_vp2.png)![Refer to caption](https://arxiv.org/html/2606.31119v1/images/system/c-73_vp3.png)![Refer to caption](https://arxiv.org/html/2606.31119v1/images/system/c-73_vp4.png)Figure A7:The c\-73 graph \(169422 nodes, 554926 edges\) from four viewpoints in\\DataFly\.
## Appendix A8System iterative improvements and future work

Expert E1 and non\-expert N2 from the usability study suggested additional interaction features\. E1 proposed automatic neighborhood highlighting \(selecting a node and its immediate neighbors\) and edge\-based tracking through transitions, while N2 suggested a “lock\-rotation” button and highlight\-on\-hover functionality\.

In response, we have implemented several improvements in the live tool\. A pause\-and\-resume function for animated transitions allows users to freeze the rotation at any viewpoint using the space key, supporting detailed inspection\. Dynamic node and label highlighting, with enlargement on hover, improves visual clarity, particularly in dense projections\.

Building on participant feedback, future work will further enhance interaction support, including ego\-network and edge highlighting, to reduce manual effort during neighbor\-tracing tasks\. To bridge the gap between expert and non\-expert users, we will introduce guided onboarding, contextual tool\-tips, and progressive disclosure of advanced features to manage interaction complexity\. Additionally, we plan to explore optimal “tours” between viewpoints—extending current linear or Grassmann geodesic transitions—to minimize both travel distance and a chosen quality metric \(e\.g\., integrated stress\), inspired by projection pursuit\[cook1995grand\]\.

## Appendix A9Additional Related Work

Our work connects to broader themes in graph and network visualization quality assessment\. Bertini and Santucci\[bertini2006visual\]proposed a systematization of visual quality metrics for information visualization, classifying them into size, visual effectiveness, and feature preservation categories, justifying the principled use of quality metrics that we adopt for evaluating viewpoints\. In a similar spirit, diagnostic approaches have been developed for specific visual representations: Gove\[gove2019gragnostics\]introduced*Gragnostics*, a set of fast, interpretable graph\-level features for comparing graph topologies, while Behrisch et al\.\[behrisch2017magnostics\]proposed*Magnostics*, an image\-based approach for ranking matrix views of networks according to the presence of visual patterns such as blocks and lines, and identified extending their diagnostics to hybrid representations such as*NodeTrix*\[henry2007nodetrix\]as future work\.*NodeTrix*, introduced by Henry et al\., combines node\-link diagrams with adjacency matrices to simultaneously convey global network structure and local community detail, highlighting the long\-standing challenge of representing dense substructures and reinforcing the case for our multi\-view approach\.