Graph Drawing Research Articles

Characterizing Planar Tanglegram Layouts and Applications to Edge Insertion Problems

Tanglegrams are formed by taking two rooted binary trees $T$ and $S$ with the same number of leaves and uniquely matching each leaf in $T$ with a leaf in $S$. They are usually represented using layouts that embed the trees and matching in the plane. Given the numerous ways to construct a layout, one problem of interest is the Tanglegram Layout Problem, which is to efficiently find a layout that minimizes the number of crossings. This parallels a similar problem involving drawings of graphs, where a common approach is to insert edges into a planar subgraph. In this paper, we explore inserting edges into a planar tanglegram. Previous results on planar tanglegrams include a Kuratowski Theorem, enumeration, and an algorithm for finding a planar layout. We build on these results and characterize all planar layouts of a planar tanglegram. We then apply this characterization to construct a quadratic-time algorithm that inserts a single edge optimally. Finally, we generalize some results to multiple edge insertion.

Unfolding Edges: Adding Context to Edges in Multivariate Graph Visualization

AbstractExisting work on visualizing multivariate graphs is primarily concerned with representing the attributes of nodes. Even though edges are the constitutive elements of networks, there have been only few attempts to visualize attributes of edges. In this work, we focus on the critical importance of edge attributes for interpreting network visualizations and building trust in the underlying data. We propose ‘unfolding of edges’ as an interactive approach to integrate multivariate edge attributes dynamically into existing node‐link diagrams. Unfolding edges is an in‐situ approach that gradually transforms basic links into detailed representations of the associated edge attributes. This approach extends focus+context, semantic zoom, and animated transitions for network visualizations to accommodate edge details on‐demand without cluttering the overall graph layout. We explore the design space for the unfolding of edges, which covers aspects of making space for the unfolding, of actually representing the edge context, and of navigating between edges. To demonstrate the utility of our approach, we present two case studies in the context of historical network analysis and computational social science. For these, web‐based prototypes were implemented based on which we conducted interviews with domain experts. The experts' feedback suggests that the proposed unfolding of edges is a useful tool for exploring rich edge information of multivariate graphs.

The peripheral TCR repertoire diversity indexes can be used as an independent indicator for long term survival rate of ovarian cancer patients.

e17558 Background: According to GLOBOCAN 2020 data, ovarian cancer (OC) ranks 19th in incidence and 15th in mortality with a rate of 1.6% and 2.1%, respectively, High-grade serous OC is the most common sub-type of malignant ovarian tumors, remains the leading cause of cancer-related death among all gynecological cancers in the developed parts of the world. In the United States in 2016, 5-year survival rate was only 10-25% in stage III or IV OC patients, and there was considerable variability in survival in an advanced stage. At present, there is no reliable molecular marker to identify patients with better or worse prognosis. As is known to all, the adaptive immune system defends the body against different foreign antigens, using two main types of antigen receptors: T-cell receptors (TCRs) and antibodies. Now we use high-throughput sequencing approach to analyze the expressed TCRβ repertoires in the peripheral blood of OC patients and healthy women donors. Methods: The experimental design and participant recruitment were approved by Medical Ethics Committee of West China Second University Hospital (Ethics Approval for Research: 2022). Total RNA was extracted from PBMC using TRIZOL reagent. High through put sequencing of the TCRβ gene, assignment of the Vβ, Dβ, and Jβ germline gene, and TCR diversity indexes analyses were conducted essentially following our previously published protocols. Statistical analysis and graph drawing were performed using R language (R 4.0.3). Results: Comparison of the TCR repertoires between the 232 OC patients (OC group) and the 250 healthy women donors (HD group) revealed significant differences. For examples, the VJ combinations of the OC group were distinct from those of the HD group. In addition, the TCR repertoire diversity indexes of the OC group were significantly lower than those of the HD group ( p < 0.001), including D50s, Simpson indexes, the number of highly expressed clones (HECs), and the frequencies of the largest TCR clones. As reported before, the TCR repertoire diversity indexes decreased gradually with increasing age in the HD group (all p < 0.001). Moreover, the TCR repertoire diversities in the survival group were significantly higher than those in the death group (D50: p = 0.0342; the number of HECs: p = 0.0414). Using as the cut off value, the OC patients with D50 < 0.032 had significantly higher mortality rates compared to the OC patients with D50 > 0.032 ( p = 0.0019). Similarly, using the numbers of HECs or the frequencies of the largest TCR clones can also distinguish OC patients in long term survival. Conclusions: Analyzing the peripheral TCR repertoires in the OC and HD revealed significant differences, and the peripheral TCR repertoire diversity indexes may be used as a potential predictor for long term survival for ovarian cancer.

An algebraic construction technique for codes over Hurwitz integers

Let $\alpha$ be a prime Hurwitz integer. $\mathcal{H}_{\alpha}$, which is the set of residual class with respect to related modulo function in the rings of Hurwitz integers, is a subset of $\mathcal{H},$ which is the set of all Hurwitz integers. In this study, we present an algebraic construction technique, which is a modulo function formed depending on two modulo operations, for codes over Hurwitz integers. We consider left congruent modulo $\alpha,$ and the domain of related modulo function is $\mathbb{Z}_{N(\alpha)},$ which is residual class ring of ordinary integers with $N(\alpha)$ elements. Therefore, we obtain the residue class rings of Hurwitz integers with $N(\alpha)$ size. In addition, we present some results for mathematical notations used in two modulo functions, and for the algebraic construction technique formed depending upon two modulo functions. Moreover, we presented graphs obtained by graph layout methods, such as spring, high-dimensional, and spiral embedding, for the set of the residual class obtained with respect to the related modulo function in the rings of Hurwitz integers.

CryptoComparator: Supporting cryptocurrencies analysis through Visual Analytics

Cryptocurrencies are a novel phenomenon in the finance world that is gaining more attention from the general public, banks, investors, and academic research lately. A characteristic of cryptocurrencies is to be the target of investments that, due to the volatility of most of the cryptocurrencies, tends to be at high risk and behave very differently from traditional currencies. A way of reducing this risk is to look at the history of existing cryptocurrencies and compare them to spot promising trends for increased gain. This paper introduces CryptoComparator, a Visual Analytics tool designed to analyze the correlations and trends of cryptocurrencies. The system exploits an initial proposal for a double elliptic graph layout, reconfigurable with three different ordering functions, to support a fast visual search of cryptocurrencies by correlation strength. Two use cases developed with a domain expert in cryptocurrency financial activities demonstrate qualitatively the support it provides for analyzing cryptocurrencies.

NeatSankey: Sankey diagrams with improved readability based on node positioning and edge bundling

Sankey diagrams are widely used to visualize event sequence data. However, when the data volume is large, its readability is affected by dense edge crossings, excessive swing amplitude, and small crossover angles, while it is computationally intensive to obtain an optimal layout. In this paper, we propose NeatSankey, a balanced method that generates Sankey diagrams smoothly. It can be laid out quickly with good readability when Sankey diagrams are very complex. At the same time, to comprehensively evaluate the readability of Sankey diagrams, we use three evaluation metrics: crossing number, swing amplitude, and layout coverage. Firstly, we use a heuristic layout algorithm and a force-directed algorithm to adjust the node layout to minimize the edge crossings and swing amplitude with edge widths considered. Secondly, to better reduce the dense confusion caused by edge crossings, we introduce a edge bundling algorithm based on attribute similarity. We present three evaluations: a comprehensive comparison of our results with state-of-the-art techniques, user studies with thirty volunteers, and a case study of two datasets. Our evaluations demonstrate the effectiveness and practicability of the NeatSankey.

Graph product structure for non-minor-closed classes

Dujmović et al. [J. ACM '20] proved that every planar graph is isomorphic to a subgraph of the strong product of a bounded treewidth graph and a path. Analogous results were obtained for graphs of bounded Euler genus or apex-minor-free graphs. These tools have been used to solve longstanding problems on queue layouts, non-repetitive colouring, p-centered colouring, and adjacency labelling. This paper proves analogous product structure theorems for various non-minor-closed classes. One noteable example is k-planar graphs (those with a drawing in the plane in which each edge is involved in at most k crossings). We prove that every k-planar graph is isomorphic to a subgraph of the strong product of a graph of treewidth O(k5) and a path. This is the first result of this type for a non-minor-closed class of graphs. It implies, amongst other results, that k-planar graphs have non-repetitive chromatic number upper-bounded by a function of k. All these results generalise for drawings of graphs on arbitrary surfaces. In fact, we work in a more general setting based on so-called shortcut systems, which are of independent interest. This leads to analogous results for certain types of map graphs, string graphs, graph powers, and nearest neighbour graphs.

Splitting Vertices in 2-Layer Graph Drawings.

Bipartite graphs model the relationships between two disjoint sets of entities in several applications and are naturally drawn as 2-layer graph drawings. In such drawings, the two sets of entities (vertices) are placed on two parallel lines (layers), and their relationships (edges) are represented by segments connecting vertices. Methods for constructing 2-layer drawings often try to minimize the number of edge crossings. We use vertex splitting to reduce the number of crossings, by replacing selected vertices on one layer by two (or more) copies and suitably distributing their incident edges among these copies. We study several optimization problems related to vertex splitting, either minimizing the number of crossings or removing all crossings with fewest splits. While we prove that some variants are ${\mathsf {NP}}$NP-complete, we obtain polynomial-time algorithms for others. We run our algorithms on a benchmark set of bipartite graphs representing the relationships between human anatomical structures and cell types.

High School Students’ User Skills Concerning Force and Motion Graphs

In this study, the aim was to examine Year 11 students’ skills of reading-interpreting and drawing graphs of force and motion and to lay bare the relationship between graph reading-interpretation and drawing graphs. Conducted in the survey model, the study was realised with the participation of 209 Year 11 students studying at Anatolian high schools in Ankara. Graph Reading and Interpretation Skills Test (GRIST) which includes 13 multiple-choice items and Graph Drawing Skills Form (GDSF) which includes 5 open-ended items were used as data collection tools. At the end of the study, it was determined that students have an intermediate level of success in reading and interpreting graphs while they have a low level of success in drawing graphs. It was concluded that students’ graph reading and interpretation skills and their graph drawing skills are related to one another.

Crossing edge minimization in radial outerplanar layered graphs using segment paths

Crossing minimization is a natural problem in graph drawing, which consists of drawing it in the plane with a minimum number of crossings on the edges. We present an algorithm to minimize the crossing edges in radial layered graphs. The algorithm determines the optimal location of nodes in their corresponding circle layer and consists of two phases. In the first phase, a counterclockwise rotation of concentric circles is performed; in the next phase, edges are converted into segment paths. Unlike other algorithms with circular drawings, our algorithm does not require the insertion or removal of nodes and edges; only rotation and path construction operations are utilized. Two versions of the algorithm were created; the exhaustive version tests all the possible paths, while the random version tests a few number of paths. The goal is to minimize the crossing edges to obtain a clear visualization using three criteria: rotation, crossing edge detection and segment path construction. The algorithm has been successfully implemented in 30 instances of graphs with 20, 35 and 50 layers, where crossings are minimized in 86–93%, 93–96% and 95–97%, respectively.

The maximum linear arrangement problem for trees under projectivity and planarity

A linear arrangement is a mapping π from the n vertices of a graph G to n distinct consecutive integers. Linear arrangements can be represented by drawing the vertices along a horizontal line and drawing the edges as semicircles above said line. In this setting, the length of an edge is defined as the absolute value of the difference between the positions of its two vertices in the arrangement, and the cost of an arrangement as the sum of all edge lengths. Here we study two variants of the Maximum Linear Arrangement problem (MaxLA), which consists of finding an arrangement that maximizes the cost. In the planar variant for free trees, vertices have to be arranged in such a way that there are no edge crossings. In the projective variant for rooted trees, arrangements have to be planar and the root of the tree cannot be covered by any edge. In this paper we present algorithms that are linear in time and space to solve planar and projective MaxLA for trees. We also prove several properties of maximum projective and planar arrangements, and show that caterpillar trees maximize planar MaxLA over all trees of a fixed size thereby generalizing a previous extremal result on trees.

GeoGraphViz: Geographically constrained 3D force‐directed graph for knowledge graph visualization

AbstractKnowledge graphs are a key technique for linking and integrating cross‐domain data, concepts, tools, and knowledge to enable data‐driven analytics. As much of the world's data have become massive in size, visualizing graph entities and their interrelationships intuitively and interactively has become a crucial task for ingesting and better utilizing graph content to support semantic reasoning, discovering hidden knowledge discovering, and better scientific understanding of geophysical and social phenomena. Despite the fact that many such phenomena (e.g., disasters) have clear spatial footprints and geographic properties, their location information is considered only as a textual label in existing graph visualization tools, limiting their capability to reveal the geospatial distribution patterns of the graph nodes. In addition, most graph visualization techniques rely on 2D graph visualization, which constrains the dimensions of information that can be presented and lacks support for graph structure examination from multiple angles. To tackle the above challenges, we developed a novel 3D map‐based graph visualization algorithm to enable interactive exploration of graph content and patterns in a spatially explicit manner. The algorithm extends a 3D force directed graph by integrating a web map, an additional geolocational force, and a force balancing variable that allows for the dynamic adjustment of the 3D graph structure and layout. This mechanism helps create a balanced graph view between the semantic forces among the graph nodes and the attractive force from a geolocation to a graph node. Our solution offers a new perspective in visualizing and understanding spatial entities and events in a knowledge graph.

Accelerating network layouts using graph neural networks

Graph layout algorithms used in network visualization represent the first and the most widely used tool to unveil the inner structure and the behavior of complex networks. Current network visualization software relies on the force-directed layout (FDL) algorithm, whose high computational complexity makes the visualization of large real networks computationally prohibitive and traps large graphs into high energy configurations, resulting in hard-to-interpret “hairball” layouts. Here we use Graph Neural Networks (GNN) to accelerate FDL, showing that deep learning can address both limitations of FDL: it offers a 10 to 100 fold improvement in speed while also yielding layouts which are more informative. We analytically derive the speedup offered by GNN, relating it to the number of outliers in the eigenspectrum of the adjacency matrix, predicting that GNNs are particularly effective for networks with communities and local regularities. Finally, we use GNN to generate a three-dimensional layout of the Internet, and introduce additional measures to assess the layout quality and its interpretability, exploring the algorithm’s ability to separate communities and the link-length distribution. The novel use of deep neural networks can help accelerate other network-based optimization problems as well, with applications from reaction-diffusion systems to epidemics.

Advantages of Using Geogebra Software when Examining the Flow and Drawing a Graph of a Function

For a long time, the term function has been treated as a mandatory topic in mathematics curricula around the world. It aims to emphasize the importance of this topic of mathematics in higher education, at technical faculties. In addition to the classic way of learning blackboard and chalk, today all over the world a variety of interactive mathematical software is often used in the study of the topic functions. The software helps students to learn how to recognize functions, to examine their properties, and draw them accurately. In this paper, the importance of the mathematical term function in higher education at technical faculties is stated. It also lists the ambiguities that students face when mastering the topic of functions. Then, software for drawing graphs of functions (material that is taught in the subject Mathematics 1 at technical faculties) is listed to help students master that material more easily. Finally, a comparison is made in the results of the final exams in Mathematics 1, before students are recommended to use GeoGebra and after using the software. A conclusion is drawn from the results and the advantages of using the mathematical software are presented.

Computing Bend-Minimum Orthogonal Drawings of Plane Series–Parallel Graphs in Linear Time

AbstractA planar orthogonal drawing of a planar 4-graph G (i.e., a planar graph with vertex-degree at most four) is a crossing-free drawing that maps each vertex of G to a distinct point of the plane and each edge of G to a polygonal chain consisting of horizontal and vertical segments. A longstanding open question in Graph Drawing, dating back over 30 years, is whether there exists a linear-time algorithm to compute an orthogonal drawing of a plane 4-graph with the minimum number of bends. The term “plane” indicates that the input graph comes together with a planar embedding, which must be preserved by the drawing (i.e., the drawing must have the same set of faces as the input graph). In this paper we positively answer the question above for the widely-studied class of series–parallel graphs. Our linear-time algorithm is based on a characterization of the planar series–parallel graphs that admit an orthogonal drawing without bends. This characterization is given in terms of the orthogonal spirality that each type of triconnected component of the graph can take; the orthogonal spirality of a component measures how much that component is “rolled-up” in an orthogonal drawing of the graph.

Bitonic st-Orderings for Upward Planar Graphs: Splits and Bends in the Variable Embedding Scenario

Bitonic st-orderings for st-planar graphs were introduced as a method to cope with several graph drawing problems. Notably, they have been used to obtain the best-known upper bound on the number of bends for upward planar polyline drawings with at most one bend per edge in polynomial area. For an st-planar graph that does not admit a bitonic st-ordering, one may split certain edges such that for the resulting graph such an ordering exists. Since each split is interpreted as a bend, one is usually interested in splitting as few edges as possible. While this optimization problem admits a linear-time algorithm in the fixed embedding setting, it remains open in the variable embedding setting. We close this gap in the literature by providing a linear-time algorithm that optimizes over all embeddings of the input st-planar graph. The best-known lower bound on the number of required splits of an st-planar graph with n vertices is n-3. However, it is possible to compute a bitonic st-ordering without any split for the st-planar graph obtained by reversing the orientation of all edges. In terms of upward planar polyline drawings in polynomial area, the former translates into n-3 bends, while the latter into no bends. We show that this idea cannot always be exploited by describing an st-planar graph that needs at least n-5 splits in both orientations. We provide analogous bounds for graphs with small degree. Finally, we further investigate the relationship between splits in bitonic st-orderings and bends in upward planar polyline drawings with polynomial area, by providing bounds on the number of bends in such drawings.

GNN at the Edge: Cost-Efficient Graph Neural Network Processing Over Distributed Edge Servers

Edge intelligence has arisen as a promising computing paradigm for supporting miscellaneous smart applications that rely on machine learning techniques. While the community has extensively investigated multi-tier edge deployment for traditional deep learning models (e.g. CNNs, RNNs), the emerging Graph Neural Networks (GNNs) are still under exploration, presenting a stark disparity to its broad edge adoptions such as traffic flow forecasting and location-based social recommendation. To bridge this gap, this paper formally studies the cost optimization for distributed GNN processing over a multi-tier heterogeneous edge network. We build a comprehensive modeling framework that can capture a variety of different cost factors, based on which we formulate a cost-efficient graph layout optimization problem that is proved to be NP-hard. Instead of trivially applying traditional data placement wisdom, we theoretically reveal the structural property of quadratic submodularity implicated in GNN’s unique computing pattern, which motivates our design of an efficient iterative solution exploiting graph cuts. Rigorous analysis shows that it provides parameterized constant approximation ratio, guaranteed convergence, and exact feasibility. To tackle potential graph topological evolution in GNN processing, we further devise an incremental update strategy and an adaptive scheduling algorithm for lightweight dynamic layout optimization. Evaluations with real-world datasets and various GNN benchmarks demonstrate that our approach achieves superior performance over de facto baselines with more than 95.8% cost reduction in a fast convergence speed.

Classification for the Concrete Syntax of Graph-Like Modeling Languages

A classification scheme for Graph-Like Modeling Languages (GLML) is presented in this paper. The novelty of this classifier lies in its application to a meta-model for GLML that deviates from the simple graph model and underlies a large number of GLML. The main goal of using this classification scheme is to support the reuse of layout algorithms for GLML. GLML are used directly or indirectly for the development of software by model-based software engineering techniques. In other domains, graph-like models are artifacts (e.g., circuit diagrams, energy flow diagrams) that serve as input for downstream specialized applications (simulators, optimizers). The concrete syntax of a language for creating, editing, and understanding models is highly important for the development of modeling tools. Layout methods for the used languages have to be implemented to achieve software tools with good usability. Developing layout algorithms is a complex topic that is covered by the specialized field of Graph Drawing. However, there is no existing procedure to determine which layout algorithm can be used for a GLML. Matching layout algorithms to GLML can be achieved by applying the presented classification scheme.

Simplifying Non-Simple Fan-Planar Drawings

A drawing of a graph is fan-planar if the edges intersecting a common edge $a$ share a vertex $A$ on the same side of $a$. More precisely, orienting $a$ arbitrarily and the other edges towards $A$ results in a consistent orientation of the crossings. So far, fan-planar drawings have only been considered in the context of simple drawings, where any two edges share at most one point, including endpoints. We show that every non-simple fan-planar drawing can be redrawn as a simple fan-planar drawing of the same graph while not introducing additional crossings. The proof is constructive and corresponds to a quadratic time algorithm. Combined with previous results on fan-planar drawings, this yields that $n$-vertex graphs having such a drawing can have at most $6.5n-20$ edges and that the recognition of such graphs is NP-hard. We thereby answer an open problem posed by Kaufmann and Ueckerdt in 2014.

Upward Planar Drawings with Three and More Slopes

The slope number of a graph $G$ is the smallest number of slopes needed for the segments representing the edges in any straight-line drawing of $G$. It serves as a measure of the visual complexity of a graph drawing. Several bounds on the slope number for particular graph classes have been established, both in the planar and the non-planar setting. Moreover, the slope number can also be defined for directed graphs and upward planar drawings. We study upward planar straight-line drawings that use only a constant number of slopes. In particular, for a fixed number $k$ of slopes, we are interested in whether a given directed graph $G$ with maximum in- and outdegree at most $k$ admits an upward planar $k$-slope drawing. We investigate this question both in the fixed and the variable embedding scenario. We show that this problem is in general NP-hard to decide for outerplanar graphs ($k = 3$) and planar graphs ($k \ge 3$). On the positive side, we can decide whether a given cactus graph admits an upward planar $k$-slope drawing and, in the affirmative, construct such a drawing in FPT time with parameter $k$. Furthermore, we can determine the minimum number of slopes required for a given tree in linear time and compute the corresponding drawing efficiently.