arXiv subjects

Peter Eades

Publications and source records attributed to Peter Eades.

18 recordsLinked to original sources

arXiv2024

GraphTrials: Visual Proofs of Graph Properties

Graph and network visualization supports exploration, analysis and communication of relational data arising in many domains: from biological and social networks, to transportation and powergrid systems. With the arrival of AI-based question-answering tools, issues of trustworthiness and explainability of generated answers motivate a greater role for visualization. In the context of graphs, we see the need for visualizations that can convince a critical audience that an assertion about the graph under analysis is valid. The requirements for such representations that convey precisely one specific graph property are quite different from standard network visualization criteria which optimize general aesthetics and readability. In this paper, we aim to provide a comprehensive introduction to visual proofs of graph properties and a foundation for further research in the area. We present a framework that defines what it means to visually prove a graph property. In the process, we introduce the notion of a visual certificate, that is, a specialized faithful graph visualization that leverages the viewer's perception, in particular, pre-attentive processing (e.g. via pop-out effects), to verify a given assertion about the represented graph. We also discuss the relationships between visual complexity, cognitive load and complexity theory, and propose a classification based on visual proof complexity. Finally, we provide examples of visual certificates for problems in different visual proof complexity classes.

cs.HC↗

arXiv2023

CelticGraph: Drawing Graphs as Celtic Knots and Links

Celtic knots are an ancient art form often attributed to Celtic cultures, used to decorate monuments and manuscripts, and to symbolise eternity and interconnectedness. This paper describes the framework CelticGraph to draw graphs as Celtic knots and links. The drawing process raises interesting combinatorial concepts in the theory of circuits in planar graphs. Further, CelticGraph uses a novel algorithm to represent edges as Bézier curves, aiming to show each link as a smooth curve with limited curvature.

cs.CG↗

arXiv2022

Shape-Faithful Graph Drawings

Shape-based metrics measure how faithfully a drawing D represents the structure of a graph G, using the proximity graph S of D. While some limited graph classes admit proximity drawings (i.e., optimally shape-faithful drawings, where S = G), algorithms for shape-faithful drawings of general graphs have not been investigated. In this paper, we present the first study for shape-faithful drawings of general graphs. First, we conduct extensive comparison experiments for popular graph layouts using the shape-based metrics, and examine the properties of highly shape-faithful drawings. Then, we present ShFR and ShSM, algorithms for shape-faithful drawings based on force-directed and stress-based algorithms, by introducing new proximity forces/stress. Experiments show that ShFR and ShSM obtain significant improvement over FR (Fruchterman-Reingold) and SM (Stress Majorization), on average 12% and 35% respectively, on shape-based metrics.

cs.DS↗

arXiv2020

New Quality Metrics for Dynamic Graph Drawing

In this paper, we present new quality metrics for dynamic graph drawings. Namely, we present a new framework for change faithfulness metrics for dynamic graph drawings, which compare the ground truth change in dynamic graphs and the geometric change in drawings. More specifically, we present two specific instances, cluster change faithfulness metrics and distance change faithfulness metrics. We first validate the effectiveness of our new metrics using deformation experiments. Then we compare various graph drawing algorithms using our metrics. Our experiments confirm that the best cluster (resp. distance) faithful graph drawing algorithms are also cluster (resp. distance) change faithful.

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arXiv2020

A Study of Mental Maps in Immersive Network Visualization

The visualization of a network influences the quality of the mental map that the viewer develops to understand the network. In this study, we investigate the effects of a 3D immersive visualization environment compared to a traditional 2D desktop environment on the comprehension of a network's structure. We compare the two visualization environments using three tasks--interpreting network structure, memorizing a set of nodes, and identifying the structural changes--commonly used for evaluating the quality of a mental map in network visualization. The results show that participants were able to interpret network structure more accurately when viewing the network in an immersive environment, particularly for larger networks. However, we found that 2D visualizations performed better than immersive visualization for tasks that required spatial memory.

cs.HC↗

arXiv2019

A Quality Metric for Symmetric Graph Drawings

Symmetry is an important aesthetic criteria in graph drawing and network visualisation. Symmetric graph drawings aim to faithfully represent automorphisms of graphs as geometric symmetries in a drawing. In this paper, we design and implement a framework for quality metrics that measure symmetry, that is, how faithfully a drawing of a graph displays automorphisms as geometric symmetries. The quality metrics are based on geometry (i.e. Euclidean distance) as well as mathematical group theory (i.e. orbits of automorphisms). More specifically, we define two varieties of symmetry quality metrics: (1) for displaying a single automorphism as a symmetry (axial or rotational) and (2) for displaying a group of automorphisms (cyclic or dihedral). We also present algorithms to compute the symmetric quality metrics in O(n log n) time for rotational symmetry and axial symmetry. We validate our symmetry quality metrics using deformation experiments. We then use the metrics to evaluate a number of established graph drawing layouts to compare how faithfully they display automorphisms of a graph as geometric symmetries.

cs.DS↗

arXiv2019

Multi-level Graph Drawing using Infomap Clustering

Infomap clustering finds the community structures that minimize the expected description length of a random walk trajectory; algorithms for infomap clustering run fast in practice for large graphs. In this paper we leverage the effectiveness of Infomap clustering combined with the multi-level graph drawing paradigm. Experiments show that our new Infomap based multi-level algorithm produces good visualization of large and complex networks, with significant improvement in quality metrics.

cs.DS↗

arXiv2019

A Quality Metric for Visualization of Clusters in Graphs

Traditionally, graph quality metrics focus on readability, but recent studies show the need for metrics which are more specific to the discovery of patterns in graphs. Cluster analysis is a popular task within graph analysis, yet there is no metric yet explicitly quantifying how well a drawing of a graph represents its cluster structure. We define a clustering quality metric measuring how well a node-link drawing of a graph represents the clusters contained in the graph. Experiments with deforming graph drawings verify that our metric effectively captures variations in the visual cluster quality of graph drawings. We then use our metric to examine how well different graph drawing algorithms visualize cluster structures in various graphs; the results con-firm that some algorithms which have been specifically designed to show cluster structures perform better than other algorithms.

cs.DS↗

arXiv2019

Gap-planar Graphs

We introduce the family of $k$-gap-planar graphs for $k \geq 0$, i.e., graphs that have a drawing in which each crossing is assigned to one of the two involved edges and each edge is assigned at most $k$ of its crossings. This definition is motivated by applications in edge casing, as a $k$-gap-planar graph can be drawn crossing-free after introducing at most $k$ local gaps per edge. We present results on the maximum density of $k$-gap-planar graphs, their relationship to other classes of beyond-planar graphs, characterization of $k$-gap-planar complete graphs, and the computational complexity of recognizing $k$-gap-planar graphs.

cs.CG↗

arXiv2018

Towards Faithful Graph Visualizations

Readability criteria have been addressed as a measurement of the quality of graph visualizations. In this paper, we argue that readability criteria are necessary but not sufficient. We propose a new kind of criteria, namely faithfulness, to evaluate the quality of graph layouts. We introduce a general model for quantify faithfulness, and contrast it with the well established readability criteria. We show examples of common visualization techniques, such as multidimensional scaling, edge bundling and several other visualization metaphors for the study of faithfulness.

cs.CG↗

arXiv2018

Polyline Drawings with Topological Constraints

Let $G$ be a simple topological graph and let $Γ$ be a polyline drawing of $G$. We say that $Γ$ \emph{partially preserves the topology} of $G$ if it has the same external boundary, the same rotation system, and the same set of crossings as $G$. Drawing $Γ$ fully preserves the topology of $G$ if the planarization of $G$ and the planarization of $Γ$ have the same planar embedding. We show that if the set of crossing-free edges of $G$ forms a connected spanning subgraph, then $G$ admits a polyline drawing that partially preserves its topology and that has curve complexity at most three (i.e., at most three bends per edge). If, however, the set of crossing-free edges of $G$ is not a connected spanning subgraph, the curve complexity may be $Ω(\sqrt{n})$. Concerning drawings that fully preserve the topology, we show that if $G$ has skewness $k$, it admits one such drawing with curve complexity at most $2k$; for skewness-1 graphs, the curve complexity can be reduced to one, which is a tight bound. We also consider optimal $2$-plane graphs and discuss trade-offs between curve complexity and crossing angle resolution of drawings that fully preserve the topology.

cs.CG↗

arXiv2018

The Weighted Barycenter Drawing Recognition Problem

We consider the question of whether a given graph drawing $Γ$ of a triconnected planar graph $G$ is a weighted barycenter drawing. We answer the question with an elegant arithmetic characterisation using the faces of $Γ$. This leads to positive answers when the graph is a Halin graph, and to a polynomial time recognition algorithm when the graph is cubic.

cs.CG↗

arXiv2018

Turning Cliques into Paths to Achieve Planarity

Motivated by hybrid graph representations, we introduce and study the following beyond-planarity problem, which we call $h$-Clique2Path Planarity: Given a graph $G$, whose vertices are partitioned into subsets of size at most $h$, each inducing a clique, remove edges from each clique so that the subgraph induced by each subset is a path, in such a way that the resulting subgraph of $G$ is planar. We study this problem when $G$ is a simple topological graph, and establish its complexity in relation to $k$-planarity. We prove that $h$-Clique2Path Planarity is NP-complete even when $h=4$ and $G$ is a simple $3$-plane graph, while it can be solved in linear time, for any $h$, when $G$ is $1$-plane.

cs.DS↗

arXiv2017

Drawing Big Graphs using Spectral Sparsification

Spectral sparsification is a general technique developed by Spielman et al. to reduce the number of edges in a graph while retaining its structural properties. We investigate the use of spectral sparsification to produce good visual representations of big graphs. We evaluate spectral sparsification approaches on real-world and synthetic graphs. We show that spectral sparsifiers are more effective than random edge sampling. Our results lead to guidelines for using spectral sparsification in big graph visualization.

cs.CG↗

arXiv2016

Simultaneous Orthogonal Planarity

We introduce and study the $\textit{OrthoSEFE}-k$ problem: Given $k$ planar graphs each with maximum degree 4 and the same vertex set, do they admit an OrthoSEFE, that is, is there an assignment of the vertices to grid points and of the edges to paths on the grid such that the same edges in distinct graphs are assigned the same path and such that the assignment induces a planar orthogonal drawing of each of the $k$ graphs? We show that the problem is NP-complete for $k \geq 3$ even if the shared graph is a Hamiltonian cycle and has sunflower intersection and for $k \geq 2$ even if the shared graph consists of a cycle and of isolated vertices. Whereas the problem is polynomial-time solvable for $k=2$ when the union graph has maximum degree five and the shared graph is biconnected. Further, when the shared graph is biconnected and has sunflower intersection, we show that every positive instance has an OrthoSEFE with at most three bends per edge.

cs.CG↗

arXiv2015

Straight-line Drawability of a Planar Graph Plus an Edge

We investigate straight-line drawings of topological graphs that consist of a planar graph plus one edge, also called almost-planar graphs. We present a characterization of such graphs that admit a straight-line drawing. The characterization enables a linear-time testing algorithm to determine whether an almost-planar graph admits a straight-line drawing, and a linear-time drawing algorithm that constructs such a drawing, if it exists. We also show that some almost-planar graphs require exponential area for a straight-line drawing.

cs.CG↗

arXiv2013

Order Preserving Matching

We introduce a new string matching problem called order-preserving matching on numeric strings where a pattern matches a text if the text contains a substring whose relative orders coincide with those of the pattern. Order-preserving matching is applicable to many scenarios such as stock price analysis and musical melody matching in which the order relations should be matched instead of the strings themselves. Solving order-preserving matching has to do with representations of order relations of a numeric string. We define prefix representation and nearest neighbor representation, which lead to efficient algorithms for order-preserving matching. We present efficient algorithms for single and multiple pattern cases. For the single pattern case, we give an O(n log m) time algorithm and optimize it further to obtain O(n + m log m) time. For the multiple pattern case, we give an O(n log m) time algorithm.

cs.DS↗

arXiv2010

A Force-Directed Method for Large Crossing Angle Graph Drawing

Recent empirical research has indicated that human graph reading performance improves when crossing angles increase. However, crossing angle has not been used as an aesthetic criterion for graph drawing algorithms so far. In this paper, we introduce a force-directed method that aims to construct graph drawings with large crossing angles. Experiments indicate that our method significantly increases crossing angles. Surprisingly, the experimental results further demonstrate that the resulting drawings produced by our method have fewer edge crossings, a shorter total edge length and more uniform edge lengths, compared to classical spring algorithms.

cs.HC↗