A Quality Metric for Symmetric Graph Drawings Amyra Meidiana* Seok-Hee Hong† Peter Eades‡ Daniel Keim§ University of Sydney University of Sydney University of Sydney University of Konstanz ABSTRACT It is important to ensure that a symmetric graph drawing accu- Symmetry is an important aesthetic criteria in graph drawing and rately represent the automorphisms of the underlying graph to the network visualisation. Symmetric graph drawings aim to faithfully greatest extent possible. However, existing symmetry detection and represent automorphisms of graphs as geometric symmetries in a quality metrics for graph drawings do not focus on this comparison drawing. between detected geometric symmetry in a graph drawing and the In this paper, we design and implement a framework for quality automorphisms of the graph. metrics that measure symmetry, that is, how faithfully a drawing In this paper, we design and implement a framework for quality of a graph displays automorphisms as geometric symmetries. The metrics for graph drawings that measure symmetry, that is, how quality metrics are based on geometry (i.e. Euclidean distance) as faithfully a drawing of a graph displays automorphisms as geometric well as mathematical group theory (i.e. orbits of automorphisms). symmetries. The quality metrics are based on geometry (i.e., Eu- More specifically, we define two varieties of symmetry quality clidean distance) as well as mathematical group theory (i.e., orbits metrics: (1) for displaying a single automorphism as a symmetry of automorphisms). (axial or rotational) and (2) for displaying a group of automorphisms More specifically, we present the following contributions: (cyclic or dihedral). We also present algorithms to compute the symmetric quality metrics in O(nlogn) time for rotational symmetry 1. We design and implement a framework for a quality metric and axial symmetry. for symmetric graph drawing, which measures the symmetry We validate our symmetry quality metrics using deformation quality of a drawing of a graph based on the comparison be- experiments. We then use the metrics to evaluate a number of tween the symmetry of the drawing and the automorphism of established graph drawing layouts to compare how faithfully they the graph. We define two varieties of the symmetry quality display automorphisms of a graph as geometric symmetries. metrics, one to measure how well a graph drawing displays a single automorphism as a symmetry (rotational or axial) and Index Terms: Human-centered computing [Visualization]: Visual- one to measure how well a graph drawing displays a group of ization design and evaluation methods automorphism simultaneously (cyclic or dihedral groups). 1 INTRODUCTION 2. We present algorithms to compute the symmetry quality met- rics in O(nlogn) time for rotational and axial symmetry and Graph drawing aims to construct a visually-informative drawing O(knlogn) for automorphism groups, where n is the number of an abstract graph in the plane. Symmetry is one of the most of vertices in the graph and k is the size of the automorphism important aesthetic criteria that represent the structure and properties group. of a graph visually [3]. Symmetric graph drawings aim to faithfully represent automorphisms of graphs as geometric symmetries in 3. We validate the single automorphism detection version of the the drawing. Also, a symmetric drawing of a graph enables an symmetry quality metrics through deformation experiments understanding of the entire graph to be built up from that of a smaller of graph drawings, showing that the scores computed by our subgraph. metric decrease when the drawings are distorted further from Symmetric drawings of a graph G are clearly related to the exact symmetry. automorphisms of G, and algorithms for constructing symmetric drawings have two steps: 4. We validate the automorphism group detection version of the symmetry quality metrics through comparing drawings dis- Step 1. Find the “appropriate” automorphisms, and playing different groups of automorphisms, showing that our Step 2. Draw the graph displaying these automorphisms as metric effectively captures the difference in symmetry quality symmetries. between drawings that display different numbers of automor- phisms as symmetries. arXiv:1910.04974v1 [cs.DS] 11 Oct 2019 The problem of determining whether a graph has a nontrivial automorphism is automorphism complete [19]. However, the prob- 5. We use our metric to compare a number of established graph lem of determining whether a graph has a nontrivial geometric drawing layouts to compare how faithfully they display auto- automorphism is NP-complete [23]. Linear time algorithms to con- morphisms of a graph as geometric symmetries. We confirm struct symmetric drawings for restricted classes of graphs exists (e.g. the effectiveness of the concentric circles layout in displaying trees [21], outerplanar graphs [22], series-parallel digraphs [15]), a graph’s automorphisms as symmetries. and planar graphs [12–14,16]. For general graphs, heuristics [7,18] and exact algorithms are available [1, 5]. 2 RELATED WORKS 2.1 Symmetries and geometric automorphisms *e-mail: [email protected] †e-mail: [email protected] An automorphism of a graph is a permutation of the vertices that ‡e-mail: [email protected] preserves the adjacency of each vertex. Suppose that a permutation §e-mail: [email protected] group A acts on a set V. We say that f 2 A fixes v 2 V if f(v) = v; if f fixes v for every f 2 A then A fixes v. If V 0 ⊆ V and f(v0) 2 V 0 for all v0 2 V 0 then f fixes V 0 (setwise). A subset V 0 of V is an orbit of A if, for each u;v 2 V 0, there is f 2 A such that f(u) = v, and V 0 does not contain a nonempty subset with this property. It is important to use a rigorous model, introduced by Eades and symmetry). However, these algorithms only give a binary answer Lin [9], for the intuitive concept of symmetry display. to whether a point set displays a symmetry and are not suitable to The symmetries of a set of points in the plane (such as a two quantify the approximate symmetry of a point set. dimensional graph drawing) form a group called the symmetry group Zabrodsky et al [27] introduces the symmetry distance method, of the set. A symmetry s of a drawing D of a graph G induces providing another method to quantify the approximate symmetry an automorphism of G, in that the restriction of s to the points of a geometric object. This method takes as input a point set P and representing vertices of G is an automorphism of G. A drawing D a symmetry s to be checked and computes a symmetric point set of a graph G displays an automorphism f of G if there is symmetry Ps realizing the input symmetry. This is done through a “folding” s of D which induces f. The symmetry group of a graph drawing transform method that “folds” points in the same orbit, averages induces an automorphism group of the graph. An automorphism their positions, and “unfolds” them into a symmetric configuration. group A of a graph G is a geometric automorphism group if there is This method minimizes the Euclidean distance between each point a drawing of G which displays every element of A. in P and its image in Ps , and the symmetry distance is given as the A non-trivial symmetry of a finite set of points in the plane is average of these Euclidean distances for every point in P. either a rotation about a point or a reflection about a line. A ge- ometric automorphism is a rotational automorphism (respectively 3 SYMMETRY QUALITY METRIC FRAMEWORK relectional/axial automorphism) if it is induced by a rotation (respec- We propose a new quality metric for graph visualization, the symme- tively reflection). Eades and Lin [9] showed that that a nontrivial try quality metric, for measuring how well the drawing of a graph geometric automorphism group is one of three kinds: displays selected geometric automorphisms of the underlying graph. Our metric is a faithfulness metric comparing the geometric symme- 1. a group of size 2 generated by an axial automorphism; try detected in a drawing of a graph with the graph’s automorphisms, the ground truth information, unlike existing metrics which only attempts to detect symmetry from the drawing. Figure 1 summarizes cyclic group k 2. a of size generated by a rotational automor- the framework used for our proposed metric. phism; 3. a dihedral group of size 2k generated by a rotational automorphism of order k and an axial automorphism. 2.2 Symmetric Graph Drawing In general, determining whether a graph can be drawn symmetrically in two dimensions is NP-complete [19]. Exact algorithms based on Branch and Cut [5] and group theory [1] are available. Linear-time algorithms are available for symmetric drawings of limited classes of graphs, such as trees [21], outerplanar graphs [22], and series-parallel digraphs [15]. Linear-time algorithms have also been presented for maximally symmetric drawings of tricon- nected [16], biconnected [12], oneconnected [13], and disconnected Figure 1: Framework for the symmetry quality metric. planar graphs [14]. For a survey on symmetric drawings of graphs in two dimensions, see [8]. Let G be a graph and f be a automorphism of G. It is important that f be a geometric automorphism of G, as otherwise it is 2.3 Graph Drawing Symmetry Quality Metrics impossible to display it as a symmetry of a drawing of G [9]. Purchase [24] defined a metric measuring the symmetry of a graph The framework computes the symmetry quality score using the drawing by checking, for each pair of vertices, whether there is a following steps: symmetric subgraph around the pair, calculating a weighted sym- metry value of the symmetric subgraph if it exists, and adding the Framework: Symmetry Quality Metric weighted symmetry value of all symmetric subgraphs.