Symmetry Detection and Classification in Drawings of Graphs

Abstract: Symmetry is a key feature observed in nature (from flowers and leaves, to butterflies and birds) and in human-made objects (from paintings and sculptures, to manufactured objects and architectural design). Rotational, translational, and especially reflectional symmetries, are also important in drawings of graphs. Detecting and classifying symmetries can be very useful in algorithms that aim to create symmetric graph drawings and in this paper we present a machine learning approach for these tasks. Specifically, we show that deep neural networks can be used to detect reflectional symmetries with 92% accuracy. We also build a multi-class classifier to distinguish between reflectional horizontal, reflectional vertical, rotational, and translational symmetries. Finally, we make available a collection of images of graph drawings with specific symmetric features that can be used in machine learning systems for training, testing and validation purposes. Our datasets, best trained ML models, source code are available online.

• The paper introduces CNN-based models achieving 92% accuracy for binary symmetry detection in graph drawings.
• The multi-class classifier distinguishes detailed symmetry types with a 99% accuracy, outperforming traditional metrics.
• The study demonstrates that deep learning effectively enhances graph aesthetics and analytical capabilities in visual representations.

Symmetry Detection and Classification in Drawings of Graphs

Introduction

The paper "Symmetry Detection and Classification in Drawings of Graphs" (1907.01004) presents a machine learning approach utilizing deep neural networks (DNNs) for the detection and classification of symmetries in graph drawings. Symmetry, an important aesthetic feature, is common in natural and man-made structures and crucial in graph drawing algorithms. The study identifies three main types of symmetries—reflectional, rotational, and translational—and develops models to classify these within graph drawings. Notably, a binary classification model achieves 92% accuracy in detecting reflectional symmetry, while a multi-class model distinguishes various symmetry types with an accuracy of 99%.

Methodology

The research employs convolutional neural networks (CNNs) to circumvent traditional algorithmic approaches in symmetry detection, which lack robust classification capabilities in graph layouts. CNNs automatically learn relevant features from input data, proving effective where conventional methods struggle, particularly in ambiguous and complex settings.

The study targets two primary objectives: binary classification of layouts into symmetric and non-symmetric, and multi-class classification to identify specific types of symmetries: horizontal, vertical, rotational, and translational. This is facilitated by a custom dataset composed of graph images covering these symmetry facets. The dataset includes variations in size, edge count, and vertex positioning to comprehensively train the machine learning models.

Results

The research outcomes demonstrate the potential of CNNs in accurately recognizing and classifying symmetries within graph drawings. The binary classification model achieves a notable accuracy of 92% in identifying reflectional symmetry. This performance is benchmarked against existing metrics such as the Purchase [purchase2002metrics] and Klapaukh [klapaukh2014empirical] metrics, illustrating superior precision and F1-scores (0.90 precision and 0.93 F1). The multi-class classifier further extends this capability, identifying specific symmetry types with a high accuracy rate of 99% across the test datasets.

Figure 1: Examples of the different layout instances in our dataset.

Figure 2: Symmetric layouts in the dataset: (a) Horizontal, (b) Vertical, (c) Translational, (d) Rotational, (e) Vertical without parallel lines, (f) Horizontal without parallel lines.

This superior performance underscores the efficacy of CNN architectures adapted to the unique challenges of graph layout symmetry detection and reinforces the value of machine learning in areas traditionally dominated by deterministic algorithms.

Implications

The implications of this study are significant for the field of graph drawing and related applications in computer graphics, vision, and beyond. By leveraging machine learning, this work equips automated systems with a powerful tool for symmetry detection, enhancing the aesthetic and functional quality of graph drawings. Practically, this can translate into improved graphical representations in software visualization, mapping systems, and even structural design processes.

Theoretically, this contributes to the broader discourse on integrating deep learning frameworks with graph theory problems, paving the way for future explorations into automated graph comprehension systems. Future developments could involve expanding the scope of symmetry types, integrating greater graph complexities, and exploring real-world dataset applications.

Conclusion

Through the application of CNNs in detecting and classifying symmetries within graph drawings, the paper advances both practical and theoretical knowledge in computational symmetry analysis. The research convincingly demonstrates machine learning's superiority over traditional methods in this domain, establishing a foundation for further innovations and implementations in graph-related applications. The dataset and models pioneered in this work are valuable resources for the continued evolution of automated symmetry detection systems.