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Symmetry
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Symmetry and Asymmetry Study in Graph Theory
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Symmetry and Asymmetry Study in Graph Theory

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Computer".

Deadline for manuscript submissions: 30 September 2025 | Viewed by 3490

Special Issue Editor

Dr. Mujiangshan Wang

E-Mail Website
Guest Editor
Institute of Advanced Computing and Digital Engineering, Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, China
Interests: combinatorics; algorithm design; graph theory; group theory; graph neural network

Special Issue Information

Dear Colleagues,

This Special Issue will explore the fundamental roles of symmetry and asymmetry in graph theory, with a particular focus on combinatorial algorithms, the integration of graph properties with uncertainty, and the fault diagnosis of nodes or links in interconnected networks. We will also examine the restricted connectivity and diagnosability of multiprocessor systems, as well as the application of graph neural networks (GNNs) to these complex problems. By addressing these topics, we aim to advance the understanding in areas such as algorithmic complexity in graph theory and TO improve the reliability and efficiency of complex networks.

Dr. Mujiangshan Wang
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • symmetry in graph theory
  • fault diagnosis of nodes or links in interconnected networks
  • combinatorial algorithms
  • graph neural networks (GNNs)
  • integration of graph properties with uncertainty
  • restricted connectivity and diagnosability of multiprocessor systems
  • algorithmic complexity in graph theory

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Published Papers (4 papers)

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Research

49 pages, 1730 KiB  
Article
Dynamic State Equations and Distributed Blockchain Control: A Differential Game Model for Optimal Emission Trajectories in Shipping Networks
by Zhongmiao Sun, Yike Xi, Baoli Shi and Jinrong Liu
Symmetry 2025, 17(6), 817; https://doi.org/10.3390/sym17060817 - 23 May 2025
Abstract
The shipping industry, a cornerstone of global trade, faces emissions reduction challenges amid tightening environmental policies. Blockchain technology, leveraging distributed symmetric architectures, enhances supply chain transparency by reducing information asymmetry, yet its dynamic interplay with emissions strategies remains underexplored. This study employs symmetry-driven [...] Read more.
The shipping industry, a cornerstone of global trade, faces emissions reduction challenges amid tightening environmental policies. Blockchain technology, leveraging distributed symmetric architectures, enhances supply chain transparency by reducing information asymmetry, yet its dynamic interplay with emissions strategies remains underexplored. This study employs symmetry-driven differential game theory to model four blockchain scenarios in port-shipping networks: no blockchain (N), port-led (PB), shipping company-led (CB), and a joint platform (FB). By solving Hamilton–Jacobi–Bellman equations, we derive optimal emissions reduction efforts, green investments, and blockchain strategies under symmetric and asymmetric decision-making frameworks. Results show blockchain adoption improves emissions reduction and service quality under cost thresholds, with port-led systems maximizing low-cost profits and shipping firms gaining asymmetrically in high-freight contexts. Joint platforms achieve symmetry in profit distribution through fee-trust synergy, enabling win–win outcomes. Integrating graph-theoretic principles, we have designed dynamic state equations for emissions and service levels, segmenting shippers by low-carbon preferences. This work bridges dynamic emissions strategies with blockchain’s network symmetry, fostering economic–environmental synergies to advance sustainable maritime supply chains. Full article
(This article belongs to the Special Issue Symmetry and Asymmetry Study in Graph Theory)
24 pages, 2062 KiB  
Article
Hybrid Optimization of Phase Masks: Integrating Non-Iterative Methods with Simulated Annealing and Validation via Tomographic Measurements
by Zhiwen Li, Chao Sun, Haihua Wang and Rui-Feng Wang
Symmetry 2025, 17(4), 530; https://doi.org/10.3390/sym17040530 - 31 Mar 2025
Cited by 2 | Viewed by 398
Abstract
The development of holography has facilitated significant advancements across a wide range of disciplines. A phase-only spatial light modulator (SLM) plays a crucial role in realizing digital holography, typically requiring a phase mask as its input. Non-iterative (NI) algorithms are widely used for [...] Read more.
The development of holography has facilitated significant advancements across a wide range of disciplines. A phase-only spatial light modulator (SLM) plays a crucial role in realizing digital holography, typically requiring a phase mask as its input. Non-iterative (NI) algorithms are widely used for phase mask generation, yet they often fall short in delivering precise solutions and lack adaptability in complex scenarios. In contrast, the Simulated Annealing (SA) algorithm provides a global optimization approach capable of addressing these limitations. This study investigates the integration of NI algorithms with the SA algorithm to enhance the optimization of phase mask generation in digital holography. Furthermore, we examine how adjusting annealing parameters, especially the cooling strategy, can significantly improve system optimization performance and symmetry. Notably, we observe a considerable improvement in the efficiency of the SA algorithm when non-iterative methods are employed to generate the initial phase mask. Our method achieves a perfect representation of the symmetry in desired light fields. The efficacy of the optimized phase masks is evaluated through optical tomographic measurements using two-dimensional mutually unbiased bases (MUBs), with the resulting average similarity reaching 0.99. These findings validate the effectiveness of our methodin optimizing phase mask generation and underscore its potential for high-precision optical mode recognition and analysis. Full article
(This article belongs to the Special Issue Symmetry and Asymmetry Study in Graph Theory)
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20 pages, 2943 KiB  
Article
The Heterogeneous Network Community Detection Model Based on Self-Attention
by Gaofeng Zhou and Rui-Feng Wang
Symmetry 2025, 17(3), 432; https://doi.org/10.3390/sym17030432 - 13 Mar 2025
Cited by 2 | Viewed by 546
Abstract
With the advancement of representation learning, graph representation learning has gained significant attention in the field of community detection for heterogeneous networks. A prominent approach in this domain involves the use of meta-paths to capture higher-order relationships between nodes, particularly when bidirectional or [...] Read more.
With the advancement of representation learning, graph representation learning has gained significant attention in the field of community detection for heterogeneous networks. A prominent approach in this domain involves the use of meta-paths to capture higher-order relationships between nodes, particularly when bidirectional or reciprocal relationships exist. However, defining effective meta-paths often requires substantial domain expertise. Moreover, these methods typically depend on additional clustering algorithms, which can limit their practical applicability. To address these challenges, context paths have been introduced as an alternative to meta-paths. When combined with a self-attention mechanism, models can dynamically assess the relative importance of different context paths. By leveraging the inherent symmetry within context paths, models enhance their ability to capture balanced relationships between nodes, thereby improving their representation of complex interactions. Building on this idea, we propose BP-GCN, a self-attention-based model for heterogeneous community detection. BP-GCN autonomously identifies node relationships within symmetric context paths, significantly improving community detection accuracy. Furthermore, the model integrates the Bernoulli–Poisson framework to establish an end-to-end detection system that eliminates the need for auxiliary clustering algorithms. Extensive experiments on multiple real-world datasets demonstrate that BP-GCN consistently outperforms existing benchmark methods. Full article
(This article belongs to the Special Issue Symmetry and Asymmetry Study in Graph Theory)
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25 pages, 2726 KiB  
Article
HybridGNN: A Self-Supervised Graph Neural Network for Efficient Maximum Matching in Bipartite Graphs
by Chun-Hu Pan, Yi Qu, Yao Yao and Mu-Jiang-Shan Wang
Symmetry 2024, 16(12), 1631; https://doi.org/10.3390/sym16121631 - 9 Dec 2024
Cited by 4 | Viewed by 1563
Abstract
Solving maximum matching problems in bipartite graphs is critical in fields such as computational biology and social network analysis. This study introduces HybridGNN, a novel Graph Neural Network model designed to efficiently address complex matching problems at scale. HybridGNN leverages a combination of [...] Read more.
Solving maximum matching problems in bipartite graphs is critical in fields such as computational biology and social network analysis. This study introduces HybridGNN, a novel Graph Neural Network model designed to efficiently address complex matching problems at scale. HybridGNN leverages a combination of Graph Attention Networks (GATv2), Graph SAGE (SAGEConv), and Graph Isomorphism Networks (GIN) layers to enhance computational efficiency and model performance. Through extensive ablation experiments, we identify that while the SAGEConv layer demonstrates suboptimal performance in terms of accuracy and F1-score, configurations incorporating GATv2 and GIN layers show significant improvements. Specifically, in six-layer GNN architectures, the combinations of GATv2 and GIN layers with ratios of 4:2 and 5:1 yield superior accuracy and F1-score. Therefore, we name these GNN configurations HybridGNN1 and HybridGNN2. Additionally, techniques such as mixed precision training, gradient accumulation, and Jumping Knowledge networks are integrated to further optimize performance. Evaluations on an email communication dataset reveal that HybridGNNs outperform traditional algorithms such as the Hopcroft–Karp algorithm, the Hungarian algorithm, and the Blossom/Edmonds’ algorithm, particularly for large and complex graphs. These findings highlight HybridGNN’s robust capability to solve maximum matching problems in bipartite graphs, making it a powerful tool for analyzing large-scale and intricate graph data. Furthermore, our study aligns with the goals of the Symmetry and Asymmetry Study in Graph Theory special issue by exploring the role of symmetry in bipartite graph structures. By leveraging GNNs, we address the challenges related to symmetry and asymmetry in graph properties, thereby improving the reliability and fault tolerance of complex networks. Full article
(This article belongs to the Special Issue Symmetry and Asymmetry Study in Graph Theory)
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