Content of SPECIAL TOPIC — Computational programs in complex systems in our journal

Published in last 1 year |  In last 2 years |  In last 3 years |  All Please wait a minute... For selected: Download citations EndNote Ris BibTeX Toggle thumbnails Select Combining deep reinforcement learning with heuristics to solve the traveling salesman problem Li Hong(洪莉), Yu Liu(刘宇), Mengqiao Xu(徐梦俏), and Wenhui Deng(邓文慧) Chin. Phys. B, 2025, 34 (1): 018705.   DOI: 10.1088/1674-1056/ad95f1 Abstract （445）   HTML （10）    PDF （675KB）（834）       Knowledge map    Recent studies employing deep learning to solve the traveling salesman problem (TSP) have mainly focused on learning construction heuristics. Such methods can improve TSP solutions, but still depend on additional programs. However, methods that focus on learning improvement heuristics to iteratively refine solutions remain insufficient. Traditional improvement heuristics are guided by a manually designed search strategy and may only achieve limited improvements. This paper proposes a novel framework for learning improvement heuristics, which automatically discovers better improvement policies for heuristics to iteratively solve the TSP. Our framework first designs a new architecture based on a transformer model to make the policy network parameterized, which introduces an action-dropout layer to prevent action selection from overfitting. It then proposes a deep reinforcement learning approach integrating a simulated annealing mechanism (named RL-SA) to learn the pairwise selected policy, aiming to improve the 2-opt algorithm's performance. The RL-SA leverages the whale optimization algorithm to generate initial solutions for better sampling efficiency and uses the Gaussian perturbation strategy to tackle the sparse reward problem of reinforcement learning. The experiment results show that the proposed approach is significantly superior to the state-of-the-art learning-based methods, and further reduces the gap between learning-based methods and highly optimized solvers in the benchmark datasets. Moreover, our pre-trained model M can be applied to guide the SA algorithm (named M-SA (ours)), which performs better than existing deep models in small-, medium-, and large-scale TSPLIB datasets. Additionally, the M-SA (ours) achieves excellent generalization performance in a real-world dataset on global liner shipping routes, with the optimization percentages in distance reduction ranging from 3.52% to 17.99%. Select Accurate prediction of essential proteins using ensemble machine learning Dezhi Lu(鲁德志), Hao Wu(吴淏), Yutong Hou(侯俞彤), Yuncheng Wu(吴云成), Yuanyuan Liu(刘媛媛), and Jinwu Wang(王金武) Chin. Phys. B, 2025, 34 (1): 018901.   DOI: 10.1088/1674-1056/ad8db2 Abstract （418）   HTML （6）    PDF （1127KB）（190）       Knowledge map    Essential proteins are crucial for biological processes and can be identified through both experimental and computational methods. While experimental approaches are highly accurate, they often demand extensive time and resources. To address these challenges, we present a computational ensemble learning framework designed to identify essential proteins more efficiently. Our method begins by using node2vec to transform proteins in the protein-protein interaction (PPI) network into continuous, low-dimensional vectors. We also extract a range of features from protein sequences, including graph-theory-based, information-based, compositional, and physiochemical attributes. Additionally, we leverage deep learning techniques to analyze high-dimensional position-specific scoring matrices (PSSMs) and capture evolutionary information. We then combine these features for classification using various machine learning algorithms. To enhance performance, we integrate the outputs of these algorithms through ensemble methods such as voting, weighted averaging, and stacking. This approach effectively addresses data imbalances and improves both robustness and accuracy. Our ensemble learning framework achieves an AUC of 0.960 and an accuracy of 0.9252, outperforming other computational methods. These results demonstrate the effectiveness of our approach in accurately identifying essential proteins and highlight its superior feature extraction capabilities. Select Dynamic modeling and analysis of brucellosis on metapopulation network: Heilongjiang as cases Xin Pei(裴鑫), Xuan-Li Wu(武绚丽), Pei Pei(裴沛), Ming-Tao Li(李明涛), Juan Zhang(张娟), and Xiu-Xiu Zhan(詹秀秀) Chin. Phys. B, 2025, 34 (1): 018904.   DOI: 10.1088/1674-1056/ad92ff Abstract （388）   HTML （6）    PDF （851KB）（341）       Knowledge map    Livestock transportation is a key factor that contributes to the spatial spread of brucellosis. To analyze the impact of sheep transportation on brucellosis transmission, we develop a human-sheep coupled brucellosis model within a metapopulation network framework. Theoretically, we examine the positively invariant set, the basic reproduction number, the existence, uniqueness, and stability of disease-free equilibrium and the existence of the endemic equilibrium of the model. For practical application, using Heilongjiang province as a case study, we simulate brucellosis transmission across 12 cities based on data using three network types: the BA network, the ER network, and homogeneous mixing network. The simulation results indicate that the network's average degree plays a role in the spread of brucellosis. For BA and ER networks, the basic reproduction number and cumulative incidence of brucellosis stabilize when the network's average degree reaches 4 or 5. In contrast, sheep transport in a homogeneous mixing network accelerates the cross-regional spread of brucellosis, whereas transportation in a BA network helps to control it effectively. Furthermore, the findings suggest that the movement of sheep is not always detrimental to controlling the spread of brucellosis. For cities with smaller sheep populations, such as Shuangyashan and Qitaihe, increasing the transport of sheep outward amplifies the spatial spread of the disease. In contrast, in cities with larger sheep populations, such as Qiqihar, Daqing, and Suihua, moderate sheep outflow can help reduce the spread. In addition, cities with large livestock populations play a dominant role in the overall transmission dynamics, underscoring the need for stricter supervision in these areas. Select Planar: A software for exact decoding quantum error correction codes with planar structure Dongyang Feng(冯东阳), Hanyan Cao(曹涵彦), and Pan Zhang(张潘) Chin. Phys. B, 2025, 34 (5): 050311.   DOI: 10.1088/1674-1056/adcb26 Abstract （228）   HTML （2）    PDF （1720KB）（241）       Knowledge map    Quantum error correction is essential for realizing fault-tolerant quantum computing, where both the efficiency and accuracy of the decoding algorithms play critical roles. In this work, we introduce the implementation of the Planar algorithm, a software framework designed for fast and exact decoding of quantum codes with a planar structure. The algorithm first converts the optimal decoding of quantum codes into a partition function computation problem of an Ising spin glass model. Then it utilizes the exact Kac-Ward formula to solve it. In this way, Planar offers the exact maximum likelihood decoding in polynomial complexity for quantum codes with a planar structure, including the surface code with independent code-capacity noise and the quantum repetition code with circuit-level noise. Unlike traditional minimum-weight decoders such as minimum-weight perfect matching (MWPM), Planar achieves theoretically optimal performance while maintaining polynomial-time efficiency. In addition, to demonstrate its capabilities, we exemplify the implementation using the rotated surface code, a commonly used quantum error correction code with a planar structure, and show that Planar achieves a threshold of approximately $ p_{\rm uc} \approx 0.109 $ under the depolarizing error model, with a time complexity scaling of $ O(N^{0.69}) $, where $ N $ is the number of spins in the Ising model. Select Text-guided diverse-expression diffusion model for molecule generation Wenchao Weng(翁文超), Hanyu Jiang(蒋涵羽), Xiangjie Kong(孔祥杰), and Giovanni Pau Chin. Phys. B, 2025, 34 (5): 050701.   DOI: 10.1088/1674-1056/adbedd Abstract （216）   HTML （0）    PDF （864KB）（238）       Knowledge map    The task of molecule generation guided by specific text descriptions has been proposed to generate molecules that match given text inputs. Mainstream methods typically use simplified molecular input line entry system (SMILES) to represent molecules and rely on diffusion models or autoregressive structures for modeling. However, the one-to-many mapping diversity when using SMILES to represent molecules causes existing methods to require complex model architectures and larger training datasets to improve performance, which affects the efficiency of model training and generation. In this paper, we propose a text-guided diverse-expression diffusion (TGDD) model for molecule generation. TGDD combines both SMILES and self-referencing embedded strings (SELFIES) into a novel diverse-expression molecular representation, enabling precise molecule mapping based on natural language. By leveraging this diverse-expression representation, TGDD simplifies the segmented diffusion generation process, achieving faster training and reduced memory consumption, while also exhibiting stronger alignment with natural language. TGDD outperforms both TGM-LDM and the autoregressive model MolT5-Base on most evaluation metrics. Select Active learning attraction basins of dynamical system Xiao-Wei Cao(曹小尾), Xiao-Lei Ru(茹小磊), and Gang Yan(严钢) Chin. Phys. B, 2025, 34 (5): 058901.   DOI: 10.1088/1674-1056/adbede Abstract （210）   HTML （0）    PDF （2571KB）（129）       Knowledge map    Dynamical systems often exhibit multiple attractors representing significantly different functioning conditions. A global map of attraction basins can offer valuable guidance for stabilizing or transitioning system states. Such a map can be constructed without prior system knowledge by identifying attractors across a sufficient number of points in the state space. However, determining the attractor for each initial state can be a laborious task. Here, we tackle the challenge of reconstructing attraction basins using as few initial points as possible. In each iteration of our approach, informative points are selected through random seeding and are driven along the current classification boundary, promoting the eventual selection of points that are both diverse and enlightening. The results across various experimental dynamical systems demonstrate that our approach requires fewer points than baseline methods while achieving comparable mapping accuracy. Additionally, the reconstructed map allows us to accurately estimate the minimum escape distance required to transition the system state to a target basin. Select Programming guide for solving constraint satisfaction problems with tensor networks Xuanzhao Gao(高煊钊), Xiaofeng Li(李晓锋), and Jinguo Liu(刘金国) Chin. Phys. B, 2025, 34 (5): 050201.   DOI: 10.1088/1674-1056/adbee2 Abstract （193）   HTML （1）    PDF （1139KB）（289）       Knowledge map    Constraint satisfaction problems (CSPs) are a class of problems that are ubiquitous in science and engineering. They feature a collection of constraints specified over subsets of variables. A CSP can be solved either directly or by reducing it to other problems. This paper introduces the Julia ecosystem for solving and analyzing CSPs with a focus on the programming practices. We introduce some important CSPs and show how these problems are reduced to each other. We also show how to transform CSPs into tensor networks, how to optimize the tensor network contraction orders, and how to extract the solution space properties by contracting the tensor networks with generic element types. Examples are given, which include computing the entropy constant, analyzing the overlap gap property, and the reduction between CSPs. Select SFFSlib: A Python library for optimizing attribute layouts from micro to macro scales in network visualization Ke-Chao Zhang(张可超), Sheng-Yue Jiang(蒋升跃), and Jing Xiao(肖婧) Chin. Phys. B, 2025, 34 (5): 058903.   DOI: 10.1088/1674-1056/adcb98 Abstract （183）   HTML （0）    PDF （16393KB）（62）       Knowledge map    Complex network modeling characterizes system relationships and structures, while network visualization enables intuitive analysis and interpretation of these patterns. However, existing network visualization tools exhibit significant limitations in representing attributes of complex networks at various scales, particularly failing to provide advanced visual representations of specific nodes and edges, community affiliation attribution, and global scalability. These limitations substantially impede the intuitive analysis and interpretation of complex network patterns through visual representation. To address these limitations, we propose SFFSlib, a multi-scale network visualization framework incorporating novel methods to highlight attribute representation in diverse network scenarios and optimize structural feature visualization. Notably, we have enhanced the visualization of pivotal details at different scales across diverse network scenarios. The visualization algorithms proposed within SFFSlib were applied to real-world datasets and benchmarked against conventional layout algorithms. The experimental results reveal that SFFSlib significantly enhances the clarity of visualizations across different scales, offering a practical solution for the advancement of network attribute representation and the overall enhancement of visualization quality. Select Trajectory tracking on the optimal path of two-dimensional quadratic barrier escaping Zengxuan Zhao(赵曾轩), Xiuying Zhang(张秀颖), Pengchen Zhao(赵鹏琛), Chunyang Wang(王春阳), Chunlei Xia(夏春雷), Mushtaq Rana Imran, and Joelous Malamula Nyasulu Chin. Phys. B, 2025, 34 (5): 050205.   DOI: 10.1088/1674-1056/adcb99 Abstract （169）   HTML （0）    PDF （1249KB）（151）       Knowledge map    The diffusion trajectory of a Brownian particle passing over the saddle point of a two-dimensional quadratic potential energy surface is tracked in detail according to the deep learning strategies. Generative adversarial networks (GANs) emanating in the category of machine learning (ML) frameworks are used to generate and assess the rationality of the data. While their optimization is based on the long short-term memory (LSTM) strategies. In addition to drawing a heat map, the optimal path of two-dimensional (2D) diffusion is simultaneously demonstrated in a stereoscopic space. The results of our simulation are completely consistent with the previous theoretical predictions. Select Analysis and design of multivalued many-to-one associative memory driven by external inputs Qiang Fang(方强) and Hao Zhang(张浩) Chin. Phys. B, 2025, 34 (8): 080701.   DOI: 10.1088/1674-1056/add24b Abstract （116）   HTML （0）    PDF （976KB）（44）       Knowledge map    This paper proposes a novel multivalued recurrent neural network model driven by external inputs, along with two innovative learning algorithms. By incorporating a multivalued activation function, the proposed model can achieve multivalued many-to-one associative memory, and the newly developed algorithms enable effective storage of many-to-one patterns in the coefficient matrix while maintaining the indispensability of inputs in many-to-one associative memory. The proposed learning algorithm addresses a critical limitation of existing models which fail to ensure completely erroneous outputs when facing partial input missing in many-to-one associative memory tasks. The methodology is rigorously derived through theoretical analysis, incorporating comprehensive verification of both the existence and global exponential stability of equilibrium points. Demonstrative examples are provided in the paper to show the effectiveness of the proposed theory. Select Evolutionary role of startups and its relevance to the success in the blockchain field based on temporal information networks Ying Wang(王颖) and Qing Guan(管青) Chin. Phys. B, 2025, 34 (8): 088901.   DOI: 10.1088/1674-1056/add247 Abstract （90）   HTML （0）    PDF （1936KB）（73）       Knowledge map    Startups form an information network that reflects their growth trajectories through information flow channels established by shared investors. However, traditional static network metrics overlook temporal dynamics and rely on single indicators to assess startups' roles in predicting future success, failing to comprehensively capture topological variations and structural diversity. To address these limitations, we construct a temporal information network using 14547 investment records from 1013 global blockchain startups between 2004 and 2020, sourced from Crunchbase. We propose two dynamic methods to characterize the information flow: temporal random walk (sTRW) for modeling information flow trajectories and temporal betweenness centrality (tTBET) for identifying key information hubs. These methods enhance walk coverage while ensuring random stability, allowing for more effective identification of influential startups. By integrating sTRW and tTBET, we develop a comprehensive metric to evaluate a startup's influence within the network. In experiments assessing startups' potential for future success—where successful startups are defined as those that have undergone M&A or IPO—incorporating this metric improves accuracy, recall, and F1 score by 0.035, 0.035, and 0.042, respectively. Our findings indicate that information flow from key startups to others diminishes as the network distance increases. Additionally, successful startups generally exhibit higher information inflows than outflows, suggesting that actively seeking investment-related information contributes to startup growth. Our research provides valuable insights for formulating startup development strategies and offers practical guidance for market regulators. Select Six-degree gravity centrality for detecting influential nodes in networks Jianbo Wang(王建波), Bohang Lin(林渤杭), Zhanwei Du(杜占玮), Ping Li(李平), and Xiao-Ke Xu(许小可) Chin. Phys. B, 2025, 34 (8): 088902.   DOI: 10.1088/1674-1056/adec62 Abstract （81）   HTML （0）    PDF （7872KB）（65）       Knowledge map    Identifying critical nodes is a pivotal research topic in network science, yet the efficient and accurate detection of highly influential nodes remains a challenge. Existing centrality measures predominantly rely on local or global topological structures, often overlooking indirect connections and their interaction strengths. This leads to imprecise assessments of node importance, limiting practical applications. To address this, we propose a novel node centrality measure, termed six-degree gravity centrality (SDGC), grounded in the six degrees of separation theory, for the precise identification of influential nodes in networks. Specifically, we introduce a set of node influence parameters—node mass, dynamic interaction distance, and attraction coefficient—to enhance the gravity model. Node mass is calculated by integrating K-shell and closeness centrality measures. The dynamic interaction distance, informed by the six-degrees of separation theory, is determined through path searches within six hops between node pairs. The attraction coefficient is derived from the difference in K-shell values between nodes. By integrating these parameters, we develop an improved gravity model to quantify node influence. Experiments conducted on nine real-world networks demonstrate that SDGC significantly outperforms nine existing classical and state-of-the-art methods in identifying the influential nodes.