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%.

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.

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.

The unique structure of signed networks, characterized by positive and negative edges, poses significant challenges for analyzing network topology. In recent years, various statistical algorithms have been developed to address this issue. However, there remains a lack of a unified framework to uncover the nontrivial properties inherent in signed network structures. To support developers, researchers, and practitioners in this field, we introduce a Python library named SNSAlib (Signed Network Structure Analysis), specifically designed to meet these analytical requirements. This library encompasses empirical signed network datasets, signed null model algorithms, signed statistics algorithms, and evaluation indicators. The primary objective of SNSAlib is to facilitate the systematic analysis of micro- and meso-structure features within signed networks, including node popularity, clustering, assortativity, embeddedness, and community structure by employing more accurate signed null models. Ultimately, it provides a robust paradigm for structure analysis of signed networks that enhances our understanding and application of signed networks.

Identifying vital nodes is one of the core issues of network science, and is crucial for epidemic prevention and control, network security maintenance, and biomedical research and development. In this paper, a new vital nodes identification method, named degree and cycle ratio (DC), is proposed by integrating degree centrality (weight $\alpha$) and cycle ratio (weight $1-\alpha$). The results show that the dynamic observations and weight $\alpha$ are nonlinear and non-monotonicity (i.e., there exists an optimal value $\alpha^*$ for $\alpha$), and that DC performs better than a single index in most networks. According to the value of $\alpha ^{\ast } $, networks are classified into degree-dominant networks ($\alpha ^{\ast }>0.5 $) and cycle-dominant networks ($\alpha ^{\ast }<0.5 $). Specifically, in most degree-dominant networks (such as Chengdu-BUS, Chongqing-BUS and Beijing-BUS), degree is dominant in the identification of vital nodes, but the identification effect can be improved by adding cycle structure information to the nodes. In most cycle-dominant networks (such as Email, Wiki and Hamsterster), the cycle ratio is dominant in the identification of vital nodes, but the effect can be notably enhanced by additional node degree information. Finally, interestingly, in Lancichinetti-Fortunato-Radicchi (LFR) synthesis networks, the cycle-dominant network is observed.

Health information spreads rapidly, which can effectively control epidemics. However, the swift dissemination of information also has potential negative impacts, which increasingly attracts attention. Message fatigue refers to the psychological response characterized by feelings of boredom and anxiety that occur after receiving an excessive amount of similar information. This phenomenon can alter individual behaviors related to epidemic prevention. Additionally, recent studies indicate that pairwise interactions alone are insufficient to describe complex social transmission processes, and higher-order structures representing group interactions are crucial. To address this, we develop a novel epidemic model that investigates the interactions between information, behavioral responses, and epidemics. Our model incorporates the impact of message fatigue on the entire transmission system. The information layer is modeled using a static simplicial network to capture group interactions, while the disease layer uses a time-varying network based on activity-driven model with attractiveness to represent the self-protection behaviors of susceptible individuals and self-isolation behaviors of infected individuals. We theoretically describe the co-evolution equations using the microscopic Markov chain approach (MMCA) and get the epidemic threshold. Experimental results show that while the negative impact of message fatigue on epidemic transmission is limited, it significantly weakens the group interactions depicted by higher-order structures. Individual behavioral responses strongly inhibit the epidemic. Our simulations using the Monte Carlo (MC) method demonstrate that greater intensity in these responses leads to clustering of susceptible individuals in the disease layer. Finally, we apply the proposed model to real networks to verify its reliability. In summary, our research results enhance the understanding of the information-epidemic coupling dynamics, and we expect to provide valuable guidance for managing future emerging epidemics.

For non-stationary complex dynamic systems, a standardized algorithm is developed to compute time correlation functions, addressing the limitations of traditional methods reliant on the stationary assumption. The proposed algorithm integrates two-point and multi-point time correlation functions into a unified framework. Further, it is verified by a practical application in complex financial systems, demonstrating its potential in various complex dynamic systems.

Independent cascade (IC) models, by simulating how one node can activate another, are important tools for studying the dynamics of information spreading in complex networks. However, traditional algorithms for the IC model implementation face significant efficiency bottlenecks when dealing with large-scale networks and multi-round simulations. To settle this problem, this study introduces a GPU-based parallel independent cascade (GPIC) algorithm, featuring an optimized representation of the network data structure and parallel task scheduling strategies. Specifically, for this GPIC algorithm, we propose a network data structure tailored for GPU processing, thereby enhancing the computational efficiency and the scalability of the IC model. In addition, we design a parallel framework that utilizes the full potential of GPU's parallel processing capabilities, thereby augmenting the computational efficiency. The results from our simulation experiments demonstrate that GPIC not only preserves accuracy but also significantly boosts efficiency, achieving a speedup factor of 129 when compared to the baseline IC method. Our experiments also reveal that when using GPIC for the independent cascade simulation, 100-200 simulation rounds are sufficient for higher-cost studies, while high precision studies benefit from 500 rounds to ensure reliable results, providing empirical guidance for applying this new algorithm to practical research.

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.

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.

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.

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.

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.

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.