Hierarchical Network Structure Research Articles

Static pinning synchronization control of self-triggered coupling dynamical networks

In this paper, a new static pinning intermittent control based on resource awareness triggering is proposed. A multi-layer control technique is used to synchronize the coupled neural network. First, a hierarchical network structure including pinned and interaction layers is induced using each pinning strategy. Second, using the ideas of average aperiodic intermittent control (AIC) rate method and constructing an auxiliary function, a new lemma is proposed for the pinning intermittent synchronization of coupled networks, where the upper/lower bound restrictions on each control width for AIC are relaxed. Third, to obtain the desired synchronization behavior, a self-triggering mechanism (STM) is proposed to execute the AIC of the pinned and interaction layers. Moreover, the proposed STM is effective for the actuation of the static pinning impulsive control. The static pinning method modifies the single pinning and switching pinning impulsive control. Finally, the proposed results are applied to Chua’s circuits, oscillators and small-world networks. Experimental results show the performance of the proposed STM which reduces 34.58% the number of control updates compared to a periodically intermittent event-triggered scheme. Further, for large-scale coupled networks, the N-dimensional Laplacian matrix LV can be decomposed into sp×sp and N−sp×N−sp dimensions by hierarchical method, thus reducing the complexity of calculation.

N/S co-doped nanocomposite of graphene oxide and graphene-like organic molecules as all-carbonaceous anode material for high-performance Li-ion batteries

In this study, to enhance the electrochemical performance of graphene-based anodes for Li-ion batteries (LIBs), we synthesized an all-carbonaceous N/S co-doped nanocomposite of graphene oxide (GO) and graphene-like small organic molecules (GOM) using a mild, eco-friendly, one-step hydrothermal method with thiourea (CH4N2S) (denoted as h-N/S-GO/GOM). The thiourea facilitated N/S co-doping and π−π bonding, which improved the interaction between hydrophilic GO and hydrophobic GOM in aqueous solution. Notably, the formation of π−π bonds between GO and GOM created pathways that enhanced electron transfer, thereby promoting efficient Li-ion transport from the electrolyte through the channels during rapid charge–discharge cycles. Additionally, the functional groups resulting from N/S co-doping increased the number of active sites within the nanocomposite. Consequently, the h-N/S-GO/GOM anode demonstrated superior electrochemical performance, achieving an average reversible capacity of 1265 mAh g−1 at 0.1 A g−1 and retaining 83.0 % of its capacity after 200 cycles. Furthermore, the nanocomposite exhibited excellent long-term cycling stability, maintaining a capacity of 688 mAh g−1 even after 1000 cycles at a high current density of 1.0 A g−1. The hierarchical network structure of the all-carbonaceous h-N/S-GO/GOM anode facilitated efficient charge transfer between the electrode and electrolyte through shorter diffusion paths for Li-ion transport and provided additional active sites, contributing to its outstanding electrical performance. The h-N/S-GO/GOM nanocomposite represents a promising alternative to traditional graphite-based anodes, offering a path toward high-performance, eco-friendly LIBs suitable for applications such as electric vehicles and energy storage systems.

A semiconducting supramolecular Co(II)-metallogel based resistive random access memory (RRAM) design with good endurance capabilities.

A highly efficient approach for synthesizing a supramolecular metallogel of Co(II) ions, denoted as CoA-TA, has been established under room temperature and atmospheric pressure conditions. This method employs the metal-coordinating organic ligand benzene-1,3,5-tricarboxylic acid as a low molecular weight gelator (LMWG) in DMF solvent. A comprehensive analysis of the mechanical properties of the resulting supramolecular Co(II)-metallogel was conducted through rheological investigation, considering angular frequency and thixotropic study. The hierarchical rocky network structure of the supramolecular Co(II)-metallogel was unveiled using field emission scanning electron microscopy (FESEM). Transmission electron microscopic (TEM) analysis showed rod-shaped structures via low-magnification high angle annular dark field (HAADF) bright field scanning transmission electron microscopic (STEM) imaging, while energy dispersive X-ray (EDX) elemental mapping confirmed its primary chemical constituents. The formation mechanism of the metallogel was examined via fourier transform infrared spectroscopy (FTIR) spectroscopy. The nature of the synthesized CoA-TA metallogel was affirmed through powder X-ray diffraction (PXRD) analysis. Furthermore, this study involved fabrication of Schottky diode structures in a metal-semiconductor-metal geometry based on cobalt(II) metallogel (CoA-TA), enabling observation of charge transport behavior. Remarkably, a resistive random access memory (RRAM) device utilizing cobalt(II) metallohydrogel (CoA-TA) demonstrated bipolar resistive switching at room temperature and under ambient conditions. The switching mechanism was investigated, revealing the formation and rupture of conductive filaments between metal electrodes that govern the resistive switching behavior. This RRAM device exhibited an impressive ON/OFF ratio (~ 414) and exceptional endurance over 5000 switching cycles. These structures offer great potential for diverse applications such as non-volatile memory design, neuromorphic computing, flexible electronics and optoelectronics. Their advantages lie in their fabrication process, reliable resistive switching behavior and overall performance stability.

Construction of parallel algorithms for the study of complex systems objects with a hierarchical-network structure

The procedure for complexite evaluation of complex systems objects that have a hierarchical-network structure has been formalized.In order to realization this procedure in real time mode, parallel computation algorithms are proposed.Speed up estimates were obtained for the mentioned algorithms, which confirm their high efficiency. The proposed parallel algorithms are oriented for execution on modern computing means with shared and distributed memory: computers with multi-core processors, clusters, hybrid architectures and in high performance distributed environments.

Sex Differences in Hierarchical and Modular Organization of Functional Brain Networks: Insights from Hierarchical Entropy and Modularity Analysis.

Existing studies have demonstrated significant sex differences in the neural mechanisms of daily life and neuropsychiatric disorders. The hierarchical organization of the functional brain network is a critical feature for assessing these neural mechanisms. But the sex differences in hierarchical organization have not been fully investigated. Here, we explore whether the hierarchical structure of the brain network differs between females and males using resting-state fMRI data. We measure the hierarchical entropy and the maximum modularity of each individual, and identify a significant negative correlation between the complexity of hierarchy and modularity in brain networks. At the mean level, females show higher modularity, whereas males exhibit a more complex hierarchy. At the consensus level, we use a co-classification matrix to perform a detailed investigation of the differences in the hierarchical organization between sexes and observe that the female group and the male group exhibit different interaction patterns of brain regions in the dorsal attention network (DAN) and visual network (VIN). Our findings suggest that the brains of females and males employ different network topologies to carry out brain functions. In addition, the negative correlation between hierarchy and modularity implies a need to balance the complexity in the hierarchical organization of the brain network, which sheds light on future studies of brain functions.

Hierarchical network structure: A novel approach to conceptualising ICD-11 Complex PTSD using a general population sample from Africa

BackgroundInvestigations have sought to model the structure of ICD-11 Complex PTSD (CPTSD) using factor analytic models, finding support for higher-order domains representing PTSD and Disturbances in Self Organisation (DSO). Network analysis has alternatively modelled CPTSD through dimensional symptom associations. MethodologyThis study investigated the structure of CPTSD leveraging a novel approach, Hierarchical Exploratory Graph Analysis, using African general population samples (N = 2524). ResultsThe hierarchical graph model was estimated identifying a structure comprising six lower-order communities, indicative of ICD-11 CPTSD symptom domains, and two higher-order communities, indicative of PTSD and DSO. Results indicate the superiority of the hierarchical model, confirming the conceptualisation of ICD-11 PTSD and CPTSD symptom groupings. LimitationsThe cross-sectional nature of these data, and novelty of the methods used prompt calls for additional investigation to support these findings. ConclusionsHierarchical Exploratory Graph Analysis may offer a valuable means to better understanding the complexity of CPTSD symptomology through a novel network modelling approach. Relationships between Sense of Threat and Affect Dysregulation may serve as bridging symptoms between PTSD and DSO difficulties. These may be prioritised as the therapeutic targets for CPTSD. This pioneering approach using EGA, offers new insights into the intricate structure of CPTSD, potentially informing the use of assessments and interventions across diverse populations.

Construction of Metaphorical Maps of Cyberspace Resources Based on Point‐Cluster Feature Generalization

ABSTRACTIn the digital age, the expansion of cyberspace has resulted in increasing complexity, making clear cyberspace visualization crucial for effective analysis and decision‐making. Current cyberspace visualizations are overly complex and fail to accurately reflect node importance. To address the challenge of complex cyberspace visualization, this study introduces the integrated centrality metric (ICM) for constructing a metaphorical map that accurately reflects node importance. The ICM, a novel node centrality measure, demonstrates superior accuracy in identifying key nodes compared to degree centrality (DC), k‐shell centrality (KC), and PageRank values. Through community partitioning and point‐cluster feature generalization, we extract a network's hierarchical structure to intuitively represent its community and backbone topology, and we construct a metaphorical map that offers a clear visualization of cyberspace. Experiments were conducted on four original networks and their extracted backbone networks to identify core nodes. The Jaccard coefficient was calculated considering the results of the three aforementioned centrality measures, ICM, and the SIR model. The results indicate that ICM achieved the best performance in both the original networks and all extracted backbone networks. This demonstrates that ICM can more precisely evaluate node importance, thereby facilitating the construction of metaphorical maps. Moreover, the proposed metaphorical map is more convenient than traditional topological maps for quickly comprehending the complex characteristics of networks.

Jellyfish-Inspired Polyurea Ionogel with Mechanical Robustness, Self-Healing, and Fluorescence Enabled by Hyperbranched Cluster Aggregates.

Ionogels are promising for soft iontronics, with their network structure playing a pivotal role in determining their performance and potential applications. However, simultaneously achieving mechanical toughness, low hysteresis, self-healing, and fluorescence using existing network structures is challenging. Drawing inspiration from jellyfish, we propose a novel hierarchical crosslinking network structure design for in situ formation of hyperbranched cluster aggregates (HCA) to fabricate polyurea ionogels to overcome these challenges. Leveraging the disparate reactivity of isocyanate groups, we induce the in situ formation of HCA through competing reactions, enhancing toughness and imparting the clustering-triggered emission of ionogel. This synergy between supramolecular interactions in the network and plasticizing effect in ionic liquid leads to reduced hysteresis of the ionogel. Furthermore, the incorporation of NCO-terminated prepolymer with dynamic oxime-urethane bonds (NPU) enables self-healing and enhances stretchability. Our investigations highlight the significant influence of HCA on ionogel performance, showcasing mechanical robustness including high strength (3.5 MPa), exceptional toughness (5.5 MJ m-3), resistance to puncture, and low hysteresis, self-healing, as well as fluorescence, surpassing conventional dynamic crosslinking approaches. This network design strategy is versatile and can meet the various demands of flexible electronics applications.

Impulsive maneuver strategy for multi-agent orbital pursuit-evasion game under sparse rewards

To address the subjectivity of dense reward designs for the orbital pursuit-evasion game with multiple optimization objectives, this paper proposes the reinforcement learning method with a hierarchical network structure to guide game strategies under sparse rewards. Initially, to overcome the convergence challenges in the reinforcement learning training process under sparse rewards, a hierarchical network structure is proposed based on the hindsight experience replay. Subsequently, considering the strict constraints imposed by orbital dynamics on spacecraft state space, the reachable domain method is introduced to refine the subgoal space in the hierarchical network, further facilitating the achievement of subgoals. Finally, by adopting the centralized training-layered execution approach, a complete multi-agent reinforcement learning method with the hierarchical network structure is established, enabling networks at each level to learn effectively in parallel within sparse reward environments. Numerical simulations indicate that, under the single-agent reinforcement learning framework, the proposed method exhibits superior stability in the late training stage and enhances exploration efficiency in the early stage by 38.89% to 55.56% to the baseline method. Under the multi-agent reinforcement learning framework, as the relative distance decreases, the subgoals generated by the hierarchical network transition from long-term to short-term, aligning with human behavioral logic.

PCP-GC-LM: single-sequence-based protein contact prediction using dual graph convolutional neural network and convolutional neural network

BackgroundRecently, the process of evolution information and the deep learning network has promoted the improvement of protein contact prediction methods. Nevertheless, still remain some bottleneck: (1) One of the bottlenecks is the prediction of orphans and other fewer evolution information proteins. (2) The other bottleneck is the method of predicting single-sequence-based proteins mainly focuses on selecting protein sequence features and tuning the neural network architecture, However, while the deeper neural networks improve prediction accuracy, there is still the problem of increasing the computational burden. Compared with other neural networks in the field of protein prediction, the graph neural network has the following advantages: due to the advantage of revealing the topology structure via graph neural network and being able to take advantage of the hierarchical structure and local connectivity of graph neural networks has certain advantages in capturing the features of different levels of abstraction in protein molecules. When using protein sequence and structure information for joint training, the dependencies between the two kinds of information can be better captured. And it can process protein molecular structures of different lengths and shapes, while traditional neural networks need to convert proteins into fixed-size vectors or matrices for processing.ResultsHere, we propose a single-sequence-based protein contact map predictor PCP-GC-LM, with dual-level graph neural networks and convolution networks. Our method performs better with other single-sequence-based predictors in different independent tests. In addition, to verify the validity of our method against complex protein structures, we will also compare it with other methods in two homodimers protein test sets (DeepHomo test dataset and CASP-CAPRI target dataset). Furthermore, we also perform ablation experiments to demonstrate the necessity of a dual graph network. In all, our framework presents new modules to accurately predict inter-chain contact maps in protein and it’s also useful to analyze interactions in other types of protein complexes.

Jellyfish‐Inspired Polyurea Ionogel with Mechanical Robustness, Self‐Healing, and Fluorescence Enabled by Hyperbranched Cluster Aggregates

AbstractIonogels are promising for soft iontronics, with their network structure playing a pivotal role in determining their performance and potential applications. However, simultaneously achieving mechanical toughness, low hysteresis, self‐healing, and fluorescence using existing network structures is challenging. Drawing inspiration from jellyfish, we propose a novel hierarchical crosslinking network structure design for in situ formation of hyperbranched cluster aggregates (HCA) to fabricate polyurea ionogels to overcome these challenges. Leveraging the disparate reactivity of isocyanate groups, we induce the in situ formation of HCA through competing reactions, enhancing toughness and imparting the clustering‐triggered emission of ionogel. This synergy between supramolecular interactions in the network and plasticizing effect in ionic liquid leads to reduced hysteresis of the ionogel. Furthermore, the incorporation of NCO‐terminated prepolymer with dynamic oxime–urethane bonds (NPU) enables self‐healing and enhances stretchability. Our investigations highlight the significant influence of HCA on ionogel performance, showcasing mechanical robustness including high strength (3.5 MPa), exceptional toughness (5.5 MJ m−3), resistance to puncture, and low hysteresis, self‐healing, as well as fluorescence, surpassing conventional dynamic crosslinking approaches. This network design strategy is versatile and can meet the various demands of flexible electronics applications.

Microplasma-assisted construction of cross-linked network hierarchical structure of NiMoO4 nanorods @NiCo-LDH nanosheets for electrochemical sensing of non-enzymatic H2O2 in food

The accumulation of small doses of hydrogen peroxide (H2O2) into food can cause many diseases in the human body, and it is urgent to develop efficient detection methods of H2O2. Herein, the hierarchical structure composite of NiCo-LDH nanosheets crosslinked NiMoO4 nanorods was grown in situ on carbon cloth (NiMoO4 NRs@NiCo-LDH NSs/CC) by micro-plasma assisted hydrothermal method. Thanks to the synergistic effect of three metals and (NiMoO4 NRs@NiCo-LDH NSs/CC) provided by nanorods/nanosheets hierarchical structure, NiMoO4 NRs@NiCo-LDH NSs/CC exposes more active sites and achieves rapid electron transfer. The H2O2 electrochemical sensor was constructed as the working electrode with a linear range of 1 μmol L−1 to 9.0 mmol L−1 and detection limit of 112 nmol L−1. In addition, the sensor has been successfully applied to the detection of H2O2 in food samples, the recovery rate is 95.2%–106.62%, RSD < 4.89%.

Highly Graphitized Coal Tar Pitch-Derived Porous Carbon as High-Performance Lithium Storage Materials.

Because of its high specific capacity and superior rate performance, porous carbon is regarded as a potential anode material for lithium-ion batteries (LIBs). However, porous carbon materials with wide pore diameter distributions suffer from low structural stability and low electrical conductivity during the application process. During this study, the calcium carbonate nanoparticle template method is used to prepare coal tar pitch-derived porous carbon (CTP-X). The coal tar pitch-derived porous carbon has a well-developed macroporous-mesoporous-microporous hierarchical porous network structure, which provides abundant active sites for Li+ storage, significantly reduces polarization and charge transfer resistance, shortens the diffusion path and promotes the rapid transport of Li+. More specifically, the CTP-2 anode shows high charge capacity (496.9 mAh g-1 at 50 mA g-1), excellent rate performance (413.6 mAh g-1 even at 500 mA g-1), and high cycling stability (capacity retention rate of about 100 % after 1,000 cycles at 2 A g-1). The clean and eco-friendly large-scale utilization of coal tar pitch will facilitate the development of high-performance anodes in the field of LIBs.

Cellulose nanocrystals built multiscale hydrogel electrolyte for highly reversible all-flexible zinc ion batteries

Hydrogel electrolytes are promising for flexible and stable zinc ion batteries (ZIBs), but they lack fatigue resistance and still face challenges in terms of mechanical stability and durability. Herein, inspired by the hierarchical structure of biological networks, multiscale anti-fatigue hydrogel electrolytes with high toughness, high strength, and rapid recovery are constructed by introducing nanoscale cellulose nanocrystals (CNCs) as building blocks into dual-network cross-linking hydroxypropyl chitosan (HPCS) modified polyacrylamide (PAM) hydrogel (denoted as PHC-gel). As a result, the PHC-gel achieves a high fatigue threshold up to 356 J m−2 and with a superior strength and stretchability up to 180 kPa and 3178 %, respectively. In addition, the assembled Zn//Zn cell exhibits long-term cycling stability of 1800 h at 1 mA cm−2/1 mAh cm−2 with a high Coulombic efficiency of 99.4 %. More importantly, the Zn//PHC-gel//MnO2 flexible cell shows remarkable flexibility and capacity retention under different stretching and deformation conditions. This work demonstrates a promising gel chemistry that improves anti-fatigue properties and controls zinc behavior, offering great potential for high-performance flexible and wearable ZIBs and beyond.

An offset-transformer hierarchical model for point cloud-based resistance spot welding quality classification

Resistance spot welding (RSW) is a widely used welding technology in automotive manufacturing, and weld nugget quality is closely related to the quality of the vehicle body. Offline random checks are largely relied on the quality inspection of weld nuggets, but they have low efficiency and high cost. To address this issue, this paper proposes a deep learning model for RSW weld nugget classification, named the offset-transformer hierarchical model (OFTFHC), which is based on the point cloud data of its appearance shape. OFTFHC uses a hierarchical network structure to gradually expand the receptive field. A local feature module is introduced to extract local features from the point cloud, effectively enabling the recognition of the fine structural features of the resistance spot weld point cloud. A residual ratio module, which is based on MLP_MA and uses max and average functions for feature enhancement, is designed to adapt to the complex spatial structure of the point cloud. The offset-transformer structure is used to learn global context features, thereby enhancing the global feature extraction capability. Through classification experiments on RSW weld nuggets across 5 categories with a total of 1050 samples, OFTFHC achieved an average accuracy of 80.6 %, outperforming existing models. This demonstrates the effectiveness and superiority of the method, making it highly suitable for weld nugget quality control in automotive automation production lines.

Multimodal functional deep learning for multiomics data.

With rapidly evolving high-throughput technologies and consistently decreasing costs, collecting multimodal omics data in large-scale studies has become feasible. Although studying multiomics provides a new comprehensive approach in understanding the complex biological mechanisms of human diseases, the high dimensionality of omics data and the complexity of the interactions among various omics levels in contributing to disease phenotypes present tremendous analytical challenges. There is a great need of novel analytical methods to address these challenges and to facilitate multiomics analyses. In this paper, we propose a multimodal functional deep learning (MFDL) method for the analysis of high-dimensional multiomics data. The MFDL method models the complex relationships between multiomics variants and disease phenotypes through the hierarchical structure of deep neural networks and handles high-dimensional omics data using the functional data analysis technique. Furthermore, MFDL leverages the structure of the multimodal model to capture interactions between different types of omics data. Through simulation studies and real-data applications, we demonstrate the advantages of MFDL in terms of prediction accuracy and its robustness to the high dimensionality and noise within the data.

Construction of chitosan/alginate aerogels with three-dimensional hierarchical pore network structure via hydrogen bonding dissolution and covalent crosslinking synergistic strategy for thermal management systems

To replace traditional petrochemical-based thermal insulation materials, in this work, the chitosan (CHI)/alginate (ALG) (CA) aerogels with three-dimensional hierarchical pore network structure were constructed by compositing CHI and ALG using a synergistic strategy of hydrogen bonding dissolution and covalent crosslinking. The structure and properties were further regulated by crosslinking the CA aerogels with epichlorohydrin (ECH). The CA aerogels exhibited various forms of covalent crosslinking, hydrogen bonding and electrostatic interactions, with hydrogen bonding content reaching 79.12 %. The CA aerogels showed an excellent three-dimensional hierarchical pore network structure, with an average pore size minimum of 15.92 nm. The structure regulation of CA aerogels obtained excellent compressive properties, with an increase of stress and strain by 137.61 % and 45.05 %, which can support a heavy object 5000 times its weight. Additionally, CA aerogels demonstrate excellent thermal insulation properties and low thermal conductivity, comparable to commercially available insulation materials. More importantly, CA aerogels have good cyclic insulation stability and thermal properties, and they have a flame retardancy rating of V-0, which shows the stability of insulation properties and excellent safety. CA aerogels provide new ideas for the development of biomass thermal insulation materials and are expected to be candidates for thermal management applications.

Mussel‐inspired self‐assembly of silver nanoclusters into multifunctional silver aerogels for enhanced catalytic and bactericidal applications

AbstractSilver nanoclusters (AgNCs) have shown broad application prospects in catalysis, sensing, and biological fields. However, the limited stability of AgNCs has become the main challenge restricting their practical application in complex environments. Herein, a mussel‐inspired, dopamine‐assisted self‐assembly approach is reported to fabricate 3D AgNC aerogels (PDA/AgNCs), which possess significantly enhanced structural stability and synergistic functional properties. The prepared AgNC aerogels display a hierarchical network structure with an ultrafine ligament size of 10.3 ± 1.2 nm and a high specific surface area of 50.7 m2 g−1. The gelation mechanism is elucidated by in‐depth characterization and time‐lapse monitoring of the gelation process vis spectroscopic and microscopic approaches. Owing to the distinct features of aerogels and the synergistic effect of AgNCs and PDA, the fabricated aerogels can not only efficiently decolorize dyes with a faster kinetic than individual AgNCs, but also exhibit remarkable broad‐spectrum antimicrobial activity. Consequently, a conceptual water‐treatment device is established by depositing PDA/AgNC aerogels on the cotton substrate, which shows good performance in both catalytic dye degradation and bacterial killing in the flowing system. This mussel‐inspired self‐assembly strategy has great potential in developing robust AgNC‐based functional materials, which also provides a new guideline for designing sophisticated materials with integrated functions and synergistic properties.

Highly Biocompatible Antibacterial Hydrogel for Wearable Sensing of Macro and Microscale Human Body Motions.

Flexible electronics, like electronic skin (e-skin), rely on stretchable conductive materials that integrate diverse components to enhance mechanical, electrical, and interfacial properties. However, poor biocompatibility, bacterial infections, and limited compatibility of functional additives within polymer matrices hinder healthcare sensors' performance. This study addresses these challenges by developing an antibacterial hydrogel using polyvinyl alcohol (PVA), konjac glucomannan (KGM), borax (B), and flower-shaped silver nanoparticles (F-AgNPs), referred as PKB/F-AgNPs hydrogel. The developed hydrogel forms a hierarchical network structure, with a tensile strength of 96 kPa, 83% self-healing efficiency within 60 minutes, and 128% cell viability in Cell Counting Kit-8 (CCK-8) assays, indicating excellent biocompatibility. It also shows strong antibacterial efficacy against Gram-negative Escherichia coli (E. coli) and Gram-positive Staphylococcus aureus (S. aureus). Blue light irradiation enhances its antibacterial activity by 1.3-fold for E. coli and 2.2-fold for S. aureus. The hydrogel's antibacterial effectiveness is assessed by monitoring changes in electrical conductivity, providing a cost-effective alternative to traditional microbial culture assays. The PKB/F-AgNPs hydrogel's flexibility and electrical conductivity enable it to function as strain sensors for detecting body movements and facial expressions. This antibacterial hydrogel underscores its potential for future human-machine interfaces and wearable electronics.

Study on the Measurement of the Level of Development of the Digital Economy and the Structure of Spatial Networks

Based on the provincial panel data from 2013-2021, the entropy weight method is used to measure the comprehensive development index of the digital economy, and the gravity model and social network analysis are used to examine the spatial correlation network structure characteristics of the digital economy. digital economy, and the gravity model and social network analysis are used to examine the spatial correlation network structure characteristics of the digital economy among provinces and regions. The study found that: (1) the overall digital economy development level of China's provinces and municipalities is on the rise, but the overall digital economy development level of China's provinces and municipalities is not as high as it should be. municipalities is on the rise, but there is a large gap in the level of digital economy development between regions, and the level of development is gradually declining from China's eastern region to the western region; (2) the spatial network structure of China's level of digital economy development has a better robustness, and development has a better robustness, and the spatial correlation on the whole is strengthened, but the speed of development is relatively slow, and at the same time, the spatial network hierarchical structure is more rigid, and regions under different geographic locations and different levels of development are likely to have spatial network hierarchical structure. development is likely to have spatial interaction effects. (3) The distribution pattern of "core-edge" is formed among provinces and regions, with the distribution pattern of "core-edge" is formed among provinces and regions, with Shanghai, Beijing, Sichuan, Anhui, Shandong, Guangdong, Jiangsu, Henan, Zhejiang, Hubei and Shaanxi being at the core of the network, with strong aggregation and radiation ability. The distribution pattern of "core-edge" is formed among provinces and regions, with Shanghai, Beijing, Sichuan, Anhui, Shandong, Jiangsu, Henan, Zhejiang, and Hubei being at the core of the network, with strong aggregation and radiation ability.