Deep learning as Ricci flow.

Deep neural networks (DNNs) are powerful tools for approximating the distribution of complex data. It is known that data passing through a trained DNN classifier undergoes a series of geometric and topological simplifications. While some progress has been made toward understanding these transformations in neural networks with smooth activation functions, an understanding in the more general setting of non-smooth activation functions, such as the rectified linear unit (ReLU), which tend to perform better, is required. Here we propose that the geometric transformations performed by DNNs during classification tasks have parallels to those expected under Hamilton's Ricci flow-a tool from differential geometry that evolves a manifold by smoothing its curvature, in order to identify its topology. To illustrate this idea, we present a computational framework to quantify the geometric changes that occur as data passes through successive layers of a DNN, and use this framework to motivate a notion of 'global Ricci network flow' that can be used to assess a DNN's ability to disentangle complex data geometries to solve classification problems. By training more than 1500 DNN classifiers of different widths and depths on synthetic and real-world data, we show that the strength of global Ricci network flow-like behaviour correlates with accuracy for well-trained DNNs, independently of depth, width and data set. Our findings motivate the use of tools from differential and discrete geometry to the problem of explainability in deep learning.

Diagnosis of the fragmentation of urban ecological network structure and its social-ecological responses.

The fragmentation of ecological network structures has become a common problem faced by cities. By establishing the urban ecological network under a specific socio-ecological system framework, we aimed to propose a quantitative index to diagnose the fragmentation of the network structure, and to construct detection model to explore the driving factors and mechanism of the network fragmentation. Using Shenzhen City as an example, we used the Floyd-Prim algorithm to generate the skeleton structure of the ecological network and construct a density discontinuity index to diagnose network fragmentation. Combined with the ecological network scenario, social-ecological system framework and a two-layer indicator system were constructed. The detection models were then established to explore the drivers of network disruption and their mode of impact. The models show that the average degree of network fragmentation in Shenzhen was 0.13, and the density of about 85% of corridor discontinuities was greater than 0.01, reflecting the serious state of structural fragmentation. Corridors with more severe structural fragmentation have poorer social-ecological coordination. The fragmentation in Shenzhen was mainly affected by the activities of actors (A) at the microlevel and the resource system (RS) at the macrolevel. The methods and the framework of socio-ecosystem analysis proposed in this paper can reveal the driving factors and influence modes of network fragmentation, providing decision-making reference for ecological restoration practice in urbanized areas.

High performance method for COPD features extraction using Complex Network.

OBJECTIVES: The paper proposes a novel methodology for the classification of Chronic Obstructive Pulmonary Disease (COPD) utilizing respiratory sound attributes. Methods: The approach involves segmenting respiratory sounds into individual breaths and conducting extensive studies on this dataset. Spectral Transforms, various Wavelet Transforms are applied to capture distinct signal features. Complex Network is also employed to extract characteristic elements, generating novel representations of spectrogram data based on graph factors, including entropy, density, and position. The normalized and enriched data is then used to develop COPD classifiers using six machine learning algorithms, fine-tuning with appropriate training details and hyperparameter tuning. Results: Our results demonstrate robust performance, with ROC curves consistently exhibiting an Area Under the Curve (AUC) > 96% across different time-frequency transformations. Notably, the Random Forest algorithm achieves an AUC of 99.67%, outperforming other algorithms. Moreover, the Wavelet Daubechies 2 (Db2) consistently approaches 98% accuracy, particularly noteworthy in conjunction with the Naive Bayes algorithm. Conclusion: This study diagnosis patients through spectrogram images extracted from lung sounds. The application of Inverse Transforms, Complex Network, and Optimized Classification Algorithms yielded results beyond expectations. This methodology provides a promising approach for accurate COPD diagnosis, leveraging Machine Learning techniques applied to respiratory sound analysis.

High free sugars, insulin resistance, and low socioeconomic indicators: the hubs in the complex network of non-communicable diseases in adolescents.

BACKGROUND: Noncommunicable diseases (NCDs) predominantly affect adults, but pathophysiological changes begin decades earlier, as a continuum, with initial events apparent in adolescence. Hence, early identification and intervention are crucial for the prevention and management of NCDs. We investigated the complex network of socioeconomic, behavioral, and metabolic factors associated with the presence of NCD in Brazilian adolescents. METHODS: We conducted a cross-sectional study nested within the São Luís segment of the Ribeirão Preto, Pelotas, and São Luís (RPS) cohort's consortium, focusing on 18-19-year-olds (n = 2515). Data were collected prospectively, from which we constructed a complex network with NCD-related factors/indicators as nodes and their co-occurrences as edges. General and sex-based models analyzed: socioeconomic status, behavioral (smoking, alcohol, and other drugs use, unhealthy diet, poor sleep, physical inactivity), and metabolic factors (overweight/obesity, elevated blood pressure, poor lipid profile). We also looked for NCDs in adolescence like asthma, abnormal spirometry, depression, suicide risk, and poor oral health. The network was characterized by degree, betweenness, eigenvector, local transitivity, Shannon entropy, and cluster coefficient. RESULTS: The adolescents had an average age of 18.3 years, 52.3% were female and 47.7% male. 99.8% of them have a diet rich in free sugars, 15% are overweight/obese and 72.3% had an elevated TyG index. High free sugar emerged as the central hub, followed by high TyG index (an early marker of insulin resistance) and low socioeconomic class. In males, low fiber intake and a high triglycerides/HDL ratio highlighted cardiometabolic concerns; in females, sedentary behavior and poor sleep marked metabolic and psychological challenges, along with caries in both sexes. CONCLUSIONS: Our findings provide insights into central health challenges during adolescence, such as high free sugars, insulin resistance, and low socioeconomic indicators, suggesting that interventions targeted at these central hubs could have a significant impact on their NCD network.

Urban synergistic carbon emissions reduction research: A perspective on spatial complexity and link prediction.

Reducing urban carbon emissions (UCEs) holds paramount importance for global sustainable development. However, the complexity of interactions among urban spatial units has impeded further research on UCEs. This study investigates synergistic emission reduction between cities by analyzing the spatial complexity within the UCEs network. The future potential for synergistic carbon emissions reduction is predicted by the link prediction algorithm. A case study conducted in the Pearl River Basin of China demonstrates that the UCEs network has a complex spatial structure, and the synergistic capacity of emission reduction among cities is enhanced. The core cities in the UCEs network, including Dongguan, Shenzhen, and Guangzhou, have spillover effects that contribute to synergistic emission reduction. Community detection reveals that the common characteristics associated with UCEs become concentrated, thereby enhancing the synergy of joint efforts between cities. The link prediction algorithm indicates a high probability of strengthened carbon emission connections in the Pearl River Delta, alongside those between upstream cities, which shows potential in forecasting synergistic emission reductions. Our research framework offers a comprehensive analysis for synergistic emission reduction from the spatial complexity of UCEs network and link prediction. It acts as a worthwhile reference for developing differentiated policies on synergistic emission reduction.

Enhancing bird habitat networks in metropolitan areas: Resilience assessment and improvement strategies - A case study from Shanghai.

Constructing bird habitat networks (BHNs) is crucial for maintaining the health and service equilibrium of urban ecosystems, especially in large metropolitan areas where the pressure of urbanization is intense. However, most existing BHNs fail to account for the dynamic changes and unique requirements of local species, leading to homogenized construction outcomes and ecological corridor objectives. This study employs a comprehensive approach to identify bird habitat patches using multiple high-quality sources, then utilize circuit theory and complex network theory to construct and assess the resilience of BHN. Our key findings showed: (1)93 bird habitat sources were identified, predominantly situated in the continuous green spaces of southern and southeastern Shanghai, whereas habitat sources in the city center and other densely built-up areas are more dispersed, highlighting them as prime targets for future ecological restoration efforts. (2) The distribution of bird habitat corridors exhibits significant spatial heterogeneity, with primary corridors predominantly spanning the southwestern and eastern parts of the study area, while secondary corridors are more abundant in the western and northern parts, forming a denser network, whereas the central area shows fewer and more isolated corridors. (3) The decline in structural and functional resilience was notably more rapid under targeted attacks than under random attacks, underscoring the need to prioritize crucial bird habitat sources on the city's periphery, especially near highly urbanized areas, in urban planning and biodiversity conservation efforts to sustain ecological balance and biodiversity. These insights provide a crucial scientific basis for urban planners, emphasizing the integration of biodiversity conservation into urban development strategies by optimizing ecological sources and corridors to balance development with ecological preservation.

Learning site-invariant features of connectomes to harmonize complex network measures.

Multi-site diffusion MRI data is often acquired on different scanners and with distinct protocols. Differences in hardware and acquisition result in data that contains site dependent information, which confounds connectome analyses aiming to combine such multi-site data. We propose a data-driven solution that isolates site-invariant information whilst maintaining relevant features of the connectome. We construct a latent space that is uncorrelated with the imaging site and highly correlated with patient age and a connectome summary measure. Here, we focus on network modularity. The proposed model is a conditional, variational autoencoder with three additional prediction tasks: one for patient age, and two for modularity trained exclusively on data from each site. This model enables us to 1) isolate site-invariant biological features, 2) learn site context, and 3) re-inject site context and project biological features to desired site domains. We tested these hypotheses by projecting 77 connectomes from two studies and protocols (Vanderbilt Memory and Aging Project (VMAP) and Biomarkers of Cognitive Decline Among Normal Individuals (BIOCARD) to a common site. We find that the resulting dataset of modularity has statistically similar means (p-value <0.05) across sites. In addition, we fit a linear model to the joint dataset and find that positive correlations between age and modularity were preserved.

Research on coal mine safety risk evolution and key hidden dangers under the perspective of complex network.

In order to find out the main causes of coal mine safety accidents and improve the pertinence of coal mine safety risk management and control, the identification and analysis of coal mine safety risks and hidden dangers are carried out based on the analysis of coal mine accident reports. Combing the complex network theory, a complex network model for the evolution of coal mine safety risks is constructed. The key elements that affect coal mine safety risk accidents are obtained through quantitative research on the characteristic indicators of the complex network model of coal mine safety risks. And the key nodes of coal mine safety risk spread network are obtained through network interference to the overall efficiency. The research results show that the complex network of coal mine safety risks illustrate the characteristics of a small-world network, and the spread of a certain risk is likely to cause coal mine safety accidents. Strengthening the risk management and control of hidden dangers with higher intermediate centrality can isolate the spread of coal mine safety risks and reduce the possibility of coal mine accidents.

Analysis of changes in river connectivity during the urbanization process: a case study of the southeastern plain of Yinzhou, China.

Affected by human activities, the naturally occurring river network in the southeastern plain of Yinzhou has gradually evolved into a natural-artificial composite water system, and changes in river connectivity due to changes in river network systems have caused water security problems, including urban flooding. To clarify the river connectivity change and its relationship with the urbanization process, this paper discusses an evaluation method for river connectivity based on complex networks and cellular automata (CA) from the perspective of complex systems, quantitatively analyzes the spatial-temporal characteristics of the structural and functional river connectivity in the study area during the 1990s-2020s, and reveals the impact of river nodes and chains on the connectivity level under the disturbance of natural or human factors. The results contained the following revelations: ① River connectivity showed a decreasing trend in the initial and rapid development stages of urbanization from the 1990s to the 2010s and a limited increasing trend in the optimization and upgrading stages from the 2010s to the 2020s. ② River network degradation and ongoing connectivity decline are found in the northeastern part of the study area. The highest river connectivity exists in Dongqianhukaifaqu. ③ The number of river nodes and chains should be maintained at approximately 80% for normal river connectivity. The nodes of high degree in the inflow area are listed in the key protection areas. ④ Changes in river connectivity are significantly correlated with the urbanization process. Changes in the functional connectivity level affect the magnitude of a flood. This study provides a theoretical basis for river network connectivity improvement and flood prevention in plain areas.

Spatiotemporal and Multilayer Trade Network Patterns of the Global Cobalt Cycle.

Recent years have witnessed increasing attempts to track trade flows of critical materials across world regions and along the life cycle for renewable energy and the low carbon transition. Previous studies often had limited spatiotemporal coverage, excluded end-use products, and modeled different life cycle stages as single-layer networks. Here, we integrated material flow analysis and complex network analysis into a multilayer framework to characterize the spatiotemporal and multilayer trade network patterns of the global cobalt cycle from 1988 to 2020. We found substantial growth and notable structural changes in global cobalt trade over the past 30 years. China, Germany, and the United States play pivotal roles in different layers and stages of the global cobalt cycle. The interlayer relationships among alloys, batteries, and materials are robust and continually strengthening, indicating a trend toward synergistic trade. However, cobalt ore-exporting countries are highly concentrated and rarely involved in later life cycle stages, resulting in the weakest relationship between the ore layer and other layers. This causes fluctuations and uncertainty in the global cobalt trade. Our model, linking industrial ecology, supply chain analysis, and network analysis, can be extended to other materials that are critical for the future green transition.

A novel longitudinal connectivity index to evaluate reticular river networks based on the combination of network maximum flow and resistance distance.

Reticular river networks, essential for ecosystems and hydrology, pose challenges in assessing longitudinal connectivity due to complex multi-path structures and variable flows, exacerbated by human-made infrastructures like sluices. Existing tools inadequately track water flow's spatiotemporal changes, highlighting the need for targeted methods to gauge connectivity within complex river network systems. The Hydraulic Capacity Connectivity Index (HCCI) was developed adopting complex network theory. This involves river networks mapping, nodes and edges construstion, weight factor definition, maximum flow and resistance distance calculation. The connectivity between nodes is represented by the product of the maximum flow and the inverse of the resistance distance. The mean connectivity of each node with all other nodes, denoted as the node connectivity capacity Ci, and the HCCI of the whole river network is defined as the mean of the Ci for all nodes. The HCCI was firstly applied to a symmetrical virtual river network to investigate the factors influencing the HCCI. The results revealed that Ci showed a radial decreasing pattern from the obstructed river reach outwards, and the boundary rivers play the most significant role in regulating the flow dynamics. Subsequently, the HCCI was applied to a real river network in the Yandu district, followed by spatiotemporal statistical analysis comparing with 1D hydraulic model's simulated river discharge. Results showed a high correlation (Pearson coefficient of 0.89) between the HCCI and monthly average river discharge at the global scale. At the local scale, the geographically weighted regression model demonstrated the strong explanatory power of Ci in predicting the distribution of river reach discharge. This suggests that the HCCI addresses multi-path connectivity assessment challenge in reticular river networks, precisely characterizing spatiotemporal flow dynamics. Furthermore, since HCCI is based on a complex network model that can calculate the connectivity between all river node pairs, it is theoretically applicable to other types of river networks, such as dendritic river networks. By identifying low-connectivity areas, HCCI can guide managers in developing scientifically sound and effective strategies for restoring river network hydrodynamics. This can help prevent water stagnation and degradation of water quality, which is beneficial for environmental protection and water resource management.

Assessing interconnectedness and systemic importance of Chinese financial institutions.

This study proposes a directed acyclic graph (DAG)-based framework for generalized variance decomposition for investigating the heterogeneous return spillovers in financial system and measuring the systemic importance of financial institutions among 34 listed Chinese financial institutions from 2011 to 2023. Findings indicate pronounced information spillovers among institutions within the same sector due to contemporaneous causal relationships. Both static and dynamic financial network analyses highlight the significance of the securities sector. Dynamic structural characteristics align with macroeconomic development and are sensitive to internal and external shocks. Systemic importance assessment reveals that market size alone doesn't determine importance, with notable disparities between banking and non-banking sectors. State-owned and joint-stock commercial banks play a vital role in banking, while local government and private capital-controlled institutions are crucial in the securities sector. This research aids regulatory efforts in maintaining a balanced regulatory environment, ensuring market efficiency, and reducing operational costs.

Data-driven and equation-free methods for neurological disorders: analysis and control of the striatum network.

The striatum as part of the basal ganglia is central to both motor, and cognitive functions. Here, we propose a large-scale biophysical network for this part of the brain, using modified Hodgkin-Huxley dynamics to model neurons, and a connectivity informed by a detailed human atlas. The model shows different spatio-temporal activity patterns corresponding to lower (presumably normal) and increased cortico-striatal activation (as found in, e.g., obsessive-compulsive disorder), depending on the intensity of the cortical inputs. By applying equation-free methods, we are able to perform a macroscopic network analysis directly from microscale simulations. We identify the mean synaptic activity as the macroscopic variable of the system, which shows similarity with local field potentials. The equation-free approach results in a numerical bifurcation and stability analysis of the macroscopic dynamics of the striatal network. The different macroscopic states can be assigned to normal/healthy and pathological conditions, as known from neurological disorders. Finally, guided by the equation-free bifurcation analysis, we propose a therapeutic close loop control scheme for the striatal network.

The relationship between structure and ecosystem services of forest and grassland based on pattern analysis method: A case study of the Mongolian Plateau.

Global warming has led to severe land desertification on the Mongolian plateau. It puts great environmental pressure on vegetation communities. This pressure leads to fragmentation of land use and landscape patterns, thus triggering changes in the spatial distribution patterns of vegetation. The spatial distribution pattern of vegetation is crucial for the performance of its ecosystem services. However, there is not enough research on the relationship between large-scale spatial distribution patterns of vegetation and ecosystem services. Therefore, this study is to construct an ecological spatial network on the Mongolian Plateau based on landscape ecology and complex network theory. Combining pattern analysis methods to analyze the network, we obtained the spatial and temporal trends of forest and grass spatial distribution patterns from 2000 to 2100, and explored the relationship between the topological properties of source patches and ecosystem services in different patterns. It was found that there are four basic patterns of spatial distribution of forest and grass in the Mongolian Plateau. The Core-Linked Ring pattern accounts for 40.74 % and exhibits the highest stability. Under the SSP5-RCP8.5 scenario, source patches are reduced by 22.76 % in 2100. Topological indicators of source patches showed significant correlations with ecosystem services. For example, the CUE of grassland patches in the Centralized Star pattern was positively correlated with betweeness centrality. The most significant improvement in WUE after optimization is 19.90 % compared to pre-optimization. The conclusion of the study shows that the spatial distribution pattern of vegetation can be used to enhance the stability of ecological spatial network and improve ecosystem services at a larger scale. It can provide a certain reference for the study of spatial patterns of vegetation distribution in arid and semi-arid areas.

[Modular pharmacology: research progress and development of analysis platform].

Based on the systematic deconstruction of multi-dimensional and multi-target biological networks, modular pharmacology explains the complex mechanism of diseases and the interactions of multi-target drugs. It has made progress in the fields of pathogenesis of disease, biological basis of disease and traditional Chinese medicine(TCM) syndrome, pharmacological mechanism of multi-target herbs, compatibility of formulas, and discovery of new drug of TCM compound. However, the complexity of multi-omics data and biological networks brings challenges to the modular deconstruction and analysis of the drug networks. Here, we constructed the "Computing Platform for Modular Pharmacology" online analysis system, which can implement the function of network construction, module identification, module discriminant analysis, hub-module analysis, intra-module and inter-module relationship analysis, and topological visualization of network based on quantitative expression profiles and protein-protein interaction(PPI) data. This tool provides a powerful tool for the research on complex diseases and multi-target drug mechanisms by means of modular pharmacology. The platform may have broad range of application in disease modular identification and correlation mechanism, interpretation of scientific principles of TCM, analysis of complex mechanisms of TCM and formulas, and discovery of multi-target drugs.

Tele-connection of embodied carbon emissions from industries in China's trade: A complex network analysis.

Regions can meet their development demands through trade, with the attendant environmental costs being shifted to other regions, and carbon emissions emitted from different industries could be transferred over long distances through the increasingly diversified trade network. However, it remains unclear how regional trade leads to the tele-connection and transfer of embodied carbon emissions form industries, and what is the structure and characteristics of the transfer. Thus, multiregional input‒output models and complex network analysis are employed to reveal the tele-connection of carbon emissions from industries in China. The results show that embodied carbon emissions from trade increased by 869.47 million tons during in five years, with North China being the largest outflow area, while the coastal regions being the inflow areas. Moreover, the secondary industry is the highest source of embodied carbon emissions, accounting for 96.68 % of the volume, and the transfer of carbon emissions mainly occurs in North and East China. In carbon emissions networks, North China holds a controlling position, as analysed by degree and strength. The first 23.3%-30% of nodes carry about 62.6%-72.4% of the entire carbon emissions flow, and the network conforms to scale-free features. Centrality further reveals that northern and coastal areas occupy core positions, with interregional carbon flows dominating the critical pathways in the network. The number of clusters evolved from three to four communities during 2012-2017 in the network, demonstrating that the carbon flow network is developing towards multipolarity and modularity. This study underscores the urgency of mitigating carbon emissions in industrial trade by identifying key nodes and cluster structures in emission networks.

Drivergene.net: A Cytoscape app for the identification of driver nodes of large-scale complex networks and case studies in discovery of drug target genes.

There are no tools to identify driver nodes of large-scale networks in approach of competition-based controllability. This study proposed a novel method for this computation of large-scale networks. It implemented the method in a new Cytoscape plug-in app called Drivergene.net. Experiments of the software on large-scale biomolecular networks have shown outstanding speed and computing power. Interestingly, 86.67% of the top 10 driver nodes found on these networks are anticancer drug target genes that reside mostly at the innermost K-cores of the networks. Finally, compared method with those of five other researchers and confirmed that the proposed method outperforms the other methods on identification of anticancer drug target genes. Taken together, Drivergene.net is a reliable tool that efficiently detects not only drug target genes from biomolecular networks but also driver nodes of large-scale complex networks. Drivergene.net with a user manual and example datasets are available https://github.com/tinhpd/Drivergene.git.

Study on the Stability of Complex Networks in the Stock Markets of Key Industries in China.

Investigating the significant "roles" within financial complex networks and their stability is of great importance for preventing financial risks. On one hand, this paper initially constructs a complex network model of the stock market based on mutual information theory and threshold methods, combined with the closing price returns of stocks. It then analyzes the basic topological characteristics of this network and examines its stability under random and targeted attacks by varying the threshold values. On the other hand, using systemic risk entropy as a metric to quantify the stability of the stock market, this paper validates the impact of the COVID-19 pandemic as a widespread, unexpected event on network stability. The research results indicate that this complex network exhibits small-world characteristics but cannot be strictly classified as a scale-free network. In this network, key roles are played by the industrial sector, media and information services, pharmaceuticals and healthcare, transportation, and utilities. Upon reducing the threshold, the network's resilience to random attacks is correspondingly strengthened. Dynamically, from 2000 to 2022, systemic risk in significant industrial share markets significantly increased. From a static perspective, the period around 2019, affected by the COVID-19 pandemic, experienced the most drastic fluctuations. Compared to the year 2000, systemic risk entropy in 2022 increased nearly sixtyfold, further indicating an increasing instability within this complex network.

A Critical Candidate Node-Based Attack Model of Network Controllability.

The controllability of complex networks is a core issue in network research. Assessing the controllability robustness of networks under destructive attacks holds significant practical importance. This paper studies the controllability of networks from the perspective of malicious attacks. A novel attack model is proposed to evaluate and challenge network controllability. This method disrupts network controllability with high precision by identifying and targeting critical candidate nodes. The model is compared with traditional attack methods, including degree-based, betweenness-based, closeness-based, pagerank-based, and hierarchical attacks. Results show that the model outperforms these methods in both disruption effectiveness and computational efficiency. Extensive experiments on both synthetic and real-world networks validate the superior performance of this approach. This study provides valuable insights for identifying key nodes crucial for maintaining network controllability. It also offers a solid framework for enhancing network resilience against malicious attacks.

Research on the Influence of Information Iterative Propagation on Complex Network Structure.

Dynamic propagation will affect the change of network structure. Different networks are affected by the iterative propagation of information to different degrees. The iterative propagation of information in the network changes the connection strength of the chain edge between nodes. Most studies on temporal networks build networks based on time characteristics, and the iterative propagation of information in the network can also reflect the time characteristics of network evolution. The change of network structure is a macromanifestation of time characteristics, whereas the dynamics in the network is a micromanifestation of time characteristics. How to concretely visualize the change of network structure influenced by the characteristics of propagation dynamics has become the focus of this article. The appearance of chain edge is the micro change of network structure, and the division of community is the macro change of network structure. Based on this, the node participation is proposed to quantify the influence of different users on the information propagation in the network, and it is simulated in different types of networks. By analyzing the iterative propagation of information, the weighted network of different networks based on the iterative propagation of information is constructed. Finally, the chain edge and community division in the network are analyzed to achieve the purpose of quantifying the influence of network propagation on complex network structure.