A review of graph and complex network theory in water distribution networks: Mathematical foundation, application and prospects

Graph theory (GT) and complex network theory play an increasingly important role in the design, operation, and management of water distribution networks (WDNs) and these tasks were originally often heavily dependent on hydraulic models. Facing the general reality of the lack of high-precision hydraulic models in water utilities, GT has become a promising surrogate or assistive technology. However, there is a lack of a systematic review of how and where the GT techniques are applied to the field of WDNs, along with an examination of potential directions that GT can contribute to addressing WDNs' challenges. This paper presents such a review and first summarizes the graph construction methods and topological properties of WDNs, which are mathematical foundations for the application of GT in WDNs. Then, main application areas, including state estimation, performance evaluation, partitioning, optimal design, optimal sensor placement, critical components identification, and interdependent networks analysis, are identified and reviewed. GT techniques can provide acceptable results and valuable insights while having a low computational burden compared with hydraulic models. Combining GT with hydraulic model significantly enhances the performance of analysis methods. Four research challenges, namely reasonable abstraction, data availability, tailored topological indicators, and integration with Graph Neural Networks (GNNs), have been identified as key areas for advancing the application and implementation of GT in WDNs. This paper would have a positive impact on promoting the use of GT for optimal design and sustainable management of WDNs.

Assessment and solutions for vulnerability of urban rail transit network based on complex network theory: A case study of Chongqing

As a typical complex network system, the operating environment of rail transit network (RTN) is complex and demanding. This study aims to accurate assess the weaknesses and vulnerability of RTN, which is crucial for ensuring its smooth operation. Taking Chongqing Rail Transit (CRT) as an example, this study developed a network topology model using the spatial L method and analyzed the network structure characteristics, along with the importance of key nodes under different indicators, based on complex network theory. Additionally, this study analyzed the geographical spatial distribution characteristics of nodes based on the topography and urban spatial structure of Chongqing. Then, this study classified the nodes in the RTN according to basic topological indicators, namely degree, betweenness centrality, network efficiency, and passenger flow volume (PFV). The results indicated six cluster of nodes, reflecting the variability in node vulnerability concerning overall influence (providing alternative paths, reducing path length), regional aggregation capacity, and transportation capacity. Finally, this study proposed targeted management strategies for different clusters of nodes and their respective geographical locations, providing necessary references for rational planning, safety protection, and sustainable construction of RTN.

Impacts of the Middle Route of China's South-to-North Water Diversion Project on the water network structure in the receiving basin

The Middle Route of South-to-North Water Diversion Project (MRSNWD) is the main skeleton of China's National Water Network, its construction has changed the structure of the original water network, and analyzing the topological change of the water network in context with MRSNWD is significant for water network planning and management. In this study, the overall network characteristics of the water network in 2010 and 2020 were analyzed based on the small-world and scale-free characteristics of complex network theory. The topological changes of the water network from a node perspective were examined using three network centrality indexes: degree centrality (DC), closeness centrality (CC), and betweenness centrality (BC), while assessing the important nodes of the water network and recognizing functional areas of cold-hot spots. The results show that the water network's centrality in the study area improved after the project construction, with the average degree of the water network increasing from 2.39 to 2.42 and the average path length decreasing from 111.81 to 97.08. The propagation efficiency and network stability also increased, with a rise in important node proportion from 9.8 to 14.4%. The nodes in the DC hotspot zone along the project route have increased by 1.5%, implying an increase in the connectivity of the water network, while MRSNWD optimizes its north-south hub propagation path. "Small-worldness" indicates that most nodes of a network can be accessed and connected over shorter paths. The water network has a significant "small-worldness" and has been enhanced by the MRSNWD's construction. Approximating the water network as a scale-free network can impact its security by identifying critical nodes. The results of this research can provide the necessary technical support and reference significance for China's National Water Network.

Network structure and stability of the river connectivity in a rapidly urbanizing region

River connectivity, which is key to the function of the river network, is deteriorating in highly urbanized areas. While previous studies have identified changes in the structure and connectivity of river networks, few studies have described the network structure of river connectivity and revealed the effects of urbanization on network structure. To this end, we detected the network structure and stability of river connectivity from a novel perspective of complex network theory. Taking the Taihu Plain as an example, we found that 1) the node degree of the river network was moderate (2.9), and the hub and connectivity were low (0.21 and 0.19, respectively). The hub and connectivity of the river network in the Yang-Cheng-Dian-Mao (YCDM) region were better than that in Wu-Cheng-Xi-Yu (WCXY) and Hang-Jia-Hu (HJH) regions. 2) The destruction of important nodes led to a dramatic decrease in the stability of river network connectivity. The river network structures on the Taihu Plain, HJH, WCXY, and YCDM were severely damaged when the removal rate of river network nodes exceeded 5 %, 7 %, 16 %, and 22 %, respectively. 3) The spatial response of the river network to urbanization was mainly negative, especially in highly urbanized regions. The effects of urbanization on network characteristics could be sorted as node degree (45 %), hub (21 %), and connectivity (18 %). Our results would provide theoretical support for the recognition, protection, and restoration of the river network in rapidly urbanizing regions.

Integrating ecosystem services and complex network theory to construct and optimize ecological security patterns: a case study of Guangdong-Hong Kong-Macao Greater Bay Area, China

The urban agglomerations' rapid expansion and population growth have led to the fragmentation of landscape patterns and the degradation of ecosystems, seriously threatening regional ecological security. Ecological security pattern (ESP) is a spatial planning approach to effectively balance the development of urbanization and ecological protection. However, previous studies have ignored the difference in the importance of ecosystem services and the spatial compactness of ecological sources. The quantitative management objectives for maintaining the resilience of ESP are also rarely discussed. In this study, taking the Guangdong-Hong Kong-Macao Greater Bay Area (GBA) as an example, ecological sources were identified by simulating multiple ES weight assignment scenarios through GeoSOS area optimization. Ecological corridors and strategic points were extracted by Linkage Mapper. The robustness analysis based on complex network theory was performed to quantify the management objectives of ESPs. The results showed that ESPs include 26,130.61 km² ecological sources (accounting for 46.6% of the area of GBA), 557 ecological corridors, and 112 ecological strategic points. In more detail, ecological sources are mainly distributed in the western and eastern mountainous areas, and ecological corridors primarily link peripheral edge areas of GBA in a circular radial shape. Compared with the current nature reserves, the identified ecological sources are more compact in landscape pattern. According to the robustness analysis, at least 23% of the important ecological sources should be strictly restricted from development activities to maintain the ESP's ability to resist ecological risks. This study also proposed corresponding differentiated ESPs management strategies. By optimizing the existing ESPs construction method and clarifying the ESPs management strategies, this study provides a completely scientific framework for the construction and management of ESPs in urban agglomerations.

Study on the rare waterbird habitat networks of a new UNESCO World Natural Heritage site based on scenario simulation

As a newly established World Natural Heritage site, the conservation of rare waterbird habitats in the Yancheng coastal wetlands has attracted wide international attention. In view of the importance of this area in international biodiversity conservation and waterbird habitat conservation, the study of the current situation of rare waterbird habitat networks with spatial isolation features of great demonstration significance to improve the habitat conservation quality of the heritage site. Based on the data obtained from habitat suitability assessments of rare waterbirds, this paper used complex network theory, ecological stepping-stone theory and circuit models to analyze the current status and robustness of rare waterbird habitat networks in the Yancheng coastal wetlands. The results showed that the stepping stones of the red-crowned cranes and the Oriental storks were mainly distributed around important habitat areas, including aquaculture ponds and reed marshes, with areas of 1275.68 hm² and 1247.74 hm², respectively, while the stepping stones of Saunders's gulls were mainly distributed within the Tiaozini habitat site, with an area of 1180.76 hm². The stability and connectivity of the habitat networks of red-crowned cranes and Oriental storks in the northern habitat area were better. At the Tiaozini habitat site, there was spatial isolation and low connectivity among the habitats and stepping stones of Saunders's gulls. In the optimal protection scenario, the stability and connectivity of the habitat network structure of red-crowned cranes and Oriental storks were the best, and the networks tended to exhibit assortativity. In the random destruction scenario, the connectivity of the habitat networks of Saunders's gulls was the lowest, and the network structure was the most fragile. Finally, the optimization and restoration patterns of habitat network based on improving the support capacity of important habitat nodes and the enhancement of the stability and connectivity of the stepping-stone networks were proposed.

Complex network analysis of volatility spillovers between global financial indicators and G20 stock markets

This paper analyses the dynamic transmission mechanism of volatility spillovers between key global financial indicators and G20 stock markets. To examine volatility spillover relations, we combine a bivariate GARCH-BEKK model with complex network theory. Specifically, we construct a volatility network of international financial markets utilising the spatial connectedness of spillovers (consisting of nodes and edges). The findings show that spillover relations between global variables and G20 markets vary significantly across five identified sub-periods. Notably, networks are much denser in crisis periods compared to non-crisis periods. In comparing two crisis periods, Global Financial Crisis (2008) and COVID-19 Crisis (2020) periods, the network statistics suggest that volatility spillovers in the latter period are more transitive and intense than the former. This suggests that financial volatility spreads more rapidly and directly through key financial indicators to the G20 stock markets. For example, oil and bonds are the largest volatility senders, while the markets of Saudi Arabia, Russia, South Africa, and Brazil are the main volatility receivers. In the former crisis, the source of financial volatility concentrates primarily in the USA, Australia, Canada, and Saudi Arabia, which are the largest volatility senders and receivers. China emerges as generally the least sensitive market to external volatility.

The drivers of collaborative innovation of the comprehensive utilization technologies of coal fly ash in China: a network analysis

Coal consumption brings a lot of coal fly ash (CFA). It requires interdisciplinary efforts in research, policy, and practice to improve the utilization of CFA. Although there have been a lot of achievements in technological innovation, the utilization of CFA is still difficult to match its output. So, it is urgent to explore how to guide its effective innovation. This paper uses social network analysis to discuss the characteristics of the collaborative innovation network of CFA comprehensive utilization technology in China. Then, this paper uses regression analysis to explore the differences in innovation performance under different research and development (R&D) backgrounds. The results show that (1) based on the network-level indicators, the collaborative innovation scale has an obvious trend of expanding. Partnerships increased from 20 to 574. Meanwhile, the network shows obvious scale-free and "small-world" characteristics, indicating that innovation resources are concentrated in a few organizations. (2) Based on the node-level indicators, the major contributor has shifted from universities and research institutions to enterprises. Enterprises account for the highest proportion (73%) and have the highest centrality (8.3). The betweenness centrality of the universities is 265, and only 14% of the organizations are universities which means universities play an important role in connecting different organizations in the network, but their participation in the collaborative innovation is insufficient. (3) Based on the collaborative relationship-level indicators, the cooperation is lack of depth. Only a small number of organizations, especially enterprises, have stable partners, showing the characteristic of "low cooperation width and high cooperation depth," which means fewer partners but more frequently collaborative innovation. (4) Based on the innovation performance, the innovation performance under the category of cooperative R&D, especially industry-academy cooperation, is better than that of independent R&D. But, industry-academy cooperation only occupied 43% of collaborative relationships in the network. Finally, this paper puts forward suggestions for governments from five aspects: decentralization, defining roles of enterprise and university, encouraging collaboration, changing the idea of the patent application, and promoting deeper cooperation.

A complex network perspective on embodiment of air pollutants from global oil refining industry

The air pollutants emitted from oil refining industry could be transferred across borders through the increasingly complex global trade network. However, the specific structural features of the network remain unclear. Based on the Multi-regional Input-output method and complex network theory, we make a first attempt to trace six oil refining air pollutants embodied in the international trade. The results show that the overall character of the global oil refining air pollutants flow network exhibits small-world behavior, and each node of the network is strongly connected. Therefore, the refining emissions mitigation measures of one node could efficiently radiate to the other nodes connected to it, which provides essential opportunities for collaborative emissions reduction among countries. Besides, the individual characteristics of each node are distinguished, several key nodes dominate the embodied emissions throughout the global oil refining air-pollutants flow network. For specific countries, the United States, China, Japan, and the United Kingdom are the hub economics in importing embodied pollutants in the network, while Russia and Canada are the key exporters. Similarly, the critical paths with large flow still come from the trade between these key nodes. Our estimated results have great policy implications for reducing air pollutants emitted from oil refining industry and also have profound implications for environmental regulation and protection in the world.

An analysis of the Chinese scheduled freighter network during the first year of the COVID-19 pandemic

COVID-19 caused the vast majority of passenger flights to be grounded, but the crisis raised the importance of the network of dedicated cargo flights and, therefore, interest in its development. This paper aims to evaluate the Chinese scheduled freighter network (CSFN) via its topological properties and to explore its changes following the COVID-19 pandemic. Using spatial analysis with the complex network theory (CNT), the paper found that the CSFN displays small-world and scale-free network properties, similar to that of air passenger network. Hangzhou, Shenzhen and Nanjing are the dominant national hubs in the CSFN because they host the headquarters of many e-commerce giant enterprises and have relatively underutilized airport capacities. The CSFN has improved since the COVID-19 pandemic, with increased network average degree, clustering coefficient, and closeness, and reduced average path. These improvements were mainly driven by major hub cities whose centralities had been strengthened with more route connections. Since China's air passenger traffic had quickly restored in the second half of 2020, we argue that the changes in the CSFN during COVID-19 were unlikely to be a result of the substitution effect between freighter and passenger aircraft. It was more likely a result of the higher air cargo demand during the pandemic and airlines' realisation of the importance of freighter operations in China.

A model to predict bottlenecks over time in a remanufacturing system under uncertainty

Bottleneck shifting prediction has been widely applied to the remanufacturing system for throughput improvement, and it would directly influence the general presentation of the remanufacturing system. However, predicting dynamic bottlenecks of remanufacturing systems is complicated due to the disturbed environment (e.g. various processing time and uncertain processing routes). This paper built a metamorphosis CNT conjunct with coupled map lattice (CML) algorithm to predict the bottleneck shifting phenomenon in remanufacturing for the first time. The CNT was applied to the articulation of remanufacturing process, while the CML algorithm was devoted to calculating the dynamic indicator of the bottleneck. We took the value-added connecting rod as the research object to illustrate the availability of the proposed method. As validated by Arena simulation, the approach presented in this paper put forward is feasible to make an accurate prediction for shifting bottlenecks in a remanufacturing system.

Comparison of topological, empirical and optimization-based approaches for locating quality detection points in water distribution networks

The positioning of quality detection points as well as the frequency of sampling is a crucial aspect for the implementation of Water Safety Plans (WSPs), which have been proposed worldwide to ensure water quality and to minimize the risk from contamination in water distribution networks (WDNs). In this regard, some international legislations and best practices about quality of drinking water suggest very fine sampling frequencies, but they do not specify where the detection points should be located in a WDN. In this paper, three different approaches, based on empiricism, optimization and topology, respectively, were applied to locate detection quality points in a WDN. The comparison highlighted that empirical approach commonly adopted by water utility practitioners is unsatisfactory. The optimization-based approach, although performing significantly better, is difficult to apply, since it requires a calibrated hydraulic model. The topological approach, based on the use of the betweenness centrality and not requiring any hydraulic information and simulation, proves to be effective, and it can be easily adopted by water utilities to identify the location for quality detection points, due to its simplicity compared with the optimization-based approach.

Vulnerability analysis of water distribution networks to accidental pipe burst

Even the best-maintained water distribution network (WDN) might suffer pipe bursts occasionally, and the utility company must reconstruct the damaged sections of the system. The affected area must be segregated by closing the corresponding isolation valves; as a result, the required amount of drinking water might not be available. This paper explores the behaviour and topology of segments, especially their criticality from the viewpoint of the whole system. A novel, objective, dimensionless, segment-based quantity is proposed to evaluate the vulnerability of both the segments and the whole WDN against a single, incidental pipe break, computed as the product of the probability of failure within the segment and the amount of unserved consumption. 27 comprehensive real-life WDNs have been examined by means of the new metric and with the help of complex network theory, exploiting the concept of the degree distribution and topology-based structural properties (e.g. network diameter, clustering coefficient). It was found that metrics based purely on topology suggest different network behaviour as vulnerability analysis, which also includes the hydraulics. The investigation of the global network vulnerabilities has revealed several critically exposed systems, and the local distributions unveiled new properties of WDNs in the case of a random pipe break.

Complex network theory for the identification and assessment of candidate protein targets

In this work we use complex network theory to provide a statistical model of the connectivity patterns of human proteins and their interaction partners. Our intention is to identify important proteins that may be predisposed to be potential candidates as drug targets for therapeutic interventions. Target proteins usually have more interaction partners than non-target proteins, but there are no hard-and-fast rules for defining the actual number of interactions. We devise a statistical measure for identifying hub proteins, we score our target proteins with gene ontology annotations. The important druggable protein targets are likely to have similar biological functions that can be assessed for their potential therapeutic value. Our system provides a statistical analysis of the local and distant neighborhood protein interactions of the potential targets using complex network measures. This approach builds a more accurate model of drug-to-target activity and therefore the likely impact on treating diseases. We integrate high quality protein interaction data from the HINT database and disease associated proteins from the DrugTarget database. Other sources include biological knowledge from Gene Ontology and drug information from DrugBank. The problem is a very challenging one since the data is highly imbalanced between target proteins and the more numerous nontargets. We use undersampling on the training data and build Random Forest classifier models which are used to identify previously unclassified target proteins. We validate and corroborate these findings from the available literature.

Functional EEG network analysis in schizophrenia: Evidence of larger segregation and deficit of modulation

OBJECTIVE

Higher mental functions depend on global cerebral functional coordination. Our aim was to study fast modulation of functional networks in schizophrenia that has not been previously assessed.

METHODS

Graph-theory was used to analyze the electroencephalographic (EEG) activity during an odd-ball task in 57 schizophrenia patients (18 first episode patients, FEPs) and 59 healthy controls. Clustering coefficient (CLC), characteristic path length (PL) and small-worldness (SW) were computed at baseline ([-300 0] ms prior to stimulus delivery) and response ([150 450] ms post-stimulus) windows. Clinical and cognitive assessments were performed.

RESULTS

CLC, PL and SW showed a significant modulation between baseline and response in controls but not in patients. Patients obtained higher CLC and SW at baseline, lower CLC and higher PL at response, and diminished modulation of CLC and SW as compared to controls. In patients, CLC and SW modulation were inversely associated to cognitive performance in executive tasks and directly associated to working memory. Similar patterns were observed in FEPs. CLC and SW during the baseline were inversely associated to their respective modulation magnitudes.

CONCLUSIONS

Our results are coherent with a hyper-segregated network at baseline (higher CLC) and a decreased modulation of the functional connectivity during cognition in schizophrenia.