Coupling between networks is widely prevalent in real systems and has dramatic effects on their resilience and functional properties. However, current theoretical models tend to assume homogeneous coupling where all the various subcomponents interact with one another, whereas real-world systems tend to have various different coupling patterns. We develop two frameworks to explore the resilience of such modular networks, including specific deterministic coupling patterns and coupling patterns where specific subnetworks are connected randomly. We find both analytically and numerically that the location of the percolation phase transition varies nonmonotonically with the fraction of interconnected nodes when the total number of interconnecting links remains fixed. Furthermore, there exists an optimal fraction $r^{*}$ of interconnected nodes where the system becomes optimally resilient and is able to withstand more damage. Our results suggest that, although the exact location of the optimal $r^{*}$ varies based on the coupling patterns, for all coupling patterns, there exists such an optimal point. Our findings provide a deeper understanding of network resilience and show how networks can be optimized based on their specific coupling patterns. Significance In real-world scenarios, many subnetworks interact with other subnetworks to form a modular interacting network system. In previous models, it was assumed that all subnetworks may be linked pairwise; however, in real systems, only some pairs of subnetworks are connected. Understanding how to develop theoretical frameworks and study system resilience of general coupling patterns of the subnetworks is of significant importance for designing and optimizing economic, social, and infrastructure networks. Here we present two frameworks for such systems having specific coupling patterns to investigate their resilience. Specifically, we find an optimal level of interaction between subnetworks, which maximizes the system’s resilience to failures. Our findings highlight the need to consider real-world coupling patterns and possible optimizations for designing resilient systems.

With the contradiction between supply and demand being intensified, the unreasonable crude oil trade structure has become increasingly prominent. Therefore, optimizing the supply and demand structure of the global crude oil trade is an urgent problem, which has become one of the crucial factors affecting the energy strategy and economic development of each country. This paper builds an optimization model that aims at minimizing crude oil trade costs based on the complex network theory. Meanwhile, an optimal solution generation approach is proposed and developed in this paper. Compared with the preoptimized trade network, the proposed model can effectively reduce the trade cost. By topological analysis of the trade network and its optimal configuration, we obtain that both the preoptimized and optimized crude oil trade networks follow power‐law distribution. By using the minimum spanning tree, we find that the major crude oil net exporting countries have the most significant influence and are at the core in the optimized trade structure. This work focuses on the sustainable development of the global crude oil trade and provides a fresh perspective for the optimal crude oil trade system. Moreover, the methodology and model may be applied in the investigation of optimization for other energy system structures.

Natural gas scarcity poses a significant risk to the global economy. The risk of production loss due to natural gas scarcity can be transferred to downstream economies through globalized supply chains. Therefore, it is important to quantify and analyze how natural gas scarcity in some regions affects the Belt and Road (B&R) economies. The embodied natural gas scarcity risks (EGSRs) of B&R economies are assessed and the EGSR transmission network is constructed. The built network shows a small-world nature. This illustrates that any interruption in key countries will quickly spread to neighboring countries, potentially affecting the global economy. The top countries, including Turkey, China, Ukraine, and India are identified in EGSR exports, which also have relatively high values of closeness centrality. The findings illustrate that the shortage of natural gas supply in these countries may have a significant impact on downstream countries or sectors and the resulting economic losses spread rapidly. These countries are critical to the resilience of the B&R economies to natural gas scarcity. The top nations, including Turkmenistan, Macedonia, and Georgia are also identified in EGSR imports, highlighting their vulnerability to natural gas scarcity. Further, the community analysis of the network provides a fresh perspective for formulating fair and reasonable allocation policies of natural gas resources and minimizing the large-scale spread of economic losses caused by natural gas scarcity.

Grain trade in Belt and Road (B&R) countries shows a mismatch between the volume and direction of grain flows and actual demand. With economic and industrial development, the water crisis has intensified, which poses a great challenge to the security of world grain supply and demand. There are few studies on the reconstruction of grain trade relations from the perspective of grain economic value. In this paper, a linear optimization model considering opportunity cost is proposed to fill the gap, and it is compared and analyzed with the optimization model considering only transportation cost. The grain supply and demand structures in both optimization results show characteristics of geographical proximity and long-tail distribution. Furthermore, the economic and water resource benefits resulting from the two optimal configurations are compared and analyzed. It is found that the economic benefits generated by grain trade in B&R countries with the consideration of opportunity cost not only cover transportation costs but also generate an economic value of about 130 trillion US dollars. Therefore, considering opportunity cost in grain trade is of great significance for strengthening cooperation and promoting the economic development of countries under the B&R framework. In terms of resource benefits, the grain trade with consideration of opportunity cost saves nearly 28 billion cubic meters of water, or about 5% of the total virtual water flow. However, about 72 billion cubic meters of water is lost for the grain trade with consideration of transportation cost. This study will help to formulate and adjust policies related to the “Belt and Road Initiative” (B&R Initiative), so as to maximize the economic benefits while optimizing the structure of grain trade and alleviating water scarcity pressures.

Water consumption has been one of the most important topics in the field of environment and economy. Even though the driving factors of water consumption have been well studied, it is still a daunting task to reveal the influence of the status of provinces in the entire supply chain. By combining the multi-regional input-output (MRIO) model and complex network theory, an inter-provincial virtual water transfer (V WT) network was constructed to analyze the overall structural characteristics of the network model and identify the structural roles of each province. The constructed inter-provincial V WT network exhibited the characteristics of a small-world network, that is, virtual water can be easily transferred from one province to another. Moreover, network analysis revealed that provinces with different positions in the V WT network played discrepant structural roles. Panel regression analysis was further used to quantify the impact of provincial structural roles on their water consumption. The results showed that water consumption in China largely depended on some structural role characteristics in the V WT network. Out-degree and out-strength characterizing the ability of direct exporting virtual water exerted significant positive influences, while in-closeness featuring the indirect virtual water importing rate had a significant negative effect on water usage. This indicated that adjusting the uneven provincial consumption structure, the direct production demand of downstream provinces and the indirect production activities in the supply chain would help reduce water consumption. Therefore, to come true the goal of water conservation in China, it would be necessary to improve the trade structure between direct and indirect exporters and importers in the entire supply chain.