Resilience of finite clusters of carbon flux network under localized attack.

The investigation into the resilience of the carbon flux network regarding its capability to sustain the normal flow and transformation of carbon under extreme climatic events, pollutant emissions, biological invasions, and other factors, and the stability of connections between its nodes, has not yet been deeply studied. In this study, we developed carbon flux network models for various regional lands using complex networks, percolation theory, and introducing time delay effects using carbon flux daily data from 2000 to 2019 for three regions: China, the mainland United States, and Europe, to measure the resilience of finite clusters with sizes greater than or equal to s of the carbon flux network under localized attack. The analysis revealed that the carbon flux networks in different regions are characterized by a degree distribution consistent with the Poisson distribution. The carbon flux network demonstrated continuous phase transition behavior under localized attack. Interestingly, numerical simulation revealed a consistent relationship between the carbon flux network and the theoretical Erdos-Rényi network model. Moreover, the carbon flux network becomes more vulnerable as s increases. In addition, we discovered that there is a general scaling relationship of critical exponent δ≈-2 between the fraction of finite clusters and s. Therefore, investigating the resilience of carbon flux networks can enable us to predict and respond to the various risks and challenges, which will help policy designers formulate appropriate response strategies and enhance carbon flux systems' stability and resilience.

Robustness of coupled networks with multiple support from functional components at different scales.

Robustness is an essential component of modern network science. Here, we investigate the robustness of coupled networks where the functionality of a node depends not only on its connectivity, here measured by the size of its connected component in its own network, but also the support provided by at least M links from another network. We here develop a theoretical framework and investigate analytically and numerically the cascading failure process when the system is under attack, deriving expressions for the proportion of functional nodes in the stable state, and the critical threshold when the system collapses. Significantly, our results show an abrupt phase transition and we derive the minimum inner and inter-connectivity density necessary for the system to remain active. We also observe that the system necessitates an increased density of links inside and across networks to prevent collapse, especially when conditions on the coupling between the networks are more stringent. Finally, we discuss the importance of our results in real-world settings and their potential use to aid decision-makers design more resilient infrastructure systems.

Percolation Theories for Quantum Networks.

Quantum networks have experienced rapid advancements in both theoretical and experimental domains over the last decade, making it increasingly important to understand their large-scale features from the viewpoint of statistical physics. This review paper discusses a fundamental question: how can entanglement be effectively and indirectly (e.g., through intermediate nodes) distributed between distant nodes in an imperfect quantum network, where the connections are only partially entangled and subject to quantum noise? We survey recent studies addressing this issue by drawing exact or approximate mappings to percolation theory, a branch of statistical physics centered on network connectivity. Notably, we show that the classical percolation frameworks do not uniquely define the network's indirect connectivity. This realization leads to the emergence of an alternative theory called "concurrence percolation", which uncovers a previously unrecognized quantum advantage that emerges at large scales, suggesting that quantum networks are more resilient than initially assumed within classical percolation contexts, offering refreshing insights into future quantum network design.

Optimization and Benefit Analysis of Grain Trade in Belt and Road Countries.

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.

Network resilience of non-hub nodes failure under memory and non-memory based attacks with limited information.

Previous studies on network robustness mainly concentrated on hub node failures with fully known network structure information. However, hub nodes are often well protected and not accessible to damage or malfunction in a real-world networked system. In addition, one can only gain insight into limited network connectivity knowledge due to large-scale properties and dynamic changes of the network itself. In particular, two different aggression patterns are present in a network attack: memory based attack, in which failed nodes are not attacked again, or non-memory based attack; that is, nodes can be repeatedly attacked. Inspired by these motivations, we propose an attack pattern with and without memory based on randomly choosing n non-hub nodes with known connectivity information. We use a network system with the Poisson and power-law degree distribution to study the network robustness after applying two failure strategies of non-hub nodes. Additionally, the critical threshold 1 - p and the size of the giant component S are determined for a network configuration model with an arbitrary degree distribution. The results indicate that the system undergoes a continuous second-order phase transition subject to the above attack strategies. We find that 1 - p gradually tends to be stable after increasing rapidly with n. Moreover, the failure of non-hub nodes with a higher degree is more destructive to the network system and makes it more vulnerable. Furthermore, from comparing the attack strategies with and without memory, the results highlight that the system shows better robustness under a non-memory based attack relative to memory based attacks for n > 1. Attacks with memory can block the system's connectivity more efficiently, which has potential applications in real-world systems. Our model sheds light on network resilience under memory and non-memory based attacks with limited information attacks and provides valuable insights into designing robust real-world systems.

Phase transition behavior of finite clusters under localized attack.

Most previous studies focused on the giant component to explore the structural robustness of complex networks under malicious attacks. As an important failure mode, localized attacks (LA) can excellently describe the local failure diffusion mechanism of many real scenarios. However, the phase transition behavior of finite clusters, as important network components, has not been clearly understood yet under LA. Here, we develop a percolation framework to theoretically and simulatively study the phase transition behavior of functional nodes belonging to the finite clusters of size greater than or equal to s(s=2,3,…) under LA in this paper. The results reveal that random network exhibits second-order phase transition behavior, the critical threshold pc increases significantly with increasing s, and the network becomes vulnerable. In particular, we find a new general scaling relationship with the critical exponent δ=-2 between the fraction of finite clusters and s. Furthermore, we apply the theoretical framework to some real networks and predict the phase transition behavior of finite clusters in real networks after they face LA. The framework and results presented in this paper are helpful to promote the design of more critical infrastructures and inspire new insights into studying phase transition behaviors for finite clusters in the network.

Three-state majority-vote model on small-world networks.

In this work, we study the opinion dynamics of the three-state majority-vote model on small-world networks of social interactions. In the majority-vote dynamics, an individual adopts the opinion of the majority of its neighbors with probability 1-q, and a different opinion with chance q, where q stands for the noise parameter. The noise q acts as a social temperature, inducing dissent among individual opinions. With probability p, we rewire the connections of the two-dimensional square lattice network, allowing long-range interactions in the society, thus yielding the small-world property present in many different real-world systems. We investigate the degree distribution, average clustering coefficient and average shortest path length to characterize the topology of the rewired networks of social interactions. By employing Monte Carlo simulations, we investigate the second-order phase transition of the three-state majority-vote dynamics, and obtain the critical noise [Formula: see text], as well as the standard critical exponents [Formula: see text], [Formula: see text], and [Formula: see text] for several values of the rewiring probability p. We conclude that the rewiring of the lattice enhances the social order in the system and drives the model to different universality classes from that of the three-state majority-vote model in two-dimensional square lattices.

Efficient network immunization under limited knowledge.

Targeted immunization of centralized nodes in large-scale networks has attracted significant attention. However, in real-world scenarios, knowledge and observations of the network may be limited, thereby precluding a full assessment of the optimal nodes to immunize (or quarantine) in order to avoid epidemic spreading such as that of the current coronavirus disease (COVID-19) epidemic. Here, we study a novel immunization strategy where only n nodes are observed at a time and the most central among these n nodes is immunized. This process can globally immunize a network. We find that even for small n (≈10) there is significant improvement in the immunization (quarantine), which is very close to the levels of immunization with full knowledge. We develop an analytical framework for our method and determine the critical percolation threshold p c and the size of the giant component P ∞ for networks with arbitrary degree distributions P(k). In the limit of n → ∞ we recover prior work on targeted immunization, whereas for n = 1 we recover the known case of random immunization. Between these two extremes, we observe that, as n increases, p c increases quickly towards its optimal value under targeted immunization with complete information. In particular, we find a new general scaling relationship between |p c (∞) - p c (n)| and n as |p c (∞) - p c (n)| ∼ n -1exp(-αn). For scale-free (SF) networks, where P(k) ∼ k -γ, 2 < γ < 3, we find that p c has a transition from zero to nonzero when n increases from n = 1 to O(log N) (where N is the size of the network). Thus, for SF networks, having knowledge of ≈log N nodes and immunizing the most optimal among them can dramatically reduce epidemic spreading. We also demonstrate our limited knowledge immunization strategy on several real-world networks and confirm that in these real networks, p c increases significantly even for small n.

Optimal resilience of modular interacting networks.

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 [Formula: see text] 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 [Formula: see text] 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.

Percolation on coupled networks with multiple effective dependency links.

The ubiquitous coupled relationship between network systems has become an essential paradigm to depict complex systems. A remarkable property in the coupled complex systems is that a functional node should have multiple external support associations in addition to maintaining the connectivity of the local network. In this paper, we develop a theoretical framework to study the structural robustness of the coupled network with multiple useful dependency links. It is defined that a functional node has the broadest connectivity within the internal network and requires at least M support link of the other network to function. In this model, we present exact analytical expressions for the process of cascading failures, the fraction of functional nodes in the stable state, and provide a calculation method of the critical threshold. The results indicate that the system undergoes an abrupt phase transition behavior after initial failure. Moreover, the minimum inner and inter-connectivity density to maintain system survival is graphically presented at different multiple effective dependency links. Furthermore, we find that the system needs more internal connection densities to avoid collapse when it requires more effective support links. These findings allow us to reveal the details of a more realistic coupled complex system and develop efficient approaches for designing resilient infrastructure.

Robustness on interdependent networks with a multiple-to-multiple dependent relationship.

Interdependent networks as an important structure of the real system not only include one-to-one dependency relationship but also include more realistic undirected multiple interdependent relationship. The study on interdependent networks plays an important role in designing more resilient real systems. Here, we mainly focus on the case of interdependent networks with a multiple-to-multiple correspondence of interdependent nodes by generalizing feedback and nonfeedback conditions. We develop a new mathematical framework and study numerically and analytically the percolation of interdependent networks with partial multiple-to-multiple dependency links by using percolation theory. By analyzing the process of cascading failure, the size of the giant component and the critical threshold are analytically obtained and testified by simulation results for couple Erdös-Re˙nyi and scale-free networks. The results imply that the system shows a discontinuous phase transition for different coupling strengths. We find that the system becomes more resilient and easy to defend by increasing the coupling strength and the connectivity density. Our model has the potential to shed light on designing more resilient real-world dependent systems.

Topological properties of the limited penetrable horizontal visibility graph family.

The limited penetrable horizontal visibility graph algorithm was recently introduced to map time series in complex networks. In this work, we extend this algorithm to create a directed-limited penetrable horizontal visibility graph and an image-limited penetrable horizontal visibility graph. We define two algorithms and provide theoretical results on the topological properties of these graphs associated with different types of real-value series. We perform several numerical simulations to check the accuracy of our theoretical results. Finally, we present an application of the directed-limited penetrable horizontal visibility graph to measure real-value time series irreversibility and an application of the image-limited penetrable horizontal visibility graph that discriminates noise from chaos. We also propose a method to measure the systematic risk using the image-limited penetrable horizontal visibility graph, and the empirical results show the effectiveness of our proposed algorithms.

Resilience of networks with community structure behaves as if under an external field.

Although detecting and characterizing community structure is key in the study of networked systems, we still do not understand how community structure affects systemic resilience and stability. We use percolation theory to develop a framework for studying the resilience of networks with a community structure. We find both analytically and numerically that interlinks (the connections among communities) affect the percolation phase transition in a way similar to an external field in a ferromagnetic- paramagnetic spin system. We also study universality class by defining the analogous critical exponents δ and γ, and we find that their values in various models and in real-world coauthor networks follow the fundamental scaling relations found in physical phase transitions. The methodology and results presented here facilitate the study of network resilience and also provide a way to understand phase transitions under external fields.

Exact results of the limited penetrable horizontal visibility graph associated to random time series and its application.

The limited penetrable horizontal visibility algorithm is an analysis tool that maps time series into complex networks and is a further development of the horizontal visibility algorithm. This paper presents exact results on the topological properties of the limited penetrable horizontal visibility graph associated with independent and identically distributed (i:i:d:) random series. We show that the i.i.d: random series maps on a limited penetrable horizontal visibility graph with exponential degree distribution, independent of the probability distribution from which the series was generated. We deduce the exact expressions of mean degree and clustering coefficient, demonstrate the long distance visibility property of the graph and perform numerical simulations to test the accuracy of our theoretical results. We then use the algorithm in several deterministic chaotic series, such as the logistic map, H´enon map, Lorenz system, energy price chaotic system and the real crude oil price. Our results show that the limited penetrable horizontal visibility algorithm is efficient to discriminate chaos from uncorrelated randomness and is able to measure the global evolution characteristics of the real time series.

Spatiotemporal Dynamics and Fitness Analysis of Global Oil Market: Based on Complex Network.

We study the overall topological structure properties of global oil trade network, such as degree, strength, cumulative distribution, information entropy and weight clustering. The structural evolution of the network is investigated as well. We find the global oil import and export networks do not show typical scale-free distribution, but display disassortative property. Furthermore, based on the monthly data of oil import values during 2005.01-2014.12, by applying random matrix theory, we investigate the complex spatiotemporal dynamic from the country level and fitness evolution of the global oil market from a demand-side analysis. Abundant information about global oil market can be obtained from deviating eigenvalues. The result shows that the oil market has experienced five different periods, which is consistent with the evolution of country clusters. Moreover, we find the changing trend of fitness function agrees with that of gross domestic product (GDP), and suggest that the fitness evolution of oil market can be predicted by forecasting GDP values. To conclude, some suggestions are provided according to the results.

Percolation on interacting networks with feedback-dependency links.

When real networks are considered, coupled networks with connectivity and feedback-dependency links are not rare but more general. Here, we develop a mathematical framework and study numerically and analytically the percolation of interacting networks with feedback-dependency links. For the case that all degree distributions of intra- and inter- connectivity links are Poissonian, we find that for a low density of inter-connectivity links, the system undergoes from second order to first order through hybrid phase transition as coupling strength increases. It implies that the average degree k of inter-connectivity links has a little influence on robustness of the system with a weak coupling strength, which corresponds to the second order transition, but for a strong coupling strength corresponds to the first order transition. That is to say, the system becomes robust as k increases. However, as the average degree k of each network increases, the system becomes robust for any coupling strength. In addition, we find that one can take less cost to design robust system as coupling strength decreases by analyzing minimum average degree kmin of maintaining system stability. Moreover, for high density of inter-connectivity links, we find that the hybrid phase transition region disappears, the first order region becomes larger and second order region becomes smaller. For the case of two coupled scale-free networks, the system also undergoes from second order to first order through hybrid transition as the coupling strength increases. We find that for a weak coupling strength, which corresponds to the second order transitions, feedback dependency links have no effect on robustness of system relative to no-feedback condition, but for strong coupling strength which corresponds to first order or hybrid phase transition, the system is more vulnerable under feedback condition comparing with no-feedback condition. Thus, for designing resilient system, designers should try to avoid the feedback dependency links, because the existence of feedback-dependency links makes the system extremely vulnerable and difficult to defend.

Robustness of network of networks under targeted attack.

The robustness of a network of networks (NON) under random attack has been studied recently [Gao et al., Phys. Rev. Lett. 107, 195701 (2011)]. Understanding how robust a NON is to targeted attacks is a major challenge when designing resilient infrastructures. We address here the question how the robustness of a NON is affected by targeted attack on high- or low-degree nodes. We introduce a targeted attack probability function that is dependent upon node degree and study the robustness of two types of NON under targeted attack: (i) a tree of n fully interdependent Erdos-Rényi or scale-free networks and (ii) a starlike network of n partially interdependent Erdos-Rényi networks. For any tree of n fully interdependent Erdos-Rényi networks and scale-free networks under targeted attack, we find that the network becomes significantly more vulnerable when nodes of higher degree have higher probability to fail. When the probability that a node will fail is proportional to its degree, for a NON composed of Erdos-Rényi networks we find analytical solutions for the mutual giant component P(∞) as a function of p, where 1-p is the initial fraction of failed nodes in each network. We also find analytical solutions for the critical fraction p(c), which causes the fragmentation of the n interdependent networks, and for the minimum average degree k[over ¯](min) below which the NON will collapse even if only a single node fails. For a starlike NON of n partially interdependent Erdos-Rényi networks under targeted attack, we find the critical coupling strength q(c) for different n. When q>q(c), the attacked system undergoes an abrupt first order type transition. When q≤q(c), the system displays a smooth second order percolation transition. We also evaluate how the central network becomes more vulnerable as the number of networks with the same coupling strength q increases. The limit of q=0 represents no dependency, and the results are consistent with the classical percolation theory of a single network under targeted attack.

Percolation of partially interdependent networks under targeted attack.

We study a system composed of two partially interdependent networks; when nodes in one network fail, they cause dependent nodes in the other network to also fail. In this paper, the percolation of partially interdependent networks under targeted attack is analyzed. We apply a general technique that maps a targeted-attack problem in interdependent networks to a random-attack problem in a transformed pair of interdependent networks. We illustrate our analytical solutions for two examples: (i) the probability for each node to fail is proportional to its degree, and (ii) each node has the same probability to fail in the initial time. We find the following: (i) For any targeted-attack problem, for the case of weak coupling, the system shows a second order phase transition, and for the strong coupling, the system shows a first order phase transition. (ii) For any coupling strength, when the high degree nodes have higher probability to fail, the system becomes more vulnerable. (iii) There exists a critical coupling strength, and when the coupling strength is greater than the critical coupling strength, the system shows a first order transition; otherwise, the system shows a second order transition.