Recent Progress on the Resilience of Complex Networks

Zou W., Senthilkumar D.V., Nagao R., Kiss I.Z., Tang Y., Koseska A., Duan J., Kurths J.

Oscillatory behaviour is essential for proper functioning of various physical and biological processes. However, diffusive coupling is capable of suppressing intrinsic oscillations due to the manifestation of the phenomena of amplitude and oscillation deaths. Here we present a scheme to revoke these quenching states in diffusively coupled dynamical networks, and demonstrate the approach in experiments with an oscillatory chemical reaction. By introducing a simple feedback factor in the diffusive coupling, we show that the stable (in)homogeneous steady states can be effectively destabilized to restore dynamic behaviours of coupled systems. Even a feeble deviation from the normal diffusive coupling drastically shrinks the death regions in the parameter space. The generality of our method is corroborated in diverse non-linear systems of diffusively coupled paradigmatic models with various death scenarios. Our study provides a general framework to strengthen the robustness of dynamic activity in diffusively coupled dynamical networks. Oscillatory behaviour is essential for proper functioning of several processes, yet quenching phenomena can lead to steady states with suppressed oscillations. Here the authors present a scheme to revoke these states in diffusively coupled networks, and demonstrate their approach on a chemical oscillator.

Berezin Y., Bashan A., Danziger M.M., Li D., Havlin S.

Many real world complex systems such as critical infrastructure networks are embedded in space and their components may depend on one another to function. They are also susceptible to geographically localized damage caused by malicious attacks or natural disasters. Here, we study a general model of spatially embedded networks with dependencies under localized attacks. We develop a theoretical and numerical approach to describe and predict the effects of localized attacks on spatially embedded systems with dependencies. Surprisingly, we find that a localized attack can cause substantially more damage than an equivalent random attack. Furthermore, we find that for a broad range of parameters, systems which appear stable are in fact metastable. Though robust to random failures—even of finite fraction—if subjected to a localized attack larger than a critical size which is independent of the system size (i.e., a zero fraction), a cascading failure emerges which leads to complete system collapse. Our results demonstrate the potential high risk of localized attacks on spatially embedded network systems with dependencies and may be useful for designing more resilient systems.

Shao S., Huang X., Stanley H.E., Havlin S.

The robustness of complex networks against node failure and malicious attack has been of interest for decades, while most of the research has focused on random attack or hub-targeted attack. In many real-world scenarios, however, attacks are neither random nor hub-targeted, but localized, where a group of neighboring nodes in a network are attacked and fail. In this paper we develop a percolation framework to analytically and numerically study the robustness of complex networks against such localized attack. In particular, we investigate this robustness in Erd\H{o}s-R\'{e}nyi networks, random-regular networks, and scale-free networks. Our results provide insight into how to better protect networks, enhance cybersecurity, and facilitate the design of more robust infrastructures.

Li D., Fu B., Wang Y., Lu G., Berezin Y., Stanley H.E., Havlin S.

Significance The transition between free flow and congestions in traffic can be observed in our daily life. Although this traffic phenomenon is well studied in highways, traffic in a network scale (representing a city) is far from being understood. A fundamental unsolved question is how the global flow in a city is being integrated from local flows. Here, we identify a fundamental mechanism of traffic organization in a network scale as a percolation process, and we show how global traffic breaks down when identified bottlenecks are congested. These bottlenecks evolve with time according to traffic dynamics and are different from structural bottleneck links found by traditional network analysis. Improvement of traffic on these bottlenecks can significantly improve the global traffic.

Shekhtman L.M., Berezin Y., Danziger M.M., Havlin S.

We present analytic and numeric results for percolation in a network formed of interdependent spatially embedded networks. We show results for a treelike and a random regular network of networks each with $(i)$ unconstrained interdependent links and $(ii)$ interdependent links restricted to a maximum length, $r$. Analytic results are given for each network of networks with unconstrained dependency links and compared with simulations. For the case of two spatially embedded networks it was found that only for $r>r_c\approx8$ does the system undergo a first order phase transition. We find that for treelike networks of networks $r_c$ significantly decreases as $n$ increases and rapidly reaches its limiting value, $r=1$. For cases where the dependencies form loops, such as in random regular networks, we show analytically and confirm through simulations, that there is a certain fraction of dependent nodes, $q_{max}$, above which the entire network structure collapses even if a single node is removed. This $q_{max}$ decreases quickly with $m$, the degree of the random regular network of networks. Our results show the extreme sensitivity of coupled spatial networks and emphasize the susceptibility of these networks to sudden collapse. The theory derived here can be used to find the robustness of any network of networks where the profile of percolation of a single network is known.

Gao J., Li D., Havlin S.

Abstract Network science has attracted much attention in recent years due to its interdisciplinary applications. We witnessed the revolution of network science in 1998 and 1999 started with small-world and scale-free networks having now thousands of high-profile publications, and it seems that since 2010 studies of ‘network of networks’ (NON), sometimes called multilayer networks or multiplex, have attracted more and more attention. The analytic framework for NON yields a novel percolation law for n interdependent networks that shows that percolation theory of single networks studied extensively in physics and mathematics in the last 50 years is a specific limit of the rich and very different general case of n coupled networks. Since then, properties and dynamics of interdependent and interconnected networks have been studied extensively, and scientists are finding many interesting results and discovering many surprising phenomena. Because most natural and engineered systems are composed of multiple subsystems and layers of connectivity, it is important to consider these features in order to improve our understanding of such complex systems. Now the study of NON has become one of the important directions in network science. In this paper, we review recent studies on the new emerging area—NON. Due to the fast growth of this field, there are many definitions of different types of NON, such as interdependent networks, interconnected networks, multilayered networks, multiplex networks and many others. There exist many datasets that can be represented as NON, such as network of different transportation networks including flight networks, railway networks and road networks, network of ecological networks including species interacting networks and food webs, network of biological networks including gene regulation network, metabolic network and protein–protein interacting network, network of social networks and so on. Among them, many interdependent networks including critical infrastructures are embedded in space, introducing spatial constraints. Thus, we also review the progress on study of spatially embedded networks. As a result of spatial constraints, such interdependent networks exhibit extreme vulnerabilities compared with their non-embedded counterparts. Such studies help us to understand, realize and hopefully mitigate the increasing risk in NON.

Kivela M., Arenas A., Barthelemy M., Gleeson J.P., Moreno Y., Porter M.A.

In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is important to take such “multilayer” features into account to try to improve our understanding of complex systems. Consequently, it is necessary to generalize “traditional” network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion. The origins of such efforts date back several decades and arose in multiple disciplines, and now the study of multilayer networks has become one of the most important directions in network science. In this paper, we discuss the history of multilayer networks (and related concepts) and review the exploding body of work on such networks. To unify the disparate terminology in the large body of recent work, we discuss a general framework for multilayer networks, construct a dictionary ∗Corresponding author: porterm@maths.ox.ac.uk 1 ar X iv :1 30 9. 72 33 v4 [ ph ys ic s. so cph ] 3 M ar 2 01 4 of terminology to relate the numerous existing concepts to each other, and provide a thorough discussion that compares, contrasts, and translates between related notions such as multilayer networks, multiplex networks, interdependent networks, networks of networks, and many others. We also survey and discuss existing data sets that can be represented as multilayer networks. We review attempts to generalize single-layer-network diagnostics to multilayer networks. We also discuss the rapidly expanding research on multilayer-network models and notions like community structure, connected components, tensor decompositions, and various types of dynamical processes on multilayer networks. We conclude with a summary and an outlook.

Zhou D., Bashan A., Cohen R., Berezin Y., Shnerb N., Havlin S.

In a system of interdependent networks, an initial failure of nodes invokes a cascade of iterative failures that may lead to a total collapse of the whole system in the form of an abrupt first-order transition. When the fraction of initial failed nodes 1-p reaches criticality p = p(c), the abrupt collapse occurs by spontaneous cascading failures. At this stage, the giant component decreases slowly in a plateau form and the number of iterations in the cascade τ diverges. The origin of this plateau and its increasing with the size of the system have been unclear. Here we find that, simultaneously with the abrupt first-order transition, a spontaneous second-order percolation occurs during the cascade of iterative failures. This sheds light on the origin of the plateau and how its length scales with the size of the system. Understanding the critical nature of the dynamical process of cascading failures may be useful for designing strategies for preventing and mitigating catastrophic collapses.

Daqing L., Yinan J., Rui K., Havlin S.

Cascading failures have become major threats to network robustness due to their potential catastrophic consequences, where local perturbations can induce global propagation of failures. Unlike failures spreading via direct contacts due to structural interdependencies, overload failures usually propagate through collective interactions among system components. Despite the critical need in developing protection or mitigation strategies in networks such as power grids and transportation, the propagation behavior of cascading failures is essentially unknown. Here we find by analyzing our collected data that jams in city traffic and faults in power grid are spatially long-range correlated with correlations decaying slowly with distance. Moreover, we find in the daily traffic, that the correlation length increases dramatically and reaches maximum, when morning or evening rush hour is approaching. Our study can impact all efforts towards improving actively system resilience ranging from evaluation of design schemes, development of protection strategies to implementation of mitigation programs.

Li D., Qin P., Wang H., Liu C., Jiang Y.

Gao J., Buldyrev S.V., Stanley H.E., Xu X., Havlin S.

Percolation theory is an approach to study vulnerability of a system. We develop analytical framework and analyze percolation properties of a network composed of interdependent networks (NetONet). Typically, percolation of a single network shows that the damage in the network due to a failure is a continuous function of the fraction of failed nodes. In sharp contrast, in NetONet, due to the cascading failures, the percolation transition may be discontinuous and even a single node failure may lead to abrupt collapse of the system. We demonstrate our general framework for a NetONet composed of $n$ classic Erd\H{o}s-R\'{e}nyi (ER) networks, where each network depends on the same number $m$ of other networks, i.e., a random regular network of interdependent ER networks. In contrast to a \emph{treelike} NetONet in which the size of the largest connected cluster (mutual component) depends on $n$, the loops in the RR NetONet cause the largest connected cluster to depend only on $m$. We also analyzed the extremely vulnerable feedback condition of coupling. In the case of ER networks, the NetONet only exhibits two phases, a second order phase transition and collapse, and there is no first phase transition regime unlike the no feedback condition. In the case of NetONet composed of RR networks, there exists a first order phase transition when $q$ is large and second order phase transition when $q$ is small. Our results can help in designing robust interdependent systems.

Majdandzic A., Podobnik B., Buldyrev S.V., Kenett D.Y., Havlin S., Eugene Stanley H.

Networks that fail can sometimes recover spontaneously—think of traffic jams suddenly easing or people waking from a coma. A model for such recoveries reveals spontaneous ‘phase flipping’ between high-activity and low-activity modes, in analogy with first-order phase transitions near a critical point. Much research has been carried out to explore the structural properties1,2,3,4,5,6,7,8,9,10 and vulnerability11,12,13,14,15,16,17,18,19 of complex networks. Of particular interest are abrupt dynamic events that cause networks to irreversibly fail13,14,15,16,17. However, in many real-world phenomena, such as brain seizures in neuroscience or sudden market crashes in finance, after an inactive period of time a significant part of the damaged network is capable of spontaneously becoming active again. The process often occurs repeatedly. To model this marked network recovery, we examine the effect of local node recoveries and stochastic contiguous spreading, and find that they can lead to the spontaneous emergence of macroscopic ‘phase-flipping’ phenomena. As the network is of finite size and is stochastic, the fraction of active nodes z switches back and forth between the two network collective modes characterized by high network activity and low network activity. Furthermore, the system exhibits a strong hysteresis behaviour analogous to phase transitions near a critical point. We present real-world network data exhibiting phase switching behaviour in accord with the predictions of the model.

Cellai D., López E., Zhou J., Gleeson J.P., Bianconi G.

From transportation networks to complex infrastructures, and to social and communication networks, a large variety of systems can be described in terms of multiplexes formed by a set of nodes interacting through different networks (layers). Multiplexes may display an increased fragility with respect to the single layers that constitute them. However, so far the overlap of the links in different layers has been mostly neglected, despite the fact that it is an ubiquitous phenomenon in most multiplexes. Here we show that the overlap among layers can improve the robustness of interdependent multiplex systems and change the critical behavior of the percolation phase transition in a complex way.

Hu Y., Zhou D., Zhang R., Han Z., Rozenblat C., Havlin S.

Real data show that interdependent networks usually involve inter-similarity. Intersimilarity means that a pair of interdependent nodes have neighbors in both networks that are also interdependent (Parshani et al \cite{PAR10B}). For example, the coupled world wide port network and the global airport network are intersimilar since many pairs of linked nodes (neighboring cities), by direct flights and direct shipping lines exist in both networks. Nodes in both networks in the same city are regarded as interdependent. If two neighboring nodes in one network depend on neighboring nodes in the another we call these links common links. The fraction of common links in the system is a measure of intersimilarity. Previous simulation results suggest that intersimilarity has considerable effect on reducing the cascading failures, however, a theoretical understanding on this effect on the cascading process is currently missing. Here, we map the cascading process with inter-similarity to a percolation of networks composed of components of common links and non common links. This transforms the percolation of inter-similar system to a regular percolation on a series of subnetworks, which can be solved analytically. We apply our analysis to the case where the network of common links is an Erd\H{o}s-R\'{e}nyi (ER) network with the average degree $K$, and the two networks of non-common links are also ER networks. We show for a fully coupled pair of ER networks, that for any $K\geq0$, although the cascade is reduced with increasing $K$, the phase transition is still discontinuous. Our analysis can be generalized to any kind of interdependent random networks system.

Radicchi F., Arenas A.

Real-world networks are rarely isolated. A model of an interdependent network of networks shows that an abrupt phase transition occurs when interconnections between independent networks are added. This study also suggests ways to minimize the danger of abrupt structural changes to real networks. Our world is linked by a complex mesh of networks through which information, people and goods flow. These networks are interdependent on each other, and present structural and dynamical features1,2,3,4,5,6 different from those observed in isolated networks7,8,9. Although examples of such dissimilar properties are becoming more abundant—such as in diffusion, robustness and competition—it is not yet clear where these differences are rooted. Here we show that the process of building independent networks into an interconnected network of networks undergoes a structurally sharp transition as the interconnections are formed. Depending on the relative importance of inter- and intra- layer connections, we find that the entire interdependent system can be tuned between two regimes: in one regime, the various layers are structurally decoupled and they act as independent entities; in the other regime, network layers are indistinguishable and the whole system behaves as a single-level network. We analytically show that the transition between the two regimes is discontinuous even for finite-size networks. Thus, any real-world interconnected system is potentially at risk of abrupt changes in its structure, which may manifest new dynamical properties.

Demmer T., Lichte D., Patriarca R., Wolf K.

Yan P., Zhang F., Zhang F., Geng L.

Urban infrastructure, the lifeline of modern society, consists of inherently multidimensional and interdependent systems that extend beyond various engineered facilities, utilities, and networks. The increasing frequency of extreme events, like floods, typhoons, power outages, and technical failures, has heightened the vulnerability of these infrastructures to cascading disasters. Over the past decade, significant attention has been devoted to understanding urban infrastructure cascading disasters. However, most of them have been limited by one-sided and one-dimensional analyses. A more systematic and scientific methodology is needed to comprehensively profile existing research on urban infrastructure cascading disasters to address this gap. This paper uses scientometric methods to investigate the state-of-the-art research in this area over the past decade. A total of 165 publications from 2014 to 2023 were retrieved from the Web of Science database for in-depth analysis. It has revealed a shift in research focus from single infrastructures to complex, interconnected systems with multidimensional dependencies. In addition, the study of disaster-causing factors has evolved from internal infrastructure failures to a focus on cascading disasters caused by extreme events, highlighting a trend of multi-factor coupling. Furthermore, predicting and modeling cascading disasters, improving infrastructure resilience, and information sharing for collaborative emergency responses have emerged as key strategies in responding to disasters. Overall, the insights gained from this study enhance our understanding of the evolution and current challenges in urban infrastructure cascading disasters. Additionally, this study offers valuable perspectives and directions for policymakers addressing extreme events in this critical area.

Huang X., Wang Z., Pang Y., Tian W.

With the intensification of global resource competition, the issue of timber supply has escalated from an economic concern to a significant strategic challenge. This study focuses on the evolution of disruption resilience in the global trade network for wood forest products, aiming to reveal the patterns of resilience dynamics under disruption risks by simulating underload cascading failure phenomena. The study provides theoretical support for enhancing the security and stability of the global wood forest product supply chain. Utilizing global trade data from the UN Comtrade Database 2023, a directed weighted complex network model was constructed, spanning upstream, midstream, and downstream sectors, with trade intensity distances serving as edge weights. By developing an underload cascading failure model, the evolution of disruption resilience was simulated under various disruption scenarios from 2002 to 2023, and the long-term impacts of critical node failures on network performance were analyzed. The results demonstrate significant spatiotemporal heterogeneity in the disruption resilience of the global wood forest product trade network. The upstream network exhibits improved resilience in total node strength but reduced global efficiency. The midstream network shows marked volatility in resilience due to external shocks, such as the global financial crisis, while the downstream network remains relatively stable. Simulations reveal that failures in core nodes (e.g., China, the United States, and Germany) disproportionately degrade global efficiency and node strength, with node centrality metrics positively correlated with network performance loss. This study elucidates the evolutionary mechanisms of disruption resilience in the wood forest product trade network under risk propagation, offering actionable insights for optimizing network robustness and supply chain stability. It is recommended that policymakers promote green supply chain initiatives, accelerate afforestation projects, and enhance domestic timber self-sufficiency to reduce reliance on imported timber, thereby strengthening node resilience and fostering sustainable forest resource utilization for economic and environmental benefits.

Seshadri A.K., Gambhir A., Debnath R.

Abstract Non-technical summary Accelerating global systemic risks impel as well as threaten low-carbon energy transitions. Polycrises can undermine low-carbon transitions, and the breakdown of low-carbon energy transitions has the potential to intensify polycrises. Identifying the systemic risks facing low-carbon transitions is critical, as is studying what systemic risks could be exacerbated by energy transitions. Given the urgency and scale of the required technological and institutional changes, integrated and interdisciplinary approaches are essential to determine how low-carbon transitions can mitigate, rather than amplify polycrisis. If done deliberately and through deliberation, low-carbon transitions could spearhead the integrative tools, methods, and strategies required to address the broader polycrisis. Technical summary The urgent need to address accelerating global systemic risks impels low-carbon energy transitions, but these same risks also pose a threat. This briefing discusses factors influencing the stability and resilience of low-carbon energy transitions over extended time-frames, necessitating a multidisciplinary approach. The collapse of these transitions could exacerbate the polycrisis, making it crucial to identify and understand the systemic risks low-carbon transitions face. Key questions addressed include: What are the systemic risks confronting low-carbon transitions? Given the unprecedented urgency and scale of required technological and institutional changes, how can low-carbon transitions mitigate, rather than amplify, global systemic risks? The article describes the role of well-designed climate policies in fostering positive outcomes, achieving political consensus, integrating fiscal and social policies, and managing new risks such as those posed by climate engineering. It emphasizes the importance of long-term strategic planning, interdisciplinary research, and inclusive decision-making. Ultimately, successful low-carbon transitions can provide tools and methods to address broader global challenges, ensuring a sustainable and equitable future amidst a backdrop of complex global interdependencies. Social media summary Low-carbon energy transitions must be approached so as to lower the risks of global polycrisis across systems.

Seif A., Radmanesh H.

Karimi S., Amiri M.J., Yavari A.R.

Abstract Habitat loss and fragmentation in forest ecosystems are serious threats that lead to reduced resilience. The integrity and stability of the ecosystem is fostered by recognizing and protecting areas that are essential to maintaining the resilience of the ecological network. Research in the field of ecological network resilience has garnered attention in recent years, although the necessity of developing various assessment methods for network resilience is evident. Taking Hyrcanian Forest ecosystem as a case study, this research aimed to identify the most important areas of the ecological network in maintaining and enhancing the resilience. To achieve this goal, first, a combination of the morphological spatial pattern analysis (MSPA) method and the assessment of the significance of ecosystem services were used to extract ecological source areas. Next, utilizing circuit theory and the least-cost path method, a network connecting sources was constructed, and pinch points were identified. After that, high-risk areas in ecological sources were found using the habitat risk assessment method. Using this integrated approach leads to the identification of valuable areas that are vulnerable to human threats and disturbances. Finally, the node removal method coupled with the calculation of network resilience indices, connectivity, and efficiency was employed to prioritize conservation areas. The results of the study indicated that the most important nodes were located in the northern edges of the forest, which have been under threat in recent years. Additionally, the region ranked moderately in terms of connectivity, indicating the importance of focusing on the conservation of forest patches before the complete fragmentation of the area. Furthermore, our findings underscore the importance of considering landscape connectivity and ecological network resilience in conservation planning for policymakers and managers aiming to protect biodiversity in the Hyrcanian Forest ecosystem.

Huang X., Wang X., Li J., Zhang M., Chen S., Xu B., Nie W.

ABSTRACTAnthropogenic interference causes ecological fragmentation and vulnerability, weakening urban ecosystems' adaptive capacity. The ecological network is based on the principles of landscape ecology, connecting resource patches through linear corridors to protect biodiversity and landscape integrity, enhance environmental carrying capacity, and improve ecosystem resilience. However, current research on ecological network resilience often relies on single methods and scales, overlooking the potential discrepancies between different approaches and scales. This study uses Zhejiang to construct ecological networks with structural, functional, and integrated approaches at provincial, urban agglomeration, and city levels. The performance of these methods in protecting structure, maintaining function, and ensuring overall resilience was compared, yielding the following results: First, the spatial output consistency of source areas across different scales for the three methods ranged from 50.48% to 97.81%. Second, the integrated approach was not optimal for all three resilience goals. The structure‐oriented method demonstrated cross‐scale applicability for the structural resilience goal, while the function‐oriented strategy performed well in maintaining functional and overall resilience. Third, the scale analysis showed consistency in results at the provincial and urban agglomeration levels when meeting the same objectives, but discrepancies at the city level. By expanding the methodologies and scale perspectives in the field of ecological network resilience, this study assesses the applicability of different scales and methods for ecological network resilience. It was found that integrated methods do not always effectively coordinate multiple protection objectives; thus, large‐scale strategies cannot be directly applied at smaller scales in practical applications. This study proposes and validates a multi‐scale, multi‐method framework for assessing the resilience of ecological networks. It reveals the potential differences between scales and methods, providing valuable theoretical insights and practical guidance for future research on ecological network resilience, particularly regarding the applicability of methods at different scales.

Budnick B., Biham O., Katzav E.

Abstract Analytical results are presented for the structure of networks that evolve via a preferential-attachment-random-deletion (PARD) model in the regime of overall network growth and in the regime of overall contraction. The phase transition between the two regimes is studied. At each time step a node addition and preferential attachment step takes place with probability P add , and a random node deletion step takes place with probability P del = 1 − P add . The balance between growth and contraction is captured by the parameter η = P add − P del , which in the regime of overall network growth satisfies 0 < η ⩽ 1 and in the regime of overall network contraction − 1 ⩽ η < 0 . Using the master equation and computer simulations we show that for − 1 < η < 0 the time-dependent degree distribution P t ( k ) converges towards a stationary form P st ( k ) which exhibits an exponential tail. This is in contrast with the power-law tail of the stationary degree distribution obtained for 0 < η ⩽ 1 . Thus, the PARD model has a phase transition at η = 0, which separates between two structurally distinct phases. At the transition, for η = 0, the degree distribution exhibits a stretched exponential tail. While the stationary degree distribution in the phase of overall growth represents an asymptotic state, in the phase of overall contraction P st ( k ) represents an intermediate asymptotic state of a finite life span, which disappears when the network vanishes.

Qiu G., Wang J., Liu J., Wang X.

Ecological infrastructure (EI), providing ecosystem services across the land-sea interface, has been proposed as a key element in sustainable terrestrial-marine ecosystem coordinated governance. Terrestrial and marine ecosystems should be regarded as an integrated unit for guaranteeing coastal ecological security. However, the existing EI construction framework focused on terrestrial ecosystems, and few studies consider the composite characteristics of the terrestrial-marine ecosystem in coastal areas. In the case study of Laizhou Bay, China, this study proposes an optimization method for multiple ecological infrastructures (MEIs) across the land-sea interface. The method is oriented towards achieving trans-regional scale cohesion, enhancing terrestrial-riverine-marine linkages, providing adequate pathways for marine ecological protection, and promoting coordinated conservation of terrestrial and marine ecosystems. The results showed that: (1) The new optimization framework synthetically considering the terrestrial multi-scale EI networks cohesion, hydrological corridors, and marine conservation network is available. (2) The preliminary ecological sources (PESs) are mainly distributed in the eastern mountainous areas, the estuary of the Yellow River, and six marine protected areas. The spatial imbalance of EI resulted in four marine protected areas in the southwest of the Bohai Sea insufficiently connected between sea-to-sea ecological sources. (3) The integrated MEIs includes four newly added ecological sources (two each for land and sea), eight trans-regional ecological corridors, 17 hydrological corridors, and 11 marine ecological corridors. Through optimization, the MEIs avoid fragmentation across multi-scale terrestrial regions, promote river-based connectivity between land and sea, and increase pathways for marine ecological protection, thereby ensuring effective circulation of regional ecological materials. (4) MEIs-conserved priority areas include 12.4 km

Ji A., He R., Chen W., Zhang L.

Jesse B., Kramer G.J., Koning V.

Abstract Background To reduce the effects of climate change, the current fossil-based energy system must transition to a low-carbon system based largely on renewables. In both academic literature and non-academic discourse concerning the energy transition, resilience is frequently mentioned as an additional objective or requirement. Despite its frequent use, resilience is a very malleable term with different meanings in different contexts. Main text This paper seeks to identify how resilience is understood in the field of the energy system and whether there are similar aspects in the different ways the term is understood. To this end, we review more than 130 papers for definitions of energy system resilience. In addition, we use different aspects to categorize and examine these. The results paint a diverse picture in terms of the definition and understanding of resilience in the energy system. However, a few definition archetypes can be identified. The first uses a straightforward approach, in which the energy system has one clearly defined equilibrium state. Here, resilience is defined in relation to the response of the energy system to a disturbance and its ability to quickly return to its equilibrium. The second type of resilience allows for different equilibriums, to which a resilient energy system can move after a disruption. Another type of resilience focuses more on the process and the actions of the system in response to disruption. Here, resilience is defined as the ability of the system to adapt and change. In the papers reviewed, we find that the operational definition of resilience often encompasses aspects of different archetypes. This diversity shows that resilience is a versatile concept with different elements. Conclusions With this paper, we aim to provide insight into how the understanding of resilience for the energy system differs depending on which aspect of the energy system is studied, and which elements might be necessary for different understandings of resilience. We conclude by providing information and recommendations on the potential usage of the term energy system resilience based on our lessons learned.

Zhang M., Li J., Wang L., Xu B., Nie W.

In the context of accelerating ecological fragmentation, it is urgent to enhance the interconnectivity of urban ecological patches to form a resilient ecological network (EN). The construction of a Natural Protected Area (NPA) system proposed in 2019 is the latest strategy implemented by China in protecting ecological spaces. However, the effectiveness of this strategy has not been adequately demonstrated. This study specifically analyzes the concrete impacts of the natural protected area system on the resilience of ecological networks (ENs). The economically developed Urban Agglomeration around Hangzhou Bay (UAHB) was chosen as an example for the argumentation. Firstly, we utilized circuit theory to construct an EN consisting of 173 ecological sources and 401 ecological corridors. Secondly, the ecological sources were categorized into three levels based on their connectivity values. Finally, a dynamic disturbance scenario simulation framework was constructed to evaluate the impact of NPA on the resilience of EN. The results indicated that: (1) The preceding 47% of ecological sources are crucial for maintaining the resilience of the EN; (2) Compared with other ecological spaces, NPAs have a 38% and 1100% greater effect on the resilience of the EN in first and second-level ecological sources, respectively, while its impact in third-level is 118% lower. This study innovatively investigates the differential impacts of hierarchical natural protected areas and natural unprotected areas on the ecological environment.

Millán A.P., Sun H., Torres J.J., Bianconi G.

Abstract Triadic interactions are higher-order interactions which occur when a set of nodes affects the interaction between two other nodes. Examples of triadic interactions are present in the brain when glia modulate the synaptic signals among neuron pairs or when interneuron axo-axonic synapses enable presynaptic inhibition and facilitation, and in ecosystems when one or more species can affect the interaction among two other species. On random graphs, triadic percolation has been recently shown to turn percolation into a fully fledged dynamical process in which the size of the giant component undergoes a route to chaos. However, in many real cases, triadic interactions are local and occur on spatially embedded networks. Here, we show that triadic interactions in spatial networks induce a very complex spatio-temporal modulation of the giant component which gives rise to triadic percolation patterns with significantly different topology. We classify the observed patterns (stripes, octopus, and small clusters) with topological data analysis and we assess their information content (entropy and complexity). Moreover, we illustrate the multistability of the dynamics of the triadic percolation patterns, and we provide a comprehensive phase diagram of the model. These results open new perspectives in percolation as they demonstrate that in presence of spatial triadic interactions, the giant component can acquire a time-varying topology. Hence, this work provides a theoretical framework that can be applied to model realistic scenarios in which the giant component is time dependent as in neuroscience.

Liu X., Yan X., Eugene Stanley H.

Complex networked systems, which range from biological systems in the natural world to infrastructure systems in the human-made world, can exhibit spontaneous recovery after a failure; for example, a brain may spontaneously return to normal after a seizure, and traffic flow can become smooth again after a jam. Previous studies on the spontaneous recovery of dynamical networks have been limited to undirected networks. However, most real-world networks are directed. To fill this gap, we build a model in which nodes may alternately fail and recover, and we develop a theoretical tool to analyze the recovery properties of directed dynamical networks. We find that the tool can accurately predict the final fraction of active nodes, and the prediction accuracy decreases as the fraction of bidirectional links in the network increases, which emphasizes the importance of directionality in network dynamics. Due to different initial states, directed dynamical networks may show alternative stable states under the same control parameter, exhibiting hysteresis behavior. In addition, for networks with finite sizes, the fraction of active nodes may jump back and forth between high and low states, mimicking repetitive failure-recovery processes. These findings could help clarify the system recovery mechanism and enable better design of networked systems with high resilience.

Zhao Y., Zhang M., Zhao D., Duo L., Lu C.

Mineral extraction in resource-based cities has caused serious damage to the original ecology, resulting in poor regional vegetation growth, reduced carbon sequestration capacity, and reduced ecosystem resilience. Especially in resource-based cities with fragile ecology, the overall anti-interference ability of the environment is relatively worse. Seeking ecological network optimization solutions that can improve vegetation growth conditions on a large scale is an effective way to enhance the resilience of regional ecosystems. This paper introduces carbon sequestration indicators and designs a differential ecological networks (ENs) optimization model (FTCC model) to achieve the goal of improving ecosystem resilience. The model identifies the patches that need to be optimized and their optimization directions based on the differences in ecological function-topology-connectivity-carbon sequestration of the patches. Finally, the resilience of the ecological network before and after optimization was compared, proving that the model is effective. The results show that the sources in the Yulin ENs form three main clusters, with connectivity between clusters relying on only a few patches. The patches in the northeastern and southwest clusters are large but their ecological functions need to be improved. After optimization, 16 new stepping stones were added, 38 new corridors were added, and the ecological function of 39 patches was enhanced. The optimized ecological network resilience was improved in terms of structure, function, and carbon sinks, and carbon sinks increased by 6364.5 tons. This study provides a reference for measures to optimize landscape space and manage ecosystem resilience enhancement in resource-based cities.