Recent Progress on the Resilience of Complex Networks

Optimization of multiple ecological infrastructures across the land-sea interface for coordination management: A case study around Laizhou Bay in China

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 km2 ecological pinch points and 6.39 km2 marine biological protective points. Focusing on these conserved priority areas provides spatial references for the implementation planning of MEIs. Compared with traditional respective ecosystem networks, the MEIs across land-sea interface optimization approach is feasible for terrestrial-marine ecosystem coordinated management.

Triadic percolation induces dynamical topological patterns in higher-order networks

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.

Optimizing the ecological network of resource-based cities to enhance the resilience of regional ecological networks

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.

Structure of networks that evolve under a combination of growth and contraction

We present analytical results for the emerging structure of networks that evolve via a combination of growth (by node addition and random attachment) and contraction (by random node deletion). To this end we consider a network model in which at each time step a node addition and random 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 the growth and contraction processes is captured by the parameter η=P_{add}-P_{del}. The case of pure network growth is described by η=1. In the case that 0<η<1, the rate of node addition exceeds the rate of node deletion and the overall process is of network growth. In the opposite case, where -1<η<0, the overall process is of network contraction, while in the special case of η=0 the expected size of the network remains fixed, apart from fluctuations. Using the master equation and the generating function formalism, we obtain a closed-form expression for the time-dependent degree distribution P_{t}(k). The degree distribution P_{t}(k) includes a term that depends on the initial degree distribution P_{0}(k), which decays as time evolves, and an asymptotic distribution P_{st}(k) which is independent of the initial condition. In the case of pure network growth (η=1), the asymptotic distribution P_{st}(k) follows an exponential distribution, while for -1<η<1 it consists of a sum of Poisson-like terms and exhibits a Poisson-like tail. In the case of overall network growth (0<η<1) the degree distribution P_{t}(k) eventually converges to P_{st}(k). In the case of overall network contraction (-1<η<0) we identify two different regimes. For -1/3<η<0 the degree distribution P_{t}(k) quickly converges towards P_{st}(k). In contrast, for -1<η<-1/3 the convergence of P_{t}(k) is initially very slow and it gets closer to P_{st}(k) only shortly before the network vanishes. Thus, the model exhibits three phase transitions: a structural transition between two functional forms of P_{st}(k) at η=1, a transition between an overall growth and overall contraction at η=0, and a dynamical transition between fast and slow convergence towards P_{st}(k) at η=-1/3. The analytical results are found to be in very good agreement with the results obtained from computer simulations.

Analysis on the Evolution and Resilience of Ecological Network Structure in Wuhan Metropolitan Area

With the accelerated urbanization and frequent occurrence of climate extremes, the regional ecosystem service level has ushered in a great challenge, and the resilience of the ecological network has gradually weakened, leading to lower ecological benefits and production levels. As a core ecologically sensitive area in the middle reaches of the Yangtze River, Wuhan metropolitan area has been expanding outward with rapid urbanization, crowding out surrounding arable and ecological land, and facing serious challenges to the sustainable development of the national space, while current cross-regional ecological protection measures need to be strengthened urgently, and exploring the structural resilience of its ecological network is of great significance to promote regional stability. In this study, Wuhan metropolitan area is taken as an example, and we explore the evolution and laws of ecological network structure from the perspective of network analysis by constructing ecological networks in Wuhan metropolitan area in 2000, 2010, and 2020. Firstly, we select regions from the ecological control line developed in China as ecological source sites, and also select multivariate data to supplement them. Then, the ecological network was established using the MCR model. Finally, network analysis was applied to discuss the evolution of network structure under multiple times and propose corresponding conservation strategies. The results show that (1) the major ecological resistance of Wuhan urban area has increased by 5.24% in 20 years. (2) The centrality and connectivity of the network nodes have increased over the 20-year period, and the overall structure of the network has stabilized and the resilience of the network has increased. (3) There is a strong link between changes in the network as a whole and local resilience. The results of the study will help analyze the relationship between the network as a whole and the region, and provide reference for optimizing the ecological network and constructing the systematic management of ecological security pattern.

Conceptualising and assessing health system resilience to shocks: a cross-disciplinary view

Health systems worldwide face major challenges in anticipating, planning for and responding to shocks from infectious disease epidemics, armed conflict, climatic and other crises. Although the literature on health system resilience has grown substantially in recent years, major uncertainties remain concerning approaches to resilience conceptualisation and measurement. This narrative review revisits literatures from a range of fields outside health to identify lessons relevant to health systems. Four key insights emerge. Firstly, shocks can only be understood by clarifying how, where and over what timescale they interact with a system of interest, and the dynamic effects they produce within it. Shock effects are contingent on historical path-dependencies, and on the presence of factors or system pathways (e.g. financing models, health workforce capabilities or supply chain designs) that may amplify or dampen impact in unexpected ways. Secondly, shocks often produce cascading effects across multiple scales, whereas the focus of much of the health resilience literature has been on macro-level, national systems. In reality, health systems bring together interconnected sub-systems across sectors and geographies, with different components, behaviours and sometimes even objectives - all influencing how a system responds to a shock. Thirdly, transformability is an integral feature of resilient social systems: cross-scale interactions help explain how systems can show both resilience and transformational capability at the same time. We illustrate these first three findings by extending the socioecological concept of adaptive cycles in social systems to health, using the example of maternal and child health service delivery. Finally, we argue that dynamic modelling approaches, under-utilised in research on health system resilience to date, have significant promise for identification of shock-moderating or shock-amplifying pathways, for understanding effects at multiple levels and ultimately for building resilience.

Accuracy of a one-dimensional reduction of dynamical systems on networks

Resilience is an ability of a system with which the system can adjust its activity to maintain its functionality when it is perturbed. To study resilience of dynamics on networks, Gao et al. [Nature (London) 530, 307 (2016)0028-083610.1038/nature16948] proposed a theoretical framework to reduce dynamical systems on networks, which are high dimensional in general, to one-dimensional dynamical systems. The accuracy of this one-dimensional reduction relies on three approximations in addition to the assumption that the network has a negligible degree correlation. In the present study, we analyze the accuracy of the one-dimensional reduction assuming networks without degree correlation. We do so mainly through examining the validity of the individual assumptions underlying the method. Across five dynamical system models, we find that the accuracy of the one-dimensional reduction hinges on the spread of the equilibrium value of the state variable across the nodes in most cases. Specifically, the one-dimensional reduction tends to be accurate when the dispersion of the node's state is small. We also find that the correlation between the node's state and the node's degree, which is common for various dynamical systems on networks, is unrelated to the accuracy of the one-dimensional reduction.

Identification of Biomarker in Brain-specific Gene Regulatory Network Using Structural Controllability Analysis

Brain tumor research has been stapled for human health while brain network research is crucial for us to understand brain activity. Here the structural controllability theory is applied to study three human brain-specific gene regulatory networks, including forebrain gene regulatory network, hindbrain gene regulatory network and neuron associated cells cancer related gene regulatory network, whose nodes are neural genes and the edges represent the gene expression regulation among the genes. The nodes are classified into two classes: critical nodes and ordinary nodes, based on the change of the number of driver nodes upon its removal. Eight topological properties (out-degree DO, in-degree DI, degree D, betweenness B, closeness CA, in-closeness CI, out-closeness CO and clustering coefficient CC) are calculated in this paper and the results prove that the critical genes have higher score of topological properties than the ordinary genes. Then two bioinformatic analysis are used to explore the biologic significance of the critical genes. On the one hand, the enrichment scores in several kinds of gene databases are calculated and reveal that the critical nodes are richer in essential genes, cancer genes and the neuron related disease genes than the ordinary nodes, which indicates that the critical nodes may be the biomarker in brain-specific gene regulatory network. On the other hand, GO analysis and KEGG pathway analysis are applied on them and the results show that the critical genes mainly take part in 14 KEGG pathways that are transcriptional misregulation in cancer, pathways in cancer and so on, which indicates that the critical genes are related to the brain tumor. Finally, by deleting the edges or routines in the network, the robustness analysis of node classification is realized, and the robustness of node classification is proved. The comparison of neuron associated cells cancer related GRN (Gene Regulatory Network) and normal brain-specific GRNs (including forebrain and hindbrain GRN) shows that the neuron-related cell cancer-related gene regulatory network is more robust than other types.

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.

Multi-Agent Systems and Complex Networks: Review and Applications in Systems Engineering

Systems engineering is an ubiquitous discipline of Engineering overlapping industrial, chemical, mechanical, manufacturing, control, software, electrical, and civil engineering. It provides tools for dealing with the complexity and dynamics related to the optimisation of physical, natural, and virtual systems management. This paper presents a review of how multi-agent systems and complex networks theory are brought together to address systems engineering and management problems. The review also encompasses current and future research directions both for theoretical fundamentals and applications in the industry. This is made by considering trends such as mesoscale, multiscale, and multilayer networks along with the state-of-art analysis on network dynamics and intelligent networks. Critical and smart infrastructure, manufacturing processes, and supply chain networks are instances of research topics for which this literature review is highly relevant.

Resilience centrality in complex networks

Resilience describes a system's ability to adjust its activity to retain the basic functionality when errors or failures occur in components (nodes) of the network. Due to the complexity of a system's structure, different components in the system exhibit diversity in the ability to affect the resilience of the system, bringing us a great challenge to protect the system from collapse. A fundamental problem is therefore to propose a physically insightful centrality index, with which to quantify the resilience contribution of a node in any systems effectively. However, existing centrality indexes are not suitable for the problem because they only consider the network structure of the system and ignore the impact of underlying dynamic characteristics. To break the limits, we derive a new centrality index: resilience centrality from the 1D dynamic equation of systems, with which we can quantify the ability of nodes to affect the resilience of the system accurately. Resilience centrality unveils the long-sought relations between the ability of nodes in a system's resilience and network structure of the system: the capacity is mainly determined by the degree and weighted nearest-neighbor degree of the node, in which weighted nearest-neighbor degree plays a prominent role. Further, we demonstrate that weighted nearest-neighbor degree has a positive impact on resilience centrality, while the effect of the degree depends on a specific parameter, average weighted degree β_{eff}, in the 1D dynamic equation. To test the performance of our approach, we construct four real networks from data, which corresponds to two complex systems with entirely different dynamic characteristics. The simulation results demonstrate the effectiveness of our resilience centrality, providing us theoretical insights into the protection of complex systems from collapse.

Influence of Interlink Topology on Multilayer Network Robustness

Cascading failures between interdependent multilayer networks are being widely studied, especially the trend of robustness caused by the interlinks between networks. However, few researchers pay attention to the effect of the interlink topology on the robustness of coupled networks, which is a critical interlink factor of multilayer networks. In this study, the method frame of multilayer network experiment simulation is given. Through numerical simulation and actual network simulation, the exhaustive method is used to enumerate all the patterns of interlink topological relations of multilayer networks (three-layer or more). The research verifies that the interlink topology affects the global robustness and that there exists a fragile interlink pattern in the patterns of interlink topologies. The star-like interlink pattern with the most uneven interlink-degree distribution leads to the weakest robustness; the pattern with average interlink-degree distribution reveals good global stability as a loop-like pattern or entire interlink pattern. In addition, the influence of interlink topology is independent. The simulation results are not affected by the network layer number and intraparameters (including the network-generated form, each layer of network node number, and average degree of each layer of network). Thus, ignoring the interlink topology may result in the actual system suddenly becoming vulnerable before the theoretical calculation point. Interlink topology as an independent factor affecting the robustness of multilayer networks should be paid more attention.

Assessing the Performance of the European Natural Gas Network for Selected Supply Disruption Scenarios Using Open-Source Information

Natural gas covers more than 20% of Europe’s primary energy demand. A potential disruption could lead to supply shortages with severe consequences for the European economy and society. History shows that such a vast and complex network system is prone to exogenous and endogenous disruptions. A dedicated large-scale dataset of the European natural gas network from publicly available information sources is assembled first. The spatial coverage, completeness and resolution allows analyzing the behavior of this geospatial infrastructure network (including consumption) and its components under likely disruptive events, such as earthquakes, and/or technical failures. Using the developed system state simulation engine, the disruption impact is mapped. The results show that storage facilities cannot in all cases compensate for a pipeline disruption. Moreover, critical pipelines, such as the Transitgas pipeline crossing the Alps and the Trans-Mediterranean pipeline bringing natural gas from Northern Africa, are identified. To analyze the pipelines with high impact on the system performance, a detailed scenario analysis using a Monte Carlo simulation resulting in supply grade mapping is conducted and presented for the case of Italy. Overall, it can be concluded that locations with a dead-end, sole supply, and without storage facility nearby, are remarkably exposed to natural gas supply losses.

Multiple phase transitions in networks of directed networks

The robustness in real-world complex systems with dependency connectivities differs from that in isolated networks. Although most complex network research has focused on interdependent undirected systems, many real-world networks-such as gene regulatory networks and traffic networks-are directed. We thus develop an analytical framework for examining the robustness of networks made up of directed networks of differing topologies. We use it to predict the phase transitions that occur during node failures and to generate the phase diagrams of a number of different systems, including treelike and random regular (RR) networks of directed Erdős-Rényi (ER) networks and scale-free networks. We find that the the phase transition and phase diagram of networks of directed networks differ from those of networks of undirected networks. For example, the RR networks of directed ER networks show a hybrid phase transition that does not occur in networks of undirected ER networks. In addition, system robustness is affected by network topology in networks of directed networks. As coupling strength q increases, treelike networks of directed ER networks change from a second-order phase transition to a first-order phase transition, and RR networks of directed ER networks change from a second-order phase transition to a hybrid phase transition, then to a first-order phase transition, and finally to a region of collapse. We also find that heterogenous network systems are more robust than homogeneous network systems. We note that there are multiple phase transitions and triple points in the phase diagram of RR networks of directed networks and this helps us understand how to increase network robustness when designing interdependent infrastructure systems.

Model-Based Approach to Engineering Resilience in Multi-UAV Systems

Multi-UAV Operations are an area of great interest in government, industry, and research community. In multi-UAV operations, a group of unmanned aerial vehicles (UAVs) are deployed to carry out missions such as search and rescue or disaster relief. As multi-UAV systems operate in an open operational environment, many disrupting events can occur. To this end, resilience of these systems is of great importance. The research performed and reported in this paper utilizes simulation-based research methodology and demonstrates that resilience of multi-UAV systems can be achieved by real-time evaluation of resilience alternatives during system operation. This evaluation is done using a dynamic utility function where priorities change as a function of context. Simulation results show that resilience response can in fact change depending on the context.

A comparative analysis of approaches to network-dismantling

Estimating, understanding, and improving the robustness of networks has many application areas such as bioinformatics, transportation, or computational linguistics. Accordingly, with the rise of network science for modeling complex systems, many methods for robustness estimation and network dismantling have been developed and applied to real-world problems. The state-of-the-art in this field is quite fuzzy, as results are published in various domain-specific venues and using different datasets. In this study, we report, to the best of our knowledge, on the analysis of the largest benchmark regarding network dismantling. We reimplemented and compared 13 competitors on 12 types of random networks, including ER, BA, and WS, with different network generation parameters. We find that network metrics, proposed more than 20 years ago, are often non-dominating competitors, while many recently proposed techniques perform well only on specific network types. Besides the solution quality, we also investigate the execution time. Moreover, we analyze the similarity of competitors, as induced by their node rankings. We compare and validate our results on real-world networks. Our study is aimed to be a reference for selecting a network dismantling method for a given network, considering accuracy requirements and run time constraints.

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.

Resumption of dynamism in damaged networks of coupled oscillators

Deterioration in dynamical activities may come up naturally or due to environmental influences in a massive portion of biological and physical systems. Such dynamical degradation may have outright effect on the substantive network performance. This requires us to provide some proper prescriptions to overcome undesired circumstances. In this paper, we present a scheme based on external feedback that can efficiently revive dynamism in damaged networks of active and inactive oscillators and thus enhance the network survivability. Both numerical and analytical investigations are performed in order to verify our claim. We also provide a comparative study on the effectiveness of this mechanism for feedbacks to the inactive group or to the active group only. Most importantly, resurrection of dynamical activity is realized even in time-delayed damaged networks, which are considered to be less persistent against deterioration in the form of inactivity in the oscillators. Furthermore, prominence in our approach is substantiated by providing evidence of enhanced network persistence in complex network topologies taking small-world and scale-free architectures, which makes the proposed remedy quite general. Besides the study in the network of Stuart-Landau oscillators, affirmative influence of external feedback has been justified in the network of chaotic Rössler systems as well.

Resilience of epidemics for SIS model on networks

Epidemic propagation on complex networks has been widely investigated, mostly with invariant parameters. However, the process of epidemic propagation is not always constant. Epidemics can be affected by various perturbations and may bounce back to its original state, which is considered resilient. Here, we study the resilience of epidemics on networks, by introducing a different infection rate λ₂ during SIS (susceptible-infected-susceptible) epidemic propagation to model perturbations (control state), whereas the infection rate is λ₁ in the rest of time. Noticing that when λ₁ is below λ_c, there is no resilience in the SIS model. Through simulations and theoretical analysis, we find that even for λ₂ < λ_c, epidemics eventually could bounce back if the control duration is below a threshold. This critical control time for epidemic resilience, i.e., cd_max, seems to be predicted by the diameter (d) of the underlying network, with the quantitative relation cd_max ∼ d^α. Our findings can help to design a better mitigation strategy for epidemics.

Maximum flow-based resilience analysis: From component to system

Resilience, the ability to withstand disruptions and recover quickly, must be considered during system design because any disruption of the system may cause considerable loss, including economic and societal. This work develops analytic maximum flow-based resilience models for series and parallel systems using Zobel's resilience measure. The two analytic models can be used to evaluate quantitatively and compare the resilience of the systems with the corresponding performance structures. For systems with identical components, the resilience of the parallel system increases with increasing number of components, while the resilience remains constant in the series system. A Monte Carlo-based simulation method is also provided to verify the correctness of our analytic resilience models and to analyze the resilience of networked systems based on that of components. A road network example is used to illustrate the analysis process, and the resilience comparison among networks with different topologies but the same components indicates that a system with redundant performance is usually more resilient than one without redundant performance. However, not all redundant capacities of components can improve the system resilience, the effectiveness of the capacity redundancy depends on where the redundant capacity is located.

Multiple-node basin stability in complex dynamical networks

Dynamical entities interacting with each other on complex networks often exhibit multistability. The stability of a desired steady regime (e.g., a synchronized state) to large perturbations is critical in the operation of many real-world networked dynamical systems such as ecosystems, power grids, the human brain, etc. This necessitates the development of appropriate quantifiers of stability of multiple stable states of such systems. Motivated by the concept of basin stability (BS) [P. J. Menck et al., Nat. Phys. 9, 89 (2013)1745-247310.1038/nphys2516], we propose here the general framework of multiple-node basin stability for gauging the global stability and robustness of networked dynamical systems in response to nonlocal perturbations simultaneously affecting multiple nodes of a system. The framework of multiple-node BS provides an estimate of the critical number of nodes that, when simultaneously perturbed, significantly reduce the capacity of the system to return to the desired stable state. Further, this methodology can be applied to estimate the minimum number of nodes of the network to be controlled or safeguarded from external perturbations to ensure proper operation of the system. Multiple-node BS can also be utilized for probing the influence of spatially localized perturbations or targeted attacks to specific parts of a network. We demonstrate the potential of multiple-node BS in assessing the stability of the synchronized state in a deterministic scale-free network of Rössler oscillators and a conceptual model of the power grid of the United Kingdom with second-order Kuramoto-type nodal dynamics.