Author Correction: Universal resilience patterns in complex networks.

In this Letter, in Fig. 3c and f the Saccharomyces cerevisiae and Escherichia coli networks were subject to both weight loss and node deletion, a combination of two types of perturbation, as opposed to weight loss only (as the labelling incorrectly indicated). The collapse in Fig. 3h was also obtained from this combined perturbation, and therefore the results displayed in Fig. 3h remain fully consistent with the theoretical framework presented in this Letter. Figure 1 to this Amendment shows the corrected Fig. 3c, f and h, in which Fig. 3c and f have been generated with weight-loss perturbations only, as originally reported, together with the originally published panels, for completeness and transparency. The codes used to generate the original and the corrected Fig. 3 are available at https://github.com/jianxigao/NuRsE . We thank Travis A. Gibson for alerting us to this error. The original Letter has not been corrected.

A spatiotemporal decay model of human mobility when facing large-scale crises.

A common feature of large-scale extreme events, such as pandemics, wildfires, and major storms is that, despite their differences in etiology and duration, they significantly change routine human movement patterns. Such changes, which can be major or minor in size and duration and which differ across contexts, affect both the consequences of the events and the ability of governments to mount effective responses. Based on naturally tracked, anonymized mobility behavior from over 90 million people in the United States, we document these mobility differences in space and over time in six large-scale crises, including wildfires, major tropical storms, winter freeze and pandemics. We introduce a model that effectively captures the high-dimensional heterogeneity in human mobility changes following large-scale extreme events. Across five different metrics and regardless of spatial resolution, the changes in human mobility behavior exhibit a consistent hyperbolic decline, a pattern we characterize as "spatiotemporal decay." When applied to the case of COVID-19, our model also uncovers significant disparities in mobility changes-individuals from wealthy areas not only reduce their mobility at higher rates at the start of the pandemic but also maintain the change longer. Residents from lower-income regions show a faster and greater hyperbolic decay, which we suggest may help account for different COVID-19 rates. Our model represents a powerful tool to understand and forecast mobility patterns post emergency, and thus to help produce more effective responses.

Polarization and tipping points.

Research has documented increasing partisan division and extremist positions that are more pronounced among political elites than among voters. Attention has now begun to focus on how polarization might be attenuated. We use a general model of opinion change to see if the self-reinforcing dynamics of influence and homophily may be characterized by tipping points that make reversibility problematic. The model applies to a legislative body or other small, densely connected organization, but does not assume country-specific institutional arrangements that would obscure the identification of fundamental regularities in the phase transitions. Agents in the model have initially random locations in a multidimensional issue space consisting of membership in one of two equal-sized parties and positions on 10 issues. Agents then update their issue positions by moving closer to nearby neighbors and farther from those with whom they disagree, depending on the agents' tolerance of disagreement and strength of party identification compared to their ideological commitment to the issues. We conducted computational experiments in which we manipulated agents' tolerance for disagreement and strength of party identification. Importantly, we also introduced exogenous shocks corresponding to events that create a shared interest against a common threat (e.g., a global pandemic). Phase diagrams of political polarization reveal difficult-to-predict transitions that can be irreversible due to asymmetric hysteresis trajectories. We conclude that future empirical research needs to pay much closer attention to the identification of tipping points and the effectiveness of possible countermeasures.

Multiple metastable network states in urban traffic.

While abrupt regime shifts between different metastable states have occurred in natural systems from many areas including ecology, biology, and climate, evidence for this phenomenon in transportation systems has been rarely observed so far. This limitation might be rooted in the fact that we lack methods to identify and analyze possible multiple states that could emerge at scales of the entire traffic network. Here, using percolation approaches, we observe such a metastable regime in traffic systems. In particular, we find multiple metastable network states, corresponding to varying levels of traffic performance, which recur over different days. Based on high-resolution global positioning system (GPS) datasets of urban traffic in the megacities of Beijing and Shanghai (each with over 50,000 road segments), we find evidence supporting the existence of tipping points separating three regimes: a global functional regime and a metastable hysteresis-like regime, followed by a global collapsed regime. We can determine the intrinsic critical points where the metastable hysteresis-like regime begins and ends and show that these critical points are very similar across different days. Our findings provide a better understanding of traffic resilience patterns and could be useful for designing early warning signals for traffic resilience management and, potentially, other complex systems.

Reconstructing Sparse Multiplex Networks with Application to Covert Networks.

Network structure provides critical information for understanding the dynamic behavior of complex systems. However, the complete structure of real-world networks is often unavailable, thus it is crucially important to develop approaches to infer a more complete structure of networks. In this paper, we integrate the configuration model for generating random networks into an Expectation-Maximization-Aggregation (EMA) framework to reconstruct the complete structure of multiplex networks. We validate the proposed EMA framework against the Expectation-Maximization (EM) framework and random model on several real-world multiplex networks, including both covert and overt ones. It is found that the EMA framework generally achieves the best predictive accuracy compared to the EM framework and the random model. As the number of layers increases, the performance improvement of EMA over EM decreases. The inferred multiplex networks can be leveraged to inform the decision-making on monitoring covert networks as well as allocating limited resources for collecting additional information to improve reconstruction accuracy. For law enforcement agencies, the inferred complete network structure can be used to develop more effective strategies for covert network interdiction.

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.

Percolation of temporal hierarchical mobility networks during COVID-19.

Percolation theory is essential for understanding disease transmission patterns on the temporal mobility networks. However, the traditional approach of the percolation process can be inefficient when analysing a large-scale, dynamic network for an extended period. Not only is it time-consuming but it is also hard to identify the connected components. Recent studies demonstrate that spatial containers restrict mobility behaviour, described by a hierarchical topology of mobility networks. Here, we leverage crowd-sourced, large-scale human mobility data to construct temporal hierarchical networks composed of over 175 000 block groups in the USA. Each daily network contains mobility between block groups within a Metropolitan Statistical Area (MSA), and long-distance travels across the MSAs. We examine percolation on both levels and demonstrate the changes of network metrics and the connected components under the influence of COVID-19. The research reveals the presence of functional subunits even with high thresholds of mobility. Finally, we locate a set of recurrent critical links that divide components resulting in the separation of core MSAs. Our findings provide novel insights into understanding the dynamical community structure of mobility networks during disruptions and could contribute to more effective infectious disease control at multiple scales. This article is part of the theme issue 'Data science approaches to infectious disease surveillance'.

Data science approaches to confronting the COVID-19 pandemic: a narrative review.

During the COVID-19 pandemic, more than ever, data science has become a powerful weapon in combating an infectious disease epidemic and arguably any future infectious disease epidemic. Computer scientists, data scientists, physicists and mathematicians have joined public health professionals and virologists to confront the largest pandemic in the century by capitalizing on the large-scale 'big data' generated and harnessed for combating the COVID-19 pandemic. In this paper, we review the newly born data science approaches to confronting COVID-19, including the estimation of epidemiological parameters, digital contact tracing, diagnosis, policy-making, resource allocation, risk assessment, mental health surveillance, social media analytics, drug repurposing and drug development. We compare the new approaches with conventional epidemiological studies, discuss lessons we learned from the COVID-19 pandemic, and highlight opportunities and challenges of data science approaches to confronting future infectious disease epidemics. This article is part of the theme issue 'Data science approaches to infectious disease surveillance'.

Universal resilience patterns in complex networks.

Resilience, a system's ability to adjust its activity to retain its basic functionality when errors, failures and environmental changes occur, is a defining property of many complex systems. Despite widespread consequences for human health, the economy and the environment, events leading to loss of resilience--from cascading failures in technological systems to mass extinctions in ecological networks--are rarely predictable and are often irreversible. These limitations are rooted in a theoretical gap: the current analytical framework of resilience is designed to treat low-dimensional models with a few interacting components, and is unsuitable for multi-dimensional systems consisting of a large number of components that interact through a complex network. Here we bridge this theoretical gap by developing a set of analytical tools with which to identify the natural control and state parameters of a multi-dimensional complex system, helping us derive effective one-dimensional dynamics that accurately predict the system's resilience. The proposed analytical framework allows us systematically to separate the roles of the system's dynamics and topology, collapsing the behaviour of different networks onto a single universal resilience function. The analytical results unveil the network characteristics that can enhance or diminish resilience, offering ways to prevent the collapse of ecological, biological or economic systems, and guiding the design of technological systems resilient to both internal failures and environmental changes.

Universal behavior of cascading failures in interdependent networks.

Catastrophic and major disasters in real-world systems, such as blackouts in power grids or global failures in critical infrastructures, are often triggered by minor events which originate a cascading failure in interdependent graphs. We present here a self-consistent theory enabling the systematic analysis of cascading failures in such networks and encompassing a broad range of dynamical systems, from epidemic spreading, to birth-death processes, to biochemical and regulatory dynamics. We offer testable predictions on breakdown scenarios, and, in particular, we unveil the conditions under which the percolation transition is of the first-order or the second-order type, as well as prove that accounting for dynamics in the nodes always accelerates the cascading process. Besides applying directly to relevant real-world situations, our results give practical hints on how to engineer more robust networked systems.

Identifying the shifting sources to predict the dynamics of COVID-19 in the U.S.

Mobility restriction is a crucial measure to control the transmission of the COVID-19. Research has shown that effective distance measured by the number of travelers instead of physical distance can capture and predict the transmission of the deadly virus. However, these efforts have been limited mainly to a single source of disease. Also, they have not been tested on finer spatial scales. Based on prior work of effective distances on the country level, we propose the multiple-source effective distance, a metric that captures the distance for the virus to propagate through the mobility network on the county level in the U.S. Then, we estimate how the change in the number of sources impacts the global mobility rate. Based on the findings, a new method is proposed to locate sources and estimate the arrival time of the virus. The new metric outperforms the original single-source effective distance in predicting the arrival time. Last, we select two potential sources and quantify the arrival time delay caused by the national emergency declaration. In doing so, we provide quantitative answers on the effectiveness of the national emergency declaration.

Concurrence Percolation in Quantum Networks.

Establishing long-distance quantum entanglement, i.e., entanglement transmission, in quantum networks (QN) is a key and timely challenge for developing efficient quantum communication. Traditional comprehension based on classical percolation assumes a necessary condition for successful entanglement transmission between any two infinitely distant nodes: they must be connected by at least a path of perfectly entangled states (singlets). Here, we relax this condition by explicitly showing that one can focus not on optimally converting singlets but on establishing concurrence-a key measure of bipartite entanglement. We thereby introduce a new statistical theory, concurrence percolation theory (ConPT), remotely analogous to classical percolation but fundamentally different, built by generalizing bond percolation in terms of "sponge-crossing" paths instead of clusters. Inspired by resistance network analysis, we determine the path connectivity by series and parallel rules and approximate higher-order rules via star-mesh transforms. Interestingly, we find that the entanglement transmission threshold predicted by ConPT is lower than the known classical-percolation-based results and is readily achievable on any series-parallel networks such as the Bethe lattice. ConPT promotes our understanding of how well quantum communication can be further systematically improved versus classical statistical predictions under the limitation of QN locality-a "quantum advantage" that is more general and efficient than expected. ConPT also shows a percolationlike universal critical behavior derived by finite-size analysis on the Bethe lattice and regular two-dimensional lattices, offering new perspectives for a theory of criticality in entanglement statistics.

A quantification method of non-failure cascading spreading in a network of networks.

The cascading spreading process in social and economic networks is more complicated than that in physical systems. These networks' multiple nodes and edges increase their structural complexity and recoverability, enabling the system to lose partial functionality rather than completely fail. However, these phenomena in social and economic networks introduce challenges to the existing network robustness models, where a node is either in a functional state or a failed state. This research uses a network of networks (NoN) to simulate multiple types of nodes and edges. A non-failure cascading process is utilized to model the nodes' self-adaptation and recoverability. The main contribution of this research is proposing a spreading model to extend the non-failure cascading process to the NoN, which can be used in predicting real-world system damage suffering from special events. The case study of this research evaluated the effect degree of crude oil trade changes on each sector from 2015 to 2016.

The impact of non-pharmaceutical interventions on the prevention and control of COVID-19 in New York City.

The emergence of coronavirus disease 2019 (COVID-19) has infected more than 62 million people worldwide. Control responses varied across countries with different outcomes in terms of epidemic size and social disruption. This study presents an age-specific susceptible-exposed-infected-recovery-death model that considers the unique characteristics of COVID-19 to examine the effectiveness of various non-pharmaceutical interventions (NPIs) in New York City (NYC). Numerical experiments from our model show that the control policies implemented in NYC reduced the number of infections by 72% [interquartile range (IQR) 53-95] and the number of deceased cases by 76% (IQR 58-96) by the end of 2020. Among all the NPIs, social distancing for the entire population and protection for the elderly in public facilities is the most effective control measure in reducing severe infections and deceased cases. School closure policy may not work as effectively as one might expect in terms of reducing the number of deceased cases. Our simulation results provide novel insights into the city-specific implementation of NPIs with minimal social disruption considering the locations and population characteristics.

A novel network control model for identifying personalized driver genes in cancer.

Although existing computational models have identified many common driver genes, it remains challenging to identify the personalized driver genes by using samples of an individual patient. Recently, the methods of exploiting the structure-based control principles of complex networks provide new clues for identifying minimum number of driver nodes to drive the state transition of large-scale complex networks from an initial state to the desired state. However, the structure-based network control methods cannot be directly applied to identify the personalized driver genes due to the unknown network dynamics of the personalized system. Here we proposed the personalized network control model (PNC) to identify the personalized driver genes by employing the structure-based network control principle on genetic data of individual patients. In PNC model, we firstly presented a paired single sample network construction method to construct the personalized state transition network for capturing the phenotype transitions between healthy and disease states. Then, we designed a novel structure-based network control method from the Feedback Vertex Sets-based control perspective to identify the personalized driver genes. The wide experimental results on 13 cancer datasets from The Cancer Genome Atlas firstly showed that PNC model outperforms current state-of-the-art methods, in terms of F-measures for identifying cancer driver genes enriched in the gold-standard cancer driver gene lists. Furthermore, these results showed that personalized driver genes can be explored by their network characteristics even when they are hidden factors in transcription and mutation profiles. Our PNC gives novel insights and useful tools into understanding the tumor heterogeneity in cancer. The PNC package and data resources used in this work can be freely downloaded from https://github.com/NWPU-903PR/PNC.

An Integrated Approach for Assessing the Impact of Large-Scale Future Floods on a Highway Transport System.

The negative impact of climate change continues to escalate flood risk. Floods directly and indirectly damage highway systems and disturb the socioeconomic order. In this study, we propose an integrated approach to quantitatively assess how floods impact the functioning of a highway system. The approach has three parts: (1) a multi-agent simulation model to represent traffic, heterogeneous user demand, and route choice in a highway network; (2) a flood simulator using future runoff scenarios generated from five global climate models, three representative concentration pathways (RCPs), and the CaMa-Flood model; and (3) an impact analyzer, which superimposes the simulated floods on the highway traffic simulation system, and quantifies the flood impact on a highway system based on car following model. This approach is illustrated with a case study of the Chinese highway network. The results show that (i) for different global climate models, the associated flood damage to a highway system is not linearly correlated with the forcing levels of RCPs, or with future years; (ii) floods in different years have variable impacts on regional connectivity; and (iii) extreme flood impacts can cause huge damages in highway networks; that is, in 2030, the estimated 84.5% of routes between provinces cannot be completed when the highway system is disturbed by a future major flood. These results have critical implications for transport sector policies and can be used to guide highway design and infrastructure protection. The approach can be extended to analyze other networks with spatial vulnerability, and it is an effective quantitative tool for reducing systemic disaster risk.

Breakdown of interdependent directed networks.

Increasing evidence shows that real-world systems interact with one another via dependency connectivities. Failing connectivities are the mechanism behind the breakdown of interacting complex systems, e.g., blackouts caused by the interdependence of power grids and communication networks. Previous research analyzing the robustness of interdependent networks has been limited to undirected networks. However, most real-world networks are directed, their in-degrees and out-degrees may be correlated, and they are often coupled to one another as interdependent directed networks. To understand the breakdown and robustness of interdependent directed networks, we develop a theoretical framework based on generating functions and percolation theory. We find that for interdependent Erdos-Rényi networks the directionality within each network increases their vulnerability and exhibits hybrid phase transitions. We also find that the percolation behavior of interdependent directed scale-free networks with and without degree correlations is so complex that two criteria are needed to quantify and compare their robustness: the percolation threshold and the integrated size of the giant component during an entire attack process. Interestingly, we find that the in-degree and out-degree correlations in each network layer increase the robustness of interdependent degree heterogeneous networks that most real networks are, but decrease the robustness of interdependent networks with homogeneous degree distribution and with strong coupling strengths. Moreover, by applying our theoretical analysis to real interdependent international trade networks, we find that the robustness of these real-world systems increases with the in-degree and out-degree correlations, confirming our theoretical analysis.

Healthcare system resilience and adaptability to pandemic disruptions in the United States.

Understanding healthcare system resilience has become paramount, particularly in the wake of the COVID-19 pandemic, which imposed unprecedented burdens on healthcare services and severely impacted public health. Resilience is defined as the system's ability to absorb, recover from and adapt to disruptions; however, despite extensive studies on this subject, we still lack empirical evidence and mathematical tools to quantify its adaptability (the ability of the system to adjust to and learn from disruptions). By analyzing millions of patients' electronic medical records across US states, we find that the COVID-19 pandemic caused two successive waves of disruptions within the healthcare systems, enabling natural experiment analysis of the adaptive capacity of each system to adapt to past disruptions. We generalized the quantification framework and found that the US healthcare systems exhibit substantial adaptability (ρ = 0.58) but only a moderate level of resilience (r = 0.70). When considering system responses across racial groups, Black and Hispanic groups were more severely impacted by pandemic disruptions than white and Asian groups. Physician abundance was the key characteristic for determining healthcare system resilience. Our results offer vital guidance in designing resilient and sustainable healthcare systems to prepare for future waves of disruptions akin to COVID-19 pandemics.

Efficient parameter inference in networked dynamical systems via steady states: A surrogate objective function approach integrating mean-field and nonlinear least squares.

In networked dynamical systems, inferring governing parameters is crucial for predicting nodal dynamics, such as gene expression levels, species abundance, or population density. While many parameter estimation techniques rely on time-series data, particularly systems that converge over extreme time ranges, only noisy steady-state data is available, requiring a new approach to infer dynamical parameters from noisy observations of steady states. However, the traditional optimization process is computationally demanding, requiring repeated simulation of coupled ordinary differential equations. To overcome these limitations, we introduce a surrogate objective function that leverages decoupled equations to compute steady states, significantly reducing computational complexity. Furthermore, by optimizing the surrogate objective function, we obtain steady states that more accurately approximate the ground truth than noisy observations and predict future equilibria when topology changes. We empirically demonstrate the effectiveness of the proposed method across ecological, gene regulatory, and epidemic networks. Our approach provides an efficient and effective way to estimate parameters from steady-state data and has the potential to improve predictions in networked dynamical systems.