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One of the core risk management tasks is to identify hidden high-risky states that may lead to system breakdown, which can provide valuable early warning knowledge. However, due to high dimensionality and nonlinear interaction embedded in large-scale complex systems like urban traffic, it remains challenging to identify hidden high-risky states from huge system state space where over 99% of possible system states are not yet visited in empirical data. Based on maximum entropy model, we infer the underlying interaction network from complicated dynamical processes of urban traffic, and construct system energy landscape. In this way, we can locate hidden high-risky states that have never been observed from real data. These states can serve as risk signals with high probability of entering hazardous minima in energy landscape, which lead to huge recovery cost. Our finding might provide insights for complex system risk management.

For real-world complex system constantly enduring perturbation, to achieve survival goal in changing yet unknown environments, the central problem is designing a self-adaptation strategy instead of fixed control strategies, which enables system to adjust its internal multi-scale structure according to environmental feedback. Inspired by thermodynamics, we develop a self-adaptive network utilizing only macroscopic information to achieve desired landscape through reconfiguring itself in unknown environments. By continuously estimating environment entropy, our designed self-adaptive network can adaptively realize desired landscape represented by topological measures. The adaptability of this network is achieved under several scenarios, including confinement on phase space and geographic constraint. The adaptation process is described by relative entropy corresponding to the Boltzmann H function, which decreases with time following unique power law distinguishing our self-adaptive network from memoryless systems. Moreover, we demonstrate the transformability of our self-adaptive network, as a critical mechanism of complex system resilience, allowing for transitions from one target landscape to another. Compared to data-driven methods, our self-adaptive network is understandable without careful choice of learning architecture and parameters. Our designed self-adaptive network could help to understand system intelligence through the lens of thermodynamics.

Urban traffic resilience has gained increased attention, with most studies adopting an engineering perspective that assumes a single optimal equilibrium and prioritizes local recovery. On the other hand, systems may possess multiple metastable states, and ecological resilience is the ability to switch between these states according to perturbations. Control strategies from these two resilience perspectives yield distinct outcomes. In fact, ecological resilience oriented control has rarely been viewed in urban traffic, despite the fact that traffic system is a complex system in highly uncertain environment with possible multiple metastable states. This absence highlights the necessity for urban traffic ecological resilience definition. To bridge this gap, we defines urban traffic ecological resilience as the ability to absorb uncertain perturbations by shifting to alternative states. The goal is to generate a system with greater adaptability, without necessarily returning to the original equilibrium. Our control framework comprises three aspects: portraying the recoverable scopes; designing alternative steady states; and controlling system to shift to alternative steady states for adapting large disturbances. Among them, the recoverable scopes are portrayed by attraction region; the alternative steady states are set close to the optimal state and outside the attraction region of the original equilibrium; the controller needs to ensure the local stability of the alternative steady states, without changing the trajectories inside the attraction region of the original equilibrium. Comparisons with classical engineering resilience oriented urban traffic resilience control schemes show that, proposed ecological resilience oriented control schemes can generate greater resilience. These results will contribute to the fundamental theory of future resilient intelligent transportation system.

Most theoretical analysis for lifetime distribution explains origins of specific distribution based on independent failure. We develop a unified framework encompassing different lifetime distribution for failure-coupled network systems. We find three types of system lifetime distributions emerged from competence between system size N and failure coupling strength $\phi$. System lifetime distribution can be describe by modified Weibull model, which degenerates into Gompertz model when N dominates and exponential distribution when $\phi$ dominates. We derive asymptotic lifetime distribution. Specially, we derive a fundamental equation of thermodynamics for failure-coupled systems. Our study will help design highly reliable systems.

Evaluating node influence is fundamental for identifying key nodes in complex networks. Existing methods typically rely on generic indicators to rank node influence across diverse networks, thereby ignoring the individualized features of each network itself. Actually, node influence stems not only from general features but the multi-scale individualized information encompassing specific network structure and task. Here we design an active learning architecture to predict node influence quantitively and precisely, which samples representative nodes based on graph entropy correlation matrix integrating multi-scale individualized information. This brings two intuitive advantages: (1) discovering potential high-influence but weak-connected nodes that are usually ignored in existing methods, (2) improving the influence maximization strategy by deducing influence interference. Significantly, our architecture demonstrates exceptional transfer learning capabilities across multiple types of networks, which can identify those key nodes with large disputation across different existing methods. Additionally, our approach, combined with a simple greedy algorithm, exhibits dominant performance in solving the influence maximization problem. This architecture holds great potential for applications in graph mining and prediction tasks.

The models of $k$-core percolation and interdependent networks (IN) have been extensively studied in their respective fields. A recent study has revealed that they share several common critical exponents. However, several newly discovered exponents in IN have not been explored in $k$-core percolation, and the origin of the similarity still remains unclear. Here, we investigate k-core percolation in random networks. We find that for k-core percolation,the fractality of the giant component fluctuations is manifested by a fractal fluctuation dimension, $\widetilde d_f = 3/4$, within a correlation \emph{size} $N'$ that scales as $N' \propto (p-p_c)^{-\widetilde\nu}$, with $\widetilde\nu = 2$, same as found in IN. Indeed, here, $\widetilde\nu \equiv d\cdot \nu'$ and $\widetilde{d}_f \equiv d'_f/d$, where $\nu'$ and $d'_f$ are respectively the same as the correlation \emph{length} exponent and the fractal fluctuation dimension observed in $d$-dimensional IN spatial networks. These two new exponents found here for $k$-core percolation demonstrate the same scaling behaviors as found for IN with the same critical exponents, reinforcing the similarity between the two models. Furthermore, we suggest that these two models are similar since both have two types of interactions: short-range (SR) connectivity and long-range (LR) influences. In IN the LR are the influences of dependency links while in k-core we find here that for $k=1$ and $k=2$ the influences are short range while for $k\geq3$ the influence is long range. In addition, analytical arguments for a universal hyper-scaling relation for the fractal fluctuation dimension of the $k$-core giant component and for IN as well as for any mixed-order transition are established.Our analysis enhances the comprehension of k-core percolation and supports the generalization of the concept of fractal fluctuations in mixed-order phase transitions.

Degeneration and adaptation are two competing sides of the same coin called resilience in the progressive processes of brain aging or diseases. Degeneration accumulates during brain aging and other cerebral activities, causing structural atrophy and dysfunction. At the same time, adaptation allows brain network reorganize to compensate for structural loss to maintain cognition function. Although hidden resilience mechanism is critical and fundamental to uncover the brain aging law, due to the lack of datasets and appropriate methodology, it remains essentially unknown how these two processes interact dynamically across brain networks. To quantitatively investigate this complex process, we analyze aging brains based on 6-year follow-up multimodal neuroimaging database from 63 persons. We reveal the critical mechanism of network resilience that various perturbation may cause fast brain structural atrophy, and then brain can reorganize its functional layout to lower its operational efficiency, which helps to slow down the structural atrophy and finally recover its functional efficiency equilibrium. This empirical finding could be explained by our theoretical model, suggesting one universal resilience dynamical function. This resilience is achieved in the brain functional network with evolving percolation and rich-club features. Our findings can help to understand the brain aging process and design possible mitigation methods to adjust interaction between degeneration and adaptation from resilience viewpoint., Comment: 24 pages, 6 figures

The importance of understanding human mobility patterns has led many studies to examine their spatial-temporal scaling laws. These studies mainly reveal that human travel can be highly non-homogeneous with power-law scaling distributions of distances and times. However, investigating and quantifying the extent of variability in time and space when traveling the same air distance has not been addressed so far. Using taxi data from five large cities, we focus on several novel measures of distance and time to explore the spatio-temporal variations of taxi travel routes relative to their typical routes during peak and nonpeak periods. To compare all trips using a single measure, we calculate the distributions of the ratios between actual travel distances and the average travel distance as well as between actual travel times and the average travel time for all origin destinations (OD) during peak and nonpeak periods. In this way, we measure the scaling of the distribution of all single trip paths with respect to their mean trip path. Our results surprisingly demonstrate very broad distributions for both the distance ratio and time ratio, characterized by a long-tail power-law distribution. Moreover, all analyzed cities have larger exponents in peak hours than in nonpeak hours. We suggest that the interesting results of shorter trip lengths and times, characterized by larger exponents during rush hours, are due to the higher availability of travelers in rush hours compared to non-rush hours...

Many systems on our planet are known to shift abruptly and irreversibly from one state to another when they are forced across a "tipping point," such as mass extinctions in ecological networks, cascading failures in infrastructure systems, and social convention changes in human and animal networks. Such a regime shift demonstrates a system's resilience that characterizes the ability of a system to adjust its activity to retain its basic functionality in the face of internal disturbances or external environmental changes. In the past 50 years, attention was almost exclusively given to low dimensional systems and calibration of their resilience functions and indicators of early warning signals without considerations for the interactions between the components. Only in recent years, taking advantages of the network theory and lavish real data sets, network scientists have directed their interest to the real-world complex networked multidimensional systems and their resilience function and early warning indicators. This report is devoted to a comprehensive review of resilience function and regime shift of complex systems in different domains, such as ecology, biology, social systems and infrastructure. We cover the related research about empirical observations, experimental studies, mathematical modeling, and theoretical analysis. We also discuss some ambiguous definitions, such as robustness, resilience, and stability., Comment: Review chapter

Passenger contact in public transit (PT) networks can be a key mediate in the spreading of infectious diseases. This paper proposes a time-varying weighted PT encounter network to model the spreading of infectious diseases through the PT systems. Social activity contacts at both local and global levels are also considered. We select the epidemiological characteristics of coronavirus disease 2019 (COVID-19) as a case study along with smart card data from Singapore to illustrate the model at the metropolitan level. A scalable and lightweight theoretical framework is derived to capture the time-varying and heterogeneous network structures, which enables to solve the problem at the whole population level with low computational costs. Different control policies from both the public health side and the transportation side are evaluated. We find that people's preventative behavior is one of the most effective measures to control the spreading of epidemics. From the transportation side, partial closure of bus routes helps to slow down but cannot fully contain the spreading of epidemics. Identifying "influential passengers" using the smart card data and isolating them at an early stage can also effectively reduce the epidemic spreading.

The new coronavirus known as COVID-19 is spread world-wide since December 2019. Without any vaccination or medicine, the means of controlling it are limited to quarantine and social distancing. Here we study the spatio-temporal propagation of the first wave of the COVID-19 virus in China and compare it to other global locations. We provide a comprehensive picture of the spatial propagation from Hubei to other provinces in China in terms of distance, population size, and human mobility and their scaling relations. Since strict quarantine has been usually applied between cities, more insight about the temporal evolution of the disease can be obtained by analyzing the epidemic within cities, especially the time evolution of the infection, death, and recovery rates which affected by policies. We study and compare the infection rate in different cities in China and provinces in Italy and find that the disease spread is characterized by a two-stages process. At early times, at order of few days, the infection rate is close to a constant probably due to the lack of means to detect infected individuals before infection symptoms are observed. Then at later times it decays approximately exponentially due to quarantines. The time evolution of the death and recovery rates also distinguish between these two stages and reflect the health system situation which could be overloaded.

While the individual travel implicates the trace of individual mobility decision, group travels signify the possible social relationship behind these traces. Different from online social network, spatial interaction between individuals is a critical yet unknown dimension to understand the collective behaviors in a city. In this paper, based on over 127 million trips in Beijing metro network, we develop a method to distinguish the group travel of friends from the encounter travel of familiar strangers. We find travels of friends are among the most predictable groups. These identified friendships are interwoven and form a friendship network, with structural properties different from encounter network. The topological role of individuals in this network is found strongly correlated with her travel predictability. The overall time savings of about 34190 minutes after redistribution of inefficient group traveler with flexible travel purposes shows the potential of designing specific traffic fares for group travel. Our identification and understanding of group travel may help to develop and organize new traffic mode in the future smart transportation., Comment: 21 pages, 5 figures

The concept of resilience can be realized in natural and engineering systems, representing the ability of system to adapt and recover from various disturbances. Although resilience is a critical property needed for understanding and managing the risks and collapses of transportation system, an accepted and useful definition of resilience for urban traffic as well as its statistical property under perturbations is still missing. Here we define city traffic resilience based on the spatio-temporal clusters of congestion in real traffic, and find that the resilience follows a scale free distribution in two-dimensional city road networks and one-dimensional highways, with different exponents, but similar exponents in different days and different cities. The traffic resilience is also revealed to have a novel scaling relation between the cluster size of the spatio-temporal jam and its recovery duration, independent of microscopic details. Our findings of universal traffic resilience can provide indication towards better understanding and designing these complex engineering systems under internal and external disturbances., Comment: 6 pages, 4 figures

Rumor spreading on online social media is presenting a significant threat to society of post-truth epoch. Extensive efforts have been devoted to rumor identification and debunking, assuming that a specific rumor propagation is a single event network and neglecting possible interdependence between different rumor cascades. Here we study the collective propagation of multiple rumors, and surprisingly find a network of users that repeatedly participate in different rumor cascades. Though these repetitive users demonstrate minor difference at the level of single propagation network, they are found to form a significantly more intensive collaboration network from multiple rumor cascades compared to news propagation. The clique-like cluster formed by repetitive rumor spreaders can serve as a high quality feature for rumor identification and blocking targets for rumor prevention. Our findings can provide a better understanding of rumor spread by viewing multiple rumor propagations as one interacting rumor ecosystem, and suggest novel methods for distinguishing and mitigation based on rumor spreading history., Comment: 22 pages, 5 figures

Social media can be a double-edged sword for society, either as a convenient channel exchanging ideas or as an unexpected conduit circulating fake news through a large population. While existing studies of fake news focus on theoretical modeling of propagation or identification methods based on machine learning, it is important to understand the realistic mechanisms between theoretical models and black-box methods. Here we track large databases of fake news and real news in both, Weibo in China and Twitter in Japan from different culture, which include their complete traces of re-postings. We find in both online social networks that fake news spreads distinctively from real news even at early stages of propagation, e.g. five hours after the first re-postings. Our finding demonstrates collective structural signals that help to understand the different propagation evolution of fake news and real news. Different from earlier studies, identifying the topological properties of the information propagation at early stages may offer novel features for early detection of fake news in social media., Comment: 35pages, 6 groups of figures

Despite the wide use of networks as a versatile tool for exploring complex social systems, little is known about how to detect and forecast abrupt changes in social systems. In this report, we develop an early warning approach based on network properties to detect such changes. By analysing three collaborative social networks---one co-stardom, one patent and one scientific collaborative network, we discover that abrupt transitions inherent in these networks can serve as a good early warning signal, indicating, respectively, the dissolution of the Soviet Union, the emergence of the "soft matter" research field, and the merging of two scientific communities. We then develop a clique growth model that explains the universal properties of these real networks and find that they belong to a new universality class, described by the Gumbel distribution.

Networks are designed to satisfy given objectives under specific requirements. While the static connectivity of networks is normally analyzed and corresponding design principles for static robustness are proposed, the challenge still remains of how to design endurable networks that maintain the required level of connectivity during its whole lifespan, against component aging. We introduce network endurance as a new concept to evaluate networks overall performance during its whole lifespan, considering both network connectivity and network duration. We develop a framework for designing an endurable network by allocating the expected lifetimes of its components, given a limited budget. Based on percolation theory and simulation, we find that the maximal network endurance can be achieved with a quantitative balance between network duration and connectivity. For different endurance requirements, we find that the optimal design can be separated into two categories: strong dependence of lifetime on node's degree leads to larger network lifetime, while weak dependence generates stronger network connectivity. Our findings could help network design, by providing a quantitative prediction of network endurance based on network topology.

Percolation transition is widely observed in networks ranging from biology to engineering. While much attention has been paid to network topologies, studies rarely focus on critical percolation phenomena driven by network dynamics. Using extensive real data, we study the critical percolation properties in city traffic dynamics. Our results suggest that two modes of different critical percolation behaviors are switching in the same network topology under different traffic dynamics. One mode of city traffic (during nonrush hours or days off) has similar critical percolation characteristics as small world networks, while the other mode (during rush hours on working days) tends to behave as a 2D lattice. This switching behavior can be understood by the fact that the high-speed urban roads during nonrush hours or days off (that are congested during rush hours) represent effective long-range connections, like in small world networks. Our results might be useful for understanding and improving traffic resilience., Comment: 8 pages, 4 figures, Daqing Li, Ziyou Gao and H. Eugene Stanley are the corresponding authors (daqingl@buaa.edu.cn, zygao@bjtu.edu.cn, hes@bu.edu)

Strengthening or destroying a network is a very important issue in designing resilient networks or in planning attacks against networks including planning strategies to immunize a network against diseases, viruses etc.. Here we develop a method for strengthening or destroying a random network with a minimum cost. We assume a correlation between the cost required to strengthen or destroy a node and the degree of the node. Accordingly, we define a cost function c(k), which is the cost of strengthening or destroying a node with degree k. Using the degrees $k$ in a network and the cost function c(k), we develop a method for defining a list of priorities of degrees, and for choosing the right group of degrees to be strengthened or destroyed that minimizes the total price of strengthening or destroying the entire network. We find that the list of priorities of degrees is universal and independent of the network's degree distribution, for all kinds of random networks. The list of priorities is the same for both strengthening a network and for destroying a network with minimum cost. However, in spite of this similarity there is a difference between their p_c - the critical fraction of nodes that has to be functional, to guarantee the existence of a giant component in the network.

Since the first reported traffic jam about a century ago, traffic congestion has been intensively studied with various methods ranging from macroscopic to microscopic viewpoint. However, due to the population growth and fast civilization, traffic congestion has become significantly worse not only leading to economic losses, but also causes environment damages. Without understanding of jams spatio-temporal propagation behavior in a city, it is impossible to develop efficient mitigation strategies to control and improve city traffic. Although some progress has been made in recent studies based on available traffic data regarding general features of traffic, the understanding of the spatio-temporal propagation of traffic jams in urban traffic is still unclear. Here we study the spatio-temporal propagation behavior of traffic jams based on collected empirical traffic data in big cities. We developed a method to identify influential jam centers and find that jams spread radially from multiple jam centers with a range of velocities. Our findings may help to predict and even control the traffic jam propagation, which could be helpful for the development of future autonomous driving technology and intelligent transportation system.

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 ${\lambda_{2}}$ during SIS (susceptible-infected-susceptible) epidemic propagation to model perturbations (control state), whereas the infection rate is ${\lambda_{1}}$ in the rest of time. Through simulations and theoretical analysis, we find that even for ${\lambda_{2}<\lambda_{c}}$, epidemics eventually could bounce back if control duration is below a threshold. This critical control time for epidemic resilience, i.e., ${cd_{max}}$ can be predicted by the diameter (${d}$) of the underlying network, with the quantitative relation ${cd_{max}\sim d^{\alpha}}$. Our findings can help to design a better mitigation strategy for epidemics., Comment: 10 pages, 5 figures

Recently, with the availability of various traffic datasets, human mobility has been studied in different contexts. Researchers attempt to understand the collective behaviors of human movement with respect to the spatio-temporal distribution in traffic dynamics, from which a gravitational scaling law characterizing the relation between the traffic flow, population and distance has been found. However, most studies focus on the integrated properties of gravitational scaling, neglecting its dynamical evolution during different hours of a day. Investigating the hourly traffic flow data of Beijing subway network, based on the hop-count distance of passengers, we find that the scaling exponent of the gravitational law is smaller in Beijing subway system compared to that reported in Seoul subway system. This means that traffic demand in Beijing is much stronger and less sensitive to the travel distance. Furthermore, we analyzed the temporal evolution of the scaling exponents in weekdays and weekends. Our findings may help to understand and improve the traffic congestion control in different subway systems.

As a dynamical complex system, traffic is characterized by a transition from free flow to congestions, which is mostly studied in highways. However, despite its importance in developing congestion mitigation strategies, the understanding of this common traffic phenomenon in a city-scale is still missing. An open question is how the traffic in the network collapses from a global efficient traffic to isolated local flows in small clusters, i.e. the question of traffic percolation. Here we study the traffic percolation properties on a lattice by simulation of an agent-based model for traffic. A critical traffic volume in this model distinguishes the free-state from congested state of traffic. Our results show that the threshold of traffic percolation decreases with increasing traffic volume and reaches a minimum value at the critical traffic volume. We show that this minimal threshold is the result of longest spatial correlation between traffic flows at the critical traffic volume. These findings may help to develop congestion mitigation strategies in a network view., Comment: The final version(with minor amendments) is pubulished by EPL(http://iopscience.iop.org/article/10.1209/0295-5075/112/38001/meta). It includes 5 pages, 4 figures