A critical phenomenon is an intrinsic feature of traffic dynamics, during which transition between isolated local flows and global flows occurs. However, very little attention has been given to the question of how the local flows in the roads are organized collectively into a global city flow. Here we characterize this organization process of traffic as "traffic percolation," where the giant cluster of local flows disintegrates when the second largest cluster reaches its maximum. We find in real-time data of city road traffic that global traffic is dynamically composed of clusters of local flows, which are connected by bottleneck links. This organization evolves during a day with different bottleneck links appearing in different hours, but similar in the same hours in different days. A small improvement of critical bottleneck roads is found to benefit significantly the global traffic, providing a method to improve city traffic with low cost. Our results may provide insights on the relation between traffic dynamics and percolation, which can be useful for efficient transportation, epidemic control, and emergency evacuation.emergence | percolation | traffic T raffic, as a large-scale and complex dynamical system, has attracted much attention, especially on its dynamical transition between free flow and congestion (1-3). The dynamics of traffic have been studied using many types of models (4-11), ranging from models in macroscopic scales based on the kinetic gas theory or fluid dynamics to approaches in microscopic scales with equations for each car in the network. However, there is still a gap between the microscopic behavior of individual vehicles and the emergence of macroscopic city traffic. Indeed, a fundamental question has rarely been addressed: how the local flows in roads interact and organize collectively into global flow throughout the city network. This knowledge is not only necessary to bridge the gap between local traffic and global traffic, but also essential for developing efficient traffic control strategies.There are mainly two obstacles in studying how the collective network dynamics of real traffic emerge from local flows. The first obstacle is the lack of valid methods to quantify the dynamical organization of traffic in the road network. The second is the lack of data on traffic dynamics in a network scale. To overcome the first obstacle, we develop here a quantitative framework based on percolation theory, which combines evolving traffic dynamics with network structure. In this framework, instead of the commonly used structural topology, only roads in the network with speed larger than a variable threshold are considered functionally connected. In this way, we can characterize and understand the formation process of traffic dynamics.To overcome the second obstacle of missing data on a network scale and understand the organization processes of real traffic in a network, we collected and analyzed velocities of more than 1,000 roads with 5-min segments records measured in a road network in a cen...

Recent studies show that in interdependent networks a very small failure in one network may lead to catastrophic consequences. Above a critical fraction of interdependent nodes, even a single node failure can invoke cascading failures that may abruptly fragment the system, while below this "critical dependency" (CD) a failure of few nodes leads only to small damage to the system. So far, the research has been focused on interdependent random networks without space limitations. However, many real systems, such as power grids and the Internet, are not random but are spatially embedded. Here we analytically and numerically analyze the stability of systems consisting of interdependent spatially embedded networks modeled as lattice networks. Surprisingly, we find that in lattice systems, in contrast to non-embedded systems, there is no CD and \textit{any} small fraction of interdependent nodes leads to an abrupt collapse. We show that this extreme vulnerability of very weakly coupled lattices is a consequence of the critical exponent describing the percolation transition of a single lattice. Our results are important for understanding the vulnerabilities and for designing robust interdependent spatial embedded networks.Comment: 13 pages, 5 figure

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

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

The connectivity pattern of networks based on ground level temperature records shows a dense stripe of links in the extra tropics of the southern hemisphere. We show that statistical categorization of these links yields a clear association with the pattern of an atmospheric Rossby wave, one of the major mechanisms associated with the weather system and with planetary scale energy transport. It is shown that alternating densities of negative and positive links are arranged in half Rossby wave distances around 3500, 7000, and 10 000 km and are aligned with the expected direction of energy flow, distribution of time delays, and the seasonality of these waves. In addition, long distance links that are associated with Rossby waves are the most dominant in the climate network.