Yehiel Berezin

The extreme vulnerability of interdependent spatially embedded networks

Networks of networks are vulnerable: a failure in one sub-network can bring the rest crashing down. Previous simulations have suggested that randomly positioned networks might offer some limited robustness under certain circumstances. Analysis now shows, however, that real-world interdependent networks, where nodes are positioned according to geographical constraints, might not be so resilient.

Percolation transition in dynamical traffic network with evolving critical bottlenecks

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

Localized attacks on spatially embedded networks with dependencies

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.Many real world complex systems such as infrastructure, communication and transportation networks are embedded in space, where entities of one system may depend on entities of other systems. These systems are subject to geographically localized failures due to malicious attacks or natural disasters. Here we study the resilience of a system composed of two interdependent spatially embedded networks to localized geographical attacks. We find that if an attack is larger than a finite (zero fraction of the system) critical size, it will spread through the entire system and lead to its complete collapse. If the attack is below the critical size, it will remain localized. In contrast, under random attack a finite fraction of the system needs to be removed to initiate system collapse. We present both numerical simulations and a theoretical approach to analyze and predict the effect of local attacks and the critical attack size. Our results demonstrate the high risk of local attacks on interdependent spatially embedded infrastructures and can be useful for designing more resilient systems.

Stability of Climate Networks with Time

The pattern of local daily fluctuations of climate fields such as temperatures and geopotential heights is not stable and hard to predict. Surprisingly, we find that the observed relations between such fluctuations in different geographical regions yields a very robust network pattern that remains highly stable during time. Using a new systematic methodology we track the origins of the network stability. It is found that about half of this network stability is due to the spatial 2D embedding of the network, and half is due to physical coupling between climate in different locations. We also find that around the equator, the contribution of the physical coupling is significantly less pronounced compared to off–equatorial regimes. Finally, we show that there is a gradual monotonic modification of the network pattern as a function of altitude difference.

Robustness of a network formed of spatially embedded networks.

We present analytic and numeric results for percolation in a network formed of interdependent spatially embedded networks. We show results for a treelike and a random regular network of networks each with (i) unconstrained dependency links and (ii) dependency links restricted to a maximum Euclidean length r. Analytic results are given for each network of networks with spatially unconstrained dependency links and compared to simulations. For the case of two fully interdependent spatially embedded networks it was found [Li et al., Phys. Rev. Lett. 108, 228702 (2012)] that the system undergoes a first-order phase transition only for r>r(c) ≈ 8. We find here that for treelike networks of networks (composed of n networks) r(c) significantly decreases as n increases and rapidly (n ≥ 11) reaches its limiting value of 1. For cases where the dependencies form loops, such as in random regular networks, we show analytically and confirm through simulations that there is a certain fraction of dependent nodes, q(max), above which the entire network structure collapses even if a single node is removed. The value of q(max) decreases quickly with m, the degree of the random regular network of networks. Our results show the extreme sensitivity of coupled spatial networks and emphasize the susceptibility of these networks to sudden collapse. The theory proposed here requires only numerical knowledge about the percolation behavior of a single network and therefore can be used to find the robustness of any network of networks where the profile of percolation of a singe network is known numerically.

Climate network structure evolves with North Atlantic Oscillation phases

We construct a network from climate records of temperature and geopotential-height in two pressure levels at different geographical sites in the North Atlantic. A link between two sites represents the cross-correlations between the records of each site. We find that within the different phases of the North Atlantic Oscillation (NAO) the correlation values of the links in the climate network are significantly different. By setting an optimized threshold on the correlation values, we find that the number of strong links in the network increases during times of positive NAO indices, and decreases during times of negative NAO indices. We find a pronounced sensitivity of the network structure to the NAO oscillations which is significantly higher compared to the observed response of spatial average of the climate records. Our results suggest a new measure that tracks the NAO pattern. Copyright c EPLA, 2012 Introduction. - A network approach has recently been applied in order to follow climate dynamics (1,2). The nodes of the climate network are geographical sites. The dynamics recorded in each site is composed of its intrinsic dynamics and the coupling with the dynamics of other sites. The cross-correlations due to the coupling between the dynamics in two different sites are represented in our network by a link between the sites (see (3) for a lab experiment that demonstrates the relation between the coupling and the correlation). The maximum value of the correlation might appear with a time delay between the two data records. The climate network approach has recently led to the discovery of several novel insights related to El-Nino dynamics (4-12).

Percolation and cascade dynamics of spatial networks with partial dependency

Recently, it has been shown that the removal of a random fraction of nodes from a system of interdependent spatial networks can lead to cascading failures which amplify the original damage and destroy the entire system, often via abrupt first-order transitions. For these distinctive phenomena to emerge, the interdependence between networks need not be total. We consider here a system of partially interdependent spatial networks (modelled as lattices) with a fraction q of the nodes interdependent and the remaining 1 − q autonomous. In our model, the dependency links between networks are of geometric length less than r. Under full dependency (q = 1), this system was shown to have a first-order percolation transition if r > rc ≈ 8. Here, we generalize this result and show that for all q > 0, there will be a first-order transition if r > rc(q). We show that rc(q) increases monotonically with decreasing q and limq→0+ rc(q) = ∞. Additionally, we present a detailed description and explanation of the cascading failures in spatially embedded interdependent networks near the percolation threshold pc. These failures follow three mechanisms depending on the value of r. Below rc the system undergoes a continuous transition similar to standard percolation on a lattice. Above rc there are two distinct first-order transitions for finite and infinite r, respectively. The cascading failure for finite r is characterized by the emergence of a critical hole which then spreads through the system while the infinite r transition is more similar to the case of random networks. Surprisingly, we find that this spreading transition can still occur even if p < pc. We present measurements of cascade dynamics which differentiate between these phase transitions and elucidate their mechanisms. These results extend previous research on spatial networks to the more realistic case of partial dependency and shed new light on the specific dynamics of dependencydriven cascading failures.

An Introduction to Interdependent Networks

Many real-world phenomena can be modelled using networks. Often, these networks interact with one another in non-trivial ways. Recently, a theory of interdependent networks has been developed which describes dependency between nodes across networks. Interdependent networks have a number of unique properties which are absent in single networks. In particular, systems of interdependent networks often undergo abrupt first-order percolation transitions induced by cascading failures. Here we present an overview of recent developments and significant findings regarding interdependent networks and networks of networks.

Interdependent Spatially Embedded Networks: Dynamics at Percolation Threshold

Spatially embedded systems of interdependent networks with full dependency (q=1) have been found to have a first-order percolation transition if the dependency link length (the maximum distance in lattice units between a node in one network and the node that it depends on in another network) is longer than a certain critical length rc ≈ 8. We find here that a similar result is valid for any finite value of q with a larger rc as q decreases. We also provide a theoretical approach which correctly predicts the relationship between rc and q. We also examine the dynamics at the percolation threshold pc for varying r and q and find that there are three different mechanisms of failure for every q depending on r. Below rc the system undergoes a continuous transition similar to standard percolation. Above rc there are two distinct first-order transitions for finite or infinite r, respectively. The transition for finite r is characterized by spreading of node failures through the system while the infinite r corresponds to a non-spatial cascading failure similar to the case of random networks. These results extend previous results on spatially embedded interdependent networks to the more realistic cases of partial dependency and shed new light on the specific dynamics of cascading failures in such systems.

The effect of spatiality on multiplex networks

Multilayer infrastructure is often interdependent, with nodes in one layer depending on nearby nodes in another layer to function. The links in each layer are often of limited length, due to the construction cost of longer links. Here, we model such systems as a multiplex network composed of two or more layers, each with links of characteristic geographic length, embedded in 2-dimensional space. This is equivalent to a system of interdependent spatially embedded networks in two dimensions in which the connectivity links are constrained in length but varied while the length of the dependency links is always zero. We find two distinct percolation transition behaviors depending on the characteristic length, ζ, of the links. When ζ is longer than a certain critical value, ζc, abrupt, first-order transitions take place, while for ζ < ζc the transition is continuous. We show that, though in single-layer networks increasing ζ decreases the percolation threshold pc, in multiplex networks it has the opposite effect: increasing pc to a maximum at ζ = ζc. By providing a more realistic topological model for spatially embedded interdependent and multiplex networks and highlighting its similarities to lattice-based models, we provide a new direction for more detailed future studies.Many multiplex networks are embedded in space, with links more likely to exist between nearby nodes than distant nodes. For example, interdependent infrastructure networks can be represented as multiplex networks, where each layer has links among nearby nodes. Here, we model the effect of spatiality on the robustness of a multiplex network embedded in 2-dimensional space, where links in each layer are of variable but constrained length. Based on empirical measurements of real-world networks, we adopt exponentially distributed link lengths with characteristic length ζ. By changing ζ, we modulate the strength of the spatial embedding. When ζ → ∞, all link lengths are equally likely, and the spatiality does not affect the topology. However, when only short links are allowed, and the topology is overwhelmingly determined by the spatial embedding. We find that, though longer links strengthen a single-layer network, they make a multi-layer network more vulnerable. We further find that when ζ is longer than a certain critical value, , abrupt, discontinuous transitions take place, while for the transition is continuous, indicating that the risk of abrupt collapse can be eliminated if the typical link length is shorter than .