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We study a realistic spatial network model constructed by randomly linking lattice sites with linklengths following an exponential distribution with a characteristic scale ζ. We find that this simple spatial network topology does not fulfill any single universality class, but exhibits a new multiuniversality with two sets of critical exponents. This bi-universality is characterized by random-like scaling laws for measurements on a scale smaller than ζ but spatial scaling for measurements on a larger scale. We further explore this topology by studying the resilience of a two-layer multiplex under localized attack. We find that for a broad range of the control parameters, our system is metastable. In this metastable region, a localized attack larger than a critical size -that does not depends on the size of the system -induces a propagating cascade of failures leading to the system collapse.

Understanding the resilience of infrastructures, such as a transportation network, has significant importance for our daily life. Recently, a homogeneous spatial network model was developed for studying spatial embedded networks with characteristic link length such as power-grids and the brain. However, although many real-world networks are spatially embedded and their links have characteristics length such as pipelines, power lines or ground transportation lines they are not homogeneous but rather heterogeneous. For example, density of links within cities are significantly higher than between cities. Here we develop and study numerically and analytically a similar realistic heterogeneous spatial modular model using percolation process to better understand the effect of heterogeneity on such networks. The model assumes that inside a city there are many lines connecting different locations, while long lines between the cities are sparse and usually directly connecting only a few nearest neighbours cities in a two dimensional plane. We find that our heterogeneous model experiences two distinct continuous transitions, one when the cities disconnect from each other and the second when each city breaks apart. This is in contrast to the homogeneous model where a single transition is found. Although the critical threshold for site percolation in 2D grid remains an open question we analytically find the critical threshold for site percolation in this model. In addition, it has been found that the homogeneous model experience a single transition having a unique phenomenon called critical stretching where a geometric crossover from random to spatial structure in different scales found to stretch non-linearly with the characteristic length at criticality. In marked contrast, we show here that the heterogeneous model does not experience such a phenomenon indicating that critical stretching strongly depends on the network structure.

We study the effect of localized attacks on a multiplex network, where each layer is a network of communities embedded in space. We assume that nodes are densely connected within a community and sparsely connected to the nodes in the neighboring communities. To investigate percolation processes in this realistic system we develop an analytical scheme, applying the finite-element method. We find, both by simulation and theory, that in many cases there is a critical size of localized damage above which it will spread and the entire system will collapse. In addition, we show that for a constant number of links, networks with less connectivity between communities are surprisingly more robust.

Réseaux spatiaux Réseaux de réseaux Réseaux couplés Résilience des infrastructuresMany complex networks have recently been recognized to involve significant interdependence between different systems. Motivation comes primarily from infrastructures like power grids and communications networks, but also includes areas such as the human brain and finance. Interdependence implies that when components in one system fail, they lead to failures in the same system or other systems. This can then lead to additional failures finally resulting in a long cascade that can cripple the entire system. Furthermore, many of these networks, in particular infrastructure networks, are embedded in space and thus have unique spatial properties that significantly decrease their resilience to failures.Here we present a review of novel results on interdependent spatial networks and how cascading processes are affected by spatial embedding. We include various aspects of spatial embedding such as cases where dependencies are spatially restricted and localized attacks on nodes contained in some spatial region of the network. In general, we find that spatial networks are more vulnerable when they are interdependent and that they are more likely to undergo abrupt failure transitions than interdependent non-embedded networks. We also present results on recovery in spatial networks, the nature of cascades due to overload failures in these networks, and some examples of percolation features found in real-world traffic networks. Finally, we conclude with an outlook on future possible research directions in this area.