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Understanding the resilience of infrastructures such as 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 present 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 this model experiences two distinct continues transitions, one when the cities disconnect from each other and the second when each city breaks apart. 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, while the homogeneous model experience a single transition having a unique phenomenon called textit{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. Here we show that the heterogeneous model does not experience such a phenomenon indicating that critical stretching strongly depends on the network structure.
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The quantitative study of traffic dynamics is crucial to ensure the efficiency of urban transportation networks. The current work investigates the spatial properties of congestion, that is, we aim to characterize the city areas where traffic bottlenecks occur. The analysis of a large amount of real road networks in previous works showed that congestion points experience spatial abrupt transitions, namely they shift away from the city center as larger urban areas are incorporated. The fundamental ingredient behind this effect is the entanglement of central and arterial roads, embedded in separated geographical regions. In this paper we extend the analysis of the conditions yielding abrupt transitions of congestion location. First, we look into the more realistic situation in which arterial and central roads, rather than lying on sharply separated regions, present spatial overlap. It results that this affects the position of bottlenecks and introduces new possible congestion areas. Secondly, we pay particular attention to the role played by the edge distribution, proving that it allows to smooth the transitions profile, and so to control the congestion displacement. Finally, we show that the aforementioned phenomenology may be recovered also as a consequence of a discontinuity in the nodes density, in a domain with uniform connectivity. Our results provide useful insights for the design and optimization of urban road networks, and the management of the daily traffic.
The structure of real-world multilayer infrastructure systems usually exhibits anisotropy due to constraints of the embedding space. For example, geographical features like mountains, rivers and shores influence the architecture of critical infrastructure networks. Moreover, such spatial networks are often non-homogeneous but rather have a modular structure with dense connections within communities and sparse connections between neighboring communities. When the networks of the different layers are interdependent, local failures and attacks may propagate throughout the system. Here we study the robustness of spatial interdependent networks which are both anisotropic and heterogeneous. We also evaluate the effect of localized attacks having different geometrical shapes. We find that anisotropic networks are more robust against localized attacks and that anisotropic attacks, surprisingly, even on isotropic structures, are more effective than isotropic attacks.
We study the effect of localized attacks on a multiplex spatial network, where each layer is a network of communities. The system is considered functional when the nodes belong to the giant component in all the multiplex layers. The communities are of linear size $zeta$, such that within them any pair of nodes are linked with same probability, and additionally nodes in nearby communities are linked with a different (typically smaller) probability. This model can represent an interdependent infrastructure system of cities where within the city there are many links while between cities there are fewer links. We develop an analytical method, similar to the finite element method applied to a network with communities, and verify our analytical results by simulations. We find, both by simulation and theory, that for different parameters of connectivity and spatiality --- there is a critical localized size of damage above which it will spread and the entire system will collapse.
Networks determine our social circles and the way we cooperate with others. We know that topological features like hubs and degree assortativity affect cooperation, and we know that cooperation is favoured if the benefit of the altruistic act divided by the cost exceeds the average number of neighbours. However, a simple rule that would predict cooperation transitions on an arbitrary network has not yet been presented. Here we show that the unique sequence of degrees in a network can be used to predict at which game parameters major shifts in the level of cooperation can be expected, including phase transitions from absorbing to mixed strategy phases. We use the evolutionary prisoners dilemma game on random and scale-free networks to demonstrate the prediction, as well as its limitations and possible pitfalls. We observe good agreements between the predictions and the results obtained with concurrent and Monte Carlo methods for the update of the strategies, thus providing a simple and fast way to estimate the outcome of evolutionary social dilemmas on arbitrary networks without the need of actually playing the game.
We study the evolution of cooperation in spatial Prisoners dilemma games with and without extortion by adopting aspiration-driven strategy updating rule. We focus explicitly on how the strategy updating manner (whether synchronous or asynchronous) and also the introduction of extortion strategy affect the collective outcome of the games. By means of Monte Carlo (MC) simulations as well as dynamical cluster techniques, we find that the involvement of extortioners facilitates the boom of cooperators in the population (and whom can always dominate the population if the temptation to defect is not too large) for both synchronous and asynchronous strategy updating, in stark contrast to the otherwise case, where cooperation is promoted for intermediate aspiration level with synchronous strategy updating, but is remarkably inhibited if the strategy updating is implemented asynchronously. We explain the results by configurational analysis and find that the presence of extortion leads to the checkerboard-like ordering of cooperators and extortioners, which enable cooperators to prevail in the population with both strategy updating manners. Moreover, extortion itself is evolutionary stable, and therefore acts as the incubator for the evolution of cooperation.
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