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Register a new userThe extreme vulnerability of interdependent spatially embedded networks
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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.
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We propose a novel algorithm that outputs the final standings of a soccer league, based on a simple dynamics that mimics a soccer tournament. In our model, a team is created with a defined potential(ability) which is updated during the tournament according to the results of previous games. The updated potential modifies a teams future winning/losing probabilities. We show that this evolutionary game is able to reproduce the statistical properties of final standings of actual editions of the Brazilian tournament (Brasileir~{a}o). However, other leagues such as the Italian and the Spanish tournaments have notoriously non-Gaussian traces and cannot be straightforwardly reproduced by this evolutionary non-Markovian model. A complete understanding of these phenomena deserves much more attention, but we suggest a simple explanation based on data collected in Brazil: Here several teams were crowned champion in previous editions corroborating that the champion typically emerges from random fluctuations that partly preserves the gaussian traces during the tournament. On the other hand, in the Italian and Spanish leagues only a few teams in recent history have won their league tournaments. These leagues are based on more robust and hierarchical structures established even before the beginning of the tournament. For the sake of completeness, we also elaborate a totally Gaussian model (which equalizes the winning, drawing, and losing probabilities) and we show that the scores of the Brasileir~{a}o cannot be reproduced. Such aspects stress that evolutionary aspects are not superfluous in our modeling. Finally, we analyse the distortions of our model in situations where a large number of teams is considered, showing the existence of a transition from a single to a double peaked histogram of the final classification scores. An interesting scaling is presented for different sized tournaments.
Cascading failures constitute an important vulnerability of interconnected systems. Here we focus on the study of such failures on networks in which the connectivity of nodes is constrained by geographical distance. Specifically, we use random geometric graphs as representative examples of such spatial networks, and study the properties of cascading failures on them in the presence of distributed flow. The key finding of this study is that the process of cascading failures is non-self-averaging on spatial networks, and thus, aggregate inferences made from analyzing an ensemble of such networks lead to incorrect conclusions when applied to a single network, no matter how large the network is. We demonstrate that this lack of self-averaging disappears with the introduction of a small fraction of long-range links into the network. We simulate the well studied preemptive node removal strategy for cascade mitigation and show that it is largely ineffective in the case of spatial networks. We introduce an altruistic strategy designed to limit the loss of network nodes in the event of a cascade triggering failure and show that it performs better than the preemptive strategy. Finally, we consider a real-world spatial network viz. a European power transmission network and validate that our findings from the study of random geometric graphs are also borne out by simulations of cascading failures on the empirical network.
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.
Computer viruses are evolving by developing spreading mechanisms based on the use of multiple vectors of propagation. The use of the social network as an extra vector of attack to penetrate the security measures in IP networks is improving the effectiveness of malware, and have therefore been used by the most aggressive viruses, like Conficker and Stuxnet. In this work we use interdependent networks to model the propagation of these kind of viruses. In particular, we study the propagation of a SIS model on interdependent networks where the state of each node is layer-independent and the dynamics in each network follows either a contact process or a reactive process, with different propagation rates. We apply this study to the case of existing multilayer networks, namely a Spanish scientific community of Statistical Physics, formed by a social network of scientific collaborations and a physical network of connected computers in each institution. We show that the interplay between layers increases dramatically the infectivity of viruses in the long term and their robustness against immunization.
In real networks complex topological features are often associated with a diversity of interactions as measured by the weights of the links. Moreover, spatial constraints may as well play an important role, resulting in a complex interplay between topology, weight, and geography. In order to study the vulnerability of such networks to intentional attacks, these attributes must be therefore considered along with the topological quantities. In order to tackle this issue, we consider the case of the world-wide airport network, which is a weighted heterogeneous network whose evolution and structure are influenced by traffic and geographical constraints. We first characterize relevant topological and weighted centrality measures and then use these quantities as selection criteria for the removal of vertices. We consider different attack strategies and different measures of the damage achieved in the network. The analysis of weighted properties shows that centrality driven attacks are capable to shatter the networks communication or transport properties even at very low level of damage in the connectivity pattern. The inclusion of weight and traffic therefore provides evidence for the extreme vulnerability of complex networks to any targeted strategy and need to be considered as key features in the finding and development of defensive strategies.
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