Towards designing robust coupled networks

Natural and technological interdependent systems have been shown to be highly vulnerable due to cascading failures and an abrupt collapse of global connectivity under initial failure. Mitigating the risk by partial disconnection endangers their functionality. Here we propose a systematic strategy of selecting a minimum number of autonomous nodes that guarantee a smooth transition in robustness. Our method which is based on betweenness is tested on various examples including the famous 2003 electrical blackout of Italy. We show that, with this strategy, the necessary number of autonomous nodes can be reduced by a factor of five compared to a random choice. We also find that the transition to abrupt collapse follows tricritical scaling characterized by a set of exponents which is independent on the protection strategy

A New Analysis Method for Simulations Using Node Categorizations

Most research concerning the influence of network structure on phenomena taking place on the network focus on relationships between global statistics of the network structure and characteristic properties of those phenomena, even though local structure has a significant effect on the dynamics of some phenomena. In the present paper, we propose a new analysis method for phenomena on networks based on a categorization of nodes. First, local statistics such as the average path length and the clustering coefficient for a node are calculated and assigned to the respective node. Then, the nodes are categorized using the self-organizing map (SOM) algorithm. Characteristic properties of the phenomena of interest are visualized for each category of nodes. The validity of our method is demonstrated using the results of two simulation models. The proposed method is useful as a research tool to understand the behavior of networks, in particular, for the large-scale networks that existing visualization techniques cannot work well.Comment: 9 pages, 8 figures. This paper will be published in Social Network Analysis and Mining(www.springerlink.com

Inter-similarity between coupled networks

Recent studies have shown that a system composed from several randomly interdependent networks is extremely vulnerable to random failure. However, real interdependent networks are usually not randomly interdependent, rather a pair of dependent nodes are coupled according to some regularity which we coin inter-similarity. For example, we study a system composed from an interdependent world wide port network and a world wide airport network and show that well connected ports tend to couple with well connected airports. We introduce two quantities for measuring the level of inter-similarity between networks (i) Inter degree-degree correlation (IDDC) (ii) Inter-clustering coefficient (ICC). We then show both by simulation models and by analyzing the port-airport system that as the networks become more inter-similar the system becomes significantly more robust to random failure.Comment: 4 pages, 3 figure

The extreme vulnerability of interdependent spatially embedded networks

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

Limited path percolation in complex networks

We study the stability of network communication after removal of $q=1-p$ links under the assumption that communication is effective only if the shortest path between nodes $i$ and $j$ after removal is shorter than $a\ell_{ij} (a\geq1)$ where $\ell_{ij}$ is the shortest path before removal. For a large class of networks, we find a new percolation transition at $\tilde{p}_c=(\kappa_o-1)^{(1-a)/a}$, where $\kappa_o\equiv /$ and $k$ is the node degree. Below $\tilde{p}_c$, only a fraction $N^{\delta}$ of the network nodes can communicate, where $\delta\equiv a(1-|\log p|/\log{(\kappa_o-1)}) < 1$, while above $\tilde{p}_c$, order $N$ nodes can communicate within the limited path length $a\ell_{ij}$. Our analytical results are supported by simulations on Erd\H{o}s-R\'{e}nyi and scale-free network models. We expect our results to influence the design of networks, routing algorithms, and immunization strategies, where short paths are most relevant.Comment: 11 pages, 3 figures, 1 tabl

Correlated multiplexity and connectivity of multiplex random networks

Nodes in a complex networked system often engage in more than one type of interactions among them; they form a multiplex network with multiple types of links. In real-world complex systems, a node's degree for one type of links and that for the other are not randomly distributed but correlated, which we term correlated multiplexity. In this paper we study a simple model of multiplex random networks and demonstrate that the correlated multiplexity can drastically affect the properties of giant component in the network. Specifically, when the degrees of a node for different interactions in a duplex Erdos-Renyi network are maximally correlated, the network contains the giant component for any nonzero link densities. In contrast, when the degrees of a node are maximally anti-correlated, the emergence of giant component is significantly delayed, yet the entire network becomes connected into a single component at a finite link density. We also discuss the mixing patterns and the cases with imperfect correlated multiplexity.Comment: Revised version, 12 pages, 6 figure

Percolation in the classical blockmodel

Classical blockmodel is known as the simplest among models of networks with community structure. The model can be also seen as an extremely simply example of interconnected networks. For this reason, it is surprising that the percolation transition in the classical blockmodel has not been examined so far, although the phenomenon has been studied in a variety of much more complicated models of interconnected and multiplex networks. In this paper we derive the self-consistent equation for the size the global percolation cluster in the classical blockmodel. We also find the condition for percolation threshold which characterizes the emergence of the giant component. We show that the discussed percolation phenomenon may cause unexpected problems in a simple optimization process of the multilevel network construction. Numerical simulations confirm the correctness of our theoretical derivations.Comment: 7 pages, 6 figure

Full Connectivity: Corners, edges and faces

We develop a cluster expansion for the probability of full connectivity of high density random networks in confined geometries. In contrast to percolation phenomena at lower densities, boundary effects, which have previously been largely neglected, are not only relevant but dominant. We derive general analytical formulas that show a persistence of universality in a different form to percolation theory, and provide numerical confirmation. We also demonstrate the simplicity of our approach in three simple but instructive examples and discuss the practical benefits of its application to different models.Comment: 28 pages, 8 figure

Investigating the topology of interacting networks - Theory and application to coupled climate subnetworks

Network theory provides various tools for investigating the structural or functional topology of many complex systems found in nature, technology and society. Nevertheless, it has recently been realised that a considerable number of systems of interest should be treated, more appropriately, as interacting networks or networks of networks. Here we introduce a novel graph-theoretical framework for studying the interaction structure between subnetworks embedded within a complex network of networks. This framework allows us to quantify the structural role of single vertices or whole subnetworks with respect to the interaction of a pair of subnetworks on local, mesoscopic and global topological scales. Climate networks have recently been shown to be a powerful tool for the analysis of climatological data. Applying the general framework for studying interacting networks, we introduce coupled climate subnetworks to represent and investigate the topology of statistical relationships between the fields of distinct climatological variables. Using coupled climate subnetworks to investigate the terrestrial atmosphere's three-dimensional geopotential height field uncovers known as well as interesting novel features of the atmosphere's vertical stratification and general circulation. Specifically, the new measure "cross-betweenness" identifies regions which are particularly important for mediating vertical wind field interactions. The promising results obtained by following the coupled climate subnetwork approach present a first step towards an improved understanding of the Earth system and its complex interacting components from a network perspective