Complex networks have been studied intensively for a decade, but research still focuses on the limited case of a single, non-interacting network. Modern systems are coupled together and therefore should be modelled as interdependent networks. A fundamental property of interdependent networks is that failure of nodes in one network may lead to failure of dependent nodes in other networks. This may happen recursively and can lead to a cascade of failures. In fact, a failure of a very small fraction of nodes in one network may lead to the complete fragmentation of a system of several interdependent networks. A dramatic real-world example of a cascade of failures ('concurrent malfunction') is the electrical blackout that affected much of Italy on 28 September 2003: the shutdown of power stations directly led to the failure of nodes in the Internet communication network, which in turn caused further breakdown of power stations. Here we develop a framework for understanding the robustness of interacting networks subject to such cascading failures. We present exact analytical solutions for the critical fraction of nodes that, on removal, will lead to a failure cascade and to a complete fragmentation of two interdependent networks. Surprisingly, a broader degree distribution increases the vulnerability of interdependent networks to random failure, which is opposite to how a single network behaves. Our findings highlight the need to consider interdependent network properties in designing robust networks.

We study a system composed from two interdependent networks A and B, where a fraction of the nodes in network A depends on the nodes of network B and a fraction of the nodes in network B depends on the nodes of network A. Due to the coupling between the networks when nodes in one network fail they cause dependent nodes in the other network to also fail. This invokes an iterative cascade of failures in both networks. When a critical fraction of nodes fail the iterative process results in a percolation phase transition that completely fragments both networks. We show both analytically and numerically that reducing the coupling between the networks leads to a change from a first order percolation phase transition to a second order percolation transition at a critical point. The scaling of the percolation order parameter near the critical point is characterized by the critical exponent β = 1.Most of the research on networks has concentrated on the limited case of a single network [1][2][3][4][5] while real world systems are composed from many interdependent networks that interact with one another [6][7][8]. As a real example , consider a power-network and an Internet communication network that are coupled together. The Internet nodes depend on the power stations for electricity while the power stations depend on the Internet for control [9].We show that introducing interactions between networks is analogous to introducing interactions among molecules in the ideal gas model. Interactions among molecules lead to the replacement of the ideal gas law by the Van der Waals equation that predicts a liquid-gas first order phase transitions line ending at a critical point characterized by a second order transition ( Fig.1(a)). Similarly, interactions between networks give rise to a first order percolation phase transition line that changes to a second order transition, as the coupling strength between the networks is reduced (Fig.1(b)). At the critical point the first order line merges with the second order line, near which the order parameter (the size of giant component) scales linearly with the distance to the critical point, leading to the critical exponent β = 1.In interdependent networks, nodes from one network depend on nodes from another network. Consequently, when nodes from one network fail they cause nodes from another network to also fail. If the connections within each network are different, this may trigger a recursive process of a cascade of failures that can completely fragments both networks. Recently, Buldyrev et al [10] studied the coupling between two N node networks A and B assuming the following restrictions: (i) Each and every node in network A depends on one node from network B and vice versa. FIG. 1: (a)The van der Waals phase diagram. Along the liquid-gas equilibrium line the order parameter (density) abruptly changes from a low value in the gas phase to a high value in the liquid phase. At the critical point(Pc, Tc) the order parameter changes continuously as function of temperature if the pressur...

Current network models assume one type of links to define the relations between the network entities. However, many real networks can only be correctly described using two different types of relations. Connectivity links that enable the nodes to function cooperatively as a network and dependency links that bind the failure of one network element to the failure of other network elements. Here we present an analytical framework for studying the robustness of networks that include both connectivity and dependency links. We show that a synergy exists between the failure of connectivity and dependency links that leads to an iterative process of cascading failures that has a devastating effect on the network stability. We present exact analytical results for the dramatic change in the network behavior when introducing dependency links. For a high density of dependency links, the network disintegrates in a form of a first-order phase transition, whereas for a low density of dependency links, the network disintegrates in a second-order transition. Moreover, opposed to networks containing only connectivity links where a broader degree distribution results in a more robust network, when both types of links are present a broad degree distribution leads to higher vulnerability.percolation | critical phenomena | complex networks | scale free networks | Erdős-Rényi networks M any friendships between individuals in a social network, numerous business connections in a financial network, or multiple cables between Internet routers are all examples of networks with a high density of connectivity links (1-11). Such networks are regarded as very stable to attacks because, even after a failure of many nodes, the network still remains connected. In contrast, dependencies between the network nodes endanger the network stability because the failure of several nodes may lead to the immediate failure of many others. As an example, consider a financial network: Each company has trading and sales connections with other companies (connectivity links). These connections enable the companies to interact with each other and function together as a global financial market. But there are also dependencies relations between companies; several companies that belong to the same owner depend on one another. If one company fails, the owner might not be able to finance the other companies that will fail too. Such dependencies jeopardize the network stability and are the possible cause of many major financial crises. Another example is an online social network (Facebook or Twitter): Each individual communicates with his friends (connectivity links), thus forming a social network through which information and rumors can spread. However, many individuals will only participate in a social network if other individuals with common interests also participate (dependency links) in that social network.The effect of failing nodes on the network stability has been studied separately for networks containing only connectivity links (12-17) and for networks containi...

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.Recently, an American Congressional Committee highlighted the intensified risk in an attack on national infrastructures, due to the growing interdependencies between different infrastructures [1]. However, despite the high significance and relevance of the subject, only a few studies on interdependent networks exist and these usually focus on the analyses of specific real network data [2][3][4][5][6]. The limited progress is mainly due to the absence of theoretical tools for analyzing interdependent systems. Very recent studies [7,8] present for the first time a framework for studying interdependency between networks and show that such interdependencies significantly increases the vulnerability of the networks to random attack. In these studies, the dependencies between the networks are assumed to be completely random, i.e., a randomly selected node from network A is connected and depends on a randomly selected node from network B and vice versa. Due to the dependencies an initial failure of even a small fraction of nodes from one network can lead to an iterative process of failures that can completely fragment both networks.However, the restriction of random interdependencies is a strong assumption that usually does not occur in many real interdependent systems. As a first example consider the two infrastructures that are mentioned both in the committee report [1] and in the studies discussed above [2,7,8]: The Italian power grid and SCADA communication networks. A power node depends on a communication node for control while a communication node depends on a power node for electricity. It is highly unlikely that a central (high degree) communication node will depend on a small (low degree) power node. Rather, it is much more common that a central communication node depends on a central power station. Moreover, coupled networks usually also poses some similarity in structure, for instance, an area that is overpopulated is bound to have many power stations as well as many communication nodes. Another real example is the world wide port and airport networks that we study in this manuscript.We find that well connected ports tend to couple to wel...

We derive an analytical expression for the critical infection rate r{c} of the susceptible-infectious-susceptible (SIS) disease spreading model on random networks. To obtain r{c}, we first calculate the probability of reinfection π, defined as the probability of a node to reinfect the node that had earlier infected it. We then derive r{c} from π using percolation theory. We show that π is governed by two effects: (i) the requirement from an infecting node to recover prior to its reinfection, which depends on the SIS disease spreading parameters, and (ii) the competition between nodes that simultaneously try to reinfect the same ancestor, which depends on the network topology.