Critical effect of dependency groups on the function of networks

Abstract

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...

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