Robustness in large-scale random networks

Abstract

We consider the issue of protection in very large networks displaying randomness in topology. We employ random graph models to describe such networks, and obtain probabilistic bounds on several parameters related to various protection schemes. In particular, we take the case of random regular networks for simplicity, where the degree of each node is the same, and consider the length of primary and backup paths in terms of the number of hops. First, for a randomly picked pair of nodes, we derive a lower bound on the average distance between the pair and discuss the tightness of the bound. In addition, noting that primary and protection paths form cycles, we obtain a lower bound on the average length of the shortest cycle around the pair. Finally, we show that the protected connections of a given maximum finite length are rare. We then generalize our network model so that different degrees are allowed according to some arbitrary distribution. Notably, we derive an upper bound on the mean number of non-finite length cycles in generalized random networks. More importantly, we show that most of the results in regular networks carry over with minor modifications, which significantly broadens the scope of networks to which our approach applies. Our main contributions are the following. First, we take an analytical approach by bringing the concept of randomness into network topologies that can provide concise rules to relate basic network parameters to robustness. Second, we establish analytical results for the length of backup paths for path and link-based protection schemes rather than for the efficiency of backup capacity, upon which most studies concentrate. Finally, we develop a unified framework for studying the issue of robustness in very general random networks with arbitrary degree distributions.