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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 reliability. In particular, we take the case of random regular networks for simplicity 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, and show that the second moment of degree over the first moment is an important shorthand for behavior of a network. Notably, 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. We present as an example the case of networks with a power-law degree distribution.