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Sensor networks are composed of a large number of low power sensor devices. For secure communication among sensors, secret keys are required to be established between them. Considering the storage limitations and the lack of post-deployment configuration information of sensors, random key predistribution schemes have been proposed. Due to limited number of keys, sensors can only share keys with a subset of the neighboring sensors. Sensors then use these neighbors to establish pairwise keys with the remaining neighbors. In order to study the communication overhead incurred due to pairwise key establishment, we derive probability models to design and analyze pairwise key establishment schemes for large-scale sensor networks. Our model applies the binomial distribution and a modified binomial distribution and analyzes the key path length in a hop-by-hop fashion. We also validate our models through a systematic validation procedure. We then show the robustness of our results and illustrate how our models can be used for addressing sensor network design problems.