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The ability to exchange secret information is critical to many commercial, governmental, and military networks. Information-theoretic security-widely accepted as the strictest notion of security-relies on channel coding techniques that exploit the inherent randomness of the propagation channels to strengthen the security of digital communications systems. Motivated by recent developments in the field, we aim to characterize the fundamental secrecy limits of wireless networks. The paper is comprised of two separate parts. In Part I, we define the intrinsically secure communications graph (iS-graph), a random graph which describes the connections that can be securely established over a large-scale network. We provide conclusive results for the local connectivity of the Poisson iS-graph, in terms of node degrees and isolation probabilities. We show how the secure connectivity of the network varies with the wireless propagation effects, the secrecy rate threshold of each link, and the noise powers of legitimate nodes and eavesdroppers. We then propose sectorized transmission and eavesdropper neutralization as viable strategies for improving the secure connectivity. Our results help clarify how the spatial density of eavesdroppers can compromise the intrinsic security of wireless networks. In Part II of the paper, we study the achievable secrecy rates and the effect of eavesdropper collusion.