  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 significantly strengthen the security
of digital communications systems. Motivated by recent developments in the
field, this paper aims at a characterization of the fundamental secrecy limits
of wireless networks. Based on a general model in which legitimate nodes and
potential eavesdroppers are randomly scattered in space, the intrinsically
secure communications graph (iS-graph) is defined from the point of view of
information-theoretic security. Conclusive results are provided for the local
connectivity of the Poisson iS-graph, in terms of node degrees and isolation
probabilities. It is shown 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. Sectorized
transmission and eavesdropper neutralization are explored as viable strategies
for improving the secure connectivity. Lastly, the maximum secrecy rate between
a node and each of its neighbours is characterized, and the case of colluding
eavesdroppers is studied. The results help clarify how the spatial density of
eavesdroppers can compromise the intrinsic security of wireless networks.
