  In wireless sensor networks, the $q$-composite key predistribution scheme is
a widely recognized way to secure communications. Although connectivity
properties of secure sensor networks with the $q$-composite scheme have been
studied in the literature, few results address physical transmission
constraints since it is challenging to analyze the network connectivity in
consideration of both the $q$-composite scheme and transmission constraints
together. These transmission constraints reflect real-world implementations of
sensor networks in which two sensors have to be within a certain distance from
each other to communicate. In this paper, we rigorously derive conditions for
connectivity in sensor networks employing the $q$-composite scheme under
transmission constraints. Furthermore, we extend the analysis to consider the
unreliability of wireless links by modeling each link being independently
active with some probability. Our results provide useful guidelines for
designing securely and reliably connected sensor networks. We also present
numerical experiments to confirm the analytical results.
