  In wireless sensor networks, where energy is scarce, it is inefficient to
have all nodes active because they consume a non-negligible amount of battery.
In this paper we consider the problem of jointly selecting sensors, relays and
links in a wireless sensor network where the active sensors need to communicate
their measurements to one or multiple access points. Information messages are
routed stochastically in order to capture the inherent reliability of the
broadcast links via multiple hops, where the nodes may be acting as sensors or
as relays. We aim at finding optimal sparse solutions where both, the
consistency between the selected subset of sensors, relays and links, and the
graph connectivity in the selected subnetwork are guaranteed. Furthermore,
active nodes should ensure a network performance in a parameter estimation
scenario. Two problems are studied: sensor and link selection; and sensor,
relay and link selection. To solve such problems, we present tractable
optimization formulations and propose two algorithms that satisfy the previous
network requirements. We also explore an extension scenario: only link
selection. Simulation results show the performance of the algorithms and
illustrate how they provide a sparse solution, which not only saves energy but
also guarantees the network requirements.
