  We consider a wireless sensor network consisting of multiple nodes that are
coordinated by a fusion center (FC) in order to estimate a common signal of
interest. In addition to being coordinated, the sensors are also able to
collaborate, i.e., share observations with other neighboring nodes, prior to
transmission. In an earlier work, we derived the energy-optimal collaboration
strategy for the single-snapshot framework, where the inference has to be made
based on observations collected at one particular instant. In this paper, we
make two important contributions. Firstly, for the single-snapshot framework,
we gain further insights into partially connected collaboration networks
(nearest-neighbor and random geometric graphs for example) through the analysis
of a family of topologies with regular structure. Secondly, we explore the
estimation problem by adding the dimension of time, where the goal is to
estimate a time-varying signal in a power-constrained network. To model the
time dynamics, we consider the stationary Gaussian process with exponential
covariance (sometimes referred to as Ornstein-Uhlenbeck process) as our
representative signal. For such a signal, we show that it is always beneficial
to sample as frequently as possible, despite the fact that the samples get
increasingly noisy due to the power-constrained nature of the problem.
Simulation results are presented to corroborate our analytical results.
