  We consider the detection of a correlated random process immersed in noise in
a wireless sensor network. Each node has an individual energy constraint and
the communication with the processing central units are affected by the path
loss propagation effect. Guided by energy efficiency concerns, we consider the
partition of the whole network into clusters, each one with a coordination node
or \emph{cluster head}. Thus, the nodes transmit their measurements to the
corresponding cluster heads, which after some processing, communicate a summary
of the received information to the fusion center, which takes the final
decision about the state of the nature. As the network has a fixed size,
communication within smaller clusters will be less affected by the path loss
effect, reducing energy consumption in the information exchange process between
nodes and cluster heads. However, this limits the capability of the network of
beneficially exploiting the spatial correlation of the process, specially when
the spatial correlation coherence of the process is of the same scale as the
clusters size. Therefore, a trade-off is established between the energy
efficiency and the beneficial use of spatial correlation. The study of this
trade-off is the main goal of this paper. We derive tight approximations of the
false alarm and miss-detection error probabilities under the Neyman-Pearson
framework for the above scenario. We also consider the application of these
results to a particular network and correlation model obtaining closed form
expressions. Finally, we validate the results for more general network and
correlation models through numerical simulations.
