  Wireless sensor networks (WSNs) are considered as a major technology enabling
the Internet of Things (IoT) paradigm. The recent emerging Graph Signal
Processing field can also contribute to enabling the IoT by providing key
tools, such as graph filters, for processing the data associated with the
sensor devices. Graph filters can be performed over WSNs in a distributed
manner by means of a certain number of communication exchanges among the nodes.
But, WSNs are often affected by interferences and noise, which leads to view
these networks as directed, random and time-varying graph topologies. Most of
existing works neglect this problem by considering an unrealistic assumption
that claims the same probability of link activation in both directions when
sending a packet between two neighboring nodes. This work focuses on the
problem of operating graph filtering in random asymmetric WSNs. We show first
that graph filtering with finite impulse response graph filters (node-invariant
and node-variant) requires having equal connectivity probabilities for all the
links in order to have an unbiased filtering, which can not be achieved in
practice in random WSNs. After this, we characterize the graph filtering error
and present an efficient strategy to conduct graph filtering tasks over random
WSNs with node-variant graph filters by maximizing accuracy, that is, ensuring
a small bias-variance trade-off. In order to enforce the desired accuracy, we
optimize the filter coefficients and design a cross-layer distributed
scheduling algorithm at the MAC layer. Extensive numerical experiments are
presented to show the efficiency of the proposed solution as well as the
cross-layer distributed scheduling algorithm for the denoising application.
