  Distributed graph filters have found applications in wireless sensor networks
(WSNs) to solve distributed tasks such as consensus, signal denoising, and
reconstruction. However, when employed over WSN, the graph filters should deal
with the network limited energy, processing, and communication capabilities.
Quantization plays a fundamental role to improve the latter but its effects on
distributed graph filtering are little understood. WSNs are also prone to
random link losses due to noise and interference. The filter output is affected
by both the quantization error and the topological randomness error, which, if
it is not properly accounted in the filter design phase, may lead to an
accumulated error through the filtering iterations and significantly degrade
the performance. In this paper, we analyze how quantization affects distributed
graph filtering over both time-invariant and time-varying graphs. We bring
insights on the quantization effects for the two most common graph filters: the
finite impulse response (FIR) and autoregressive moving average (ARMA) graph
filter. We devise theoretical performance guarantees on the filter performance
when the quantization stepsize is fixed or changes dynamically over the
filtering iterations. For FIR filters, we show that a dynamic quantization
stepsize leads to more control on the quantization noise than the
fixed-stepsize quantization. For ARMA graph filters, we show that decreasing
the quantization stepsize over the iterations reduces the quantization noise to
zero at the steady-state. In addition, we propose robust filter design
strategies that minimize the quantization noise for both time-invariant and
time-varying networks. Numerical experiments on synthetic and two real data
sets corroborate our findings and show the different trade-offs between
quantization bits, filter order, and robustness to topological randomness.
