  Distributed graph signal processing algorithms require the network nodes to
communicate by exchanging messages in order to achieve a common objective.
These messages have a finite precision in realistic networks, which may
necessitate to implement message quantization. Quantization, in turn, may
generate distortion and performance penalty in the distributed processing
tasks. This paper proposes a novel method for distributed graph filtering that
minimizes the error due to message quantization without compromising the
communication costs. It first bounds the exchanged messages and then allocates
a limited bit budget in an optimized way to the different messages and network
nodes. In particular, our novel quantization algorithm adapts to both the
network topology and the message importance in a distributed processing task.
Our results show that the proposed method is effective in minimizing the error
due to quantization and that it permits to outperform baseline distributed
algorithms when the bit budget is limited. They further show that errors
produced in nodes with high eccentricity or in the first steps of the
distributed algorithm contribute more to the global error. Also, sparse and
irregular graphs require more irregular bit distribution. Our method provides
one of the first quantization solutions for distributed graph processing, which
is able to adapt to the target task, the graph properties and the communication
constraints.
