  Knowing the pressure at all times in each node of a water distribution system
(WDS) facilitates safe and efficient operation. Yet, complete measurement data
cannot be collected due to the limited number of instruments in a real-life
WDS. The data-driven methodology of reconstructing all the nodal pressures by
observing only a limited number of nodes is presented in the paper. The
reconstruction method is based on K-localized spectral graph filters, wherewith
graph convolution on water networks is possible. The effect of the number of
layers, layer depth and the degree of the Chebyshev-polynomial applied in the
kernel is discussed taking into account the peculiarities of the application.
In addition, a weighting method is shown, wherewith information on friction
loss can be embed into the spectral graph filters through the adjacency matrix.
The performance of the proposed model is presented on 3 WDSs at different
number of nodes observed compared to the total number of nodes. The weighted
connections prove no benefit over the binary connections, but the proposed
model reconstructs the nodal pressure with at most 5% relative error on average
at an observation ratio of 5% at least. The results are achieved with shallow
graph neural networks by following the considerations discussed in the paper.
