  In this work, we seek to improve the velocity reconstruction of clusters by
using Graph Neural Networks -- a type of deep neural network designed to
analyze sparse, unstructured data. In comparison to the Convolutional Neural
Network (CNN) which is built for structured data such as regular grids, GNN is
particularly suitable for analyzing galaxy catalogs. In our GNNs, galaxies as
represented as nodes that are connected with edges. The galaxy positions and
properties -- stellar mass, star formation rate, and total number of galaxies
within 100~\mpc -- are combined to predict the line-of-sight velocity of the
clusters. To train our networks, we use mock SDSS galaxies and clusters
constructed from the Magneticum hydrodynamic simulations. Our GNNs reach a
precision in reconstructed line-of-sight velocity of $\Delta v$=163 km/s,
outperforming by $\approx$10\% the perturbation theory~($\Delta v$=181 km/s) or
the CNN~($\Delta v$=179 km/s). The stellar mass provides additional
information, improving the precision by $\approx$6\% beyond the position-only
GNN, while other properties add little information. Our GNNs remain capable of
reconstructing the velocity field when redshift-space distortion is included,
with $\Delta v$=210 km/s which is again 10\% better than CNN with RSD. Finally,
we find that even with an impressive, nearly 70\% increase in galaxy number
density from SDSS to DESI, our GNNs only show an underwhelming 2\% improvement,
in line with previous works using other methods. Our work demonstrates that,
while the efficiency in velocity reconstruction may have plateaued already at
SDSS number density, further improvements are still hopeful with new
reconstruction models such as the GNNs studied here.
