  While many existing graph neural networks (GNNs) have been proven to perform
$\ell_2$-based graph smoothing that enforces smoothness globally, in this work
we aim to further enhance the local smoothness adaptivity of GNNs via
$\ell_1$-based graph smoothing. As a result, we introduce a family of GNNs
(Elastic GNNs) based on $\ell_1$ and $\ell_2$-based graph smoothing. In
particular, we propose a novel and general message passing scheme into GNNs.
This message passing algorithm is not only friendly to back-propagation
training but also achieves the desired smoothing properties with a theoretical
convergence guarantee. Experiments on semi-supervised learning tasks
demonstrate that the proposed Elastic GNNs obtain better adaptivity on
benchmark datasets and are significantly robust to graph adversarial attacks.
The implementation of Elastic GNNs is available at
\url{https://github.com/lxiaorui/ElasticGNN}.
