  Message-Passing Neural Networks (MPNNs), the most prominent Graph Neural
Network (GNN) framework, celebrate much success in the analysis of
graph-structured data. Concurrently, the sparsification of Neural Network
models attracts a great amount of academic and industrial interest. In this
paper we conduct a structured, empirical study of the effect of sparsification
on the trainable part of MPNNs known as the Update step. To this end, we design
a series of models to successively sparsify the linear transform in the Update
step. Specifically, we propose the ExpanderGNN model with a tuneable
sparsification rate and the Activation-Only GNN, which has no linear transform
in the Update step. In agreement with a growing trend in the literature the
sparsification paradigm is changed by initialising sparse neural network
architectures rather than expensively sparsifying already trained
architectures. Our novel benchmark models enable a better understanding of the
influence of the Update step on model performance and outperform existing
simplified benchmark models such as the Simple Graph Convolution. The
ExpanderGNNs, and in some cases the Activation-Only models, achieve performance
on par with their vanilla counterparts on several downstream tasks, while
containing significantly fewer trainable parameters. Our code is publicly
available at: https://github.com/ChangminWu/ExpanderGNN.
