  Graph pooling is a central component of a myriad of graph neural network
(GNN) architectures. As an inheritance from traditional CNNs, most approaches
formulate graph pooling as a cluster assignment problem, extending the idea of
local patches in regular grids to graphs. Despite the wide adherence to this
design choice, no work has rigorously evaluated its influence on the success of
GNNs. In this paper, we build upon representative GNNs and introduce variants
that challenge the need for locality-preserving representations, either using
randomization or clustering on the complement graph. Strikingly, our
experiments demonstrate that using these variants does not result in any
decrease in performance. To understand this phenomenon, we study the interplay
between convolutional layers and the subsequent pooling ones. We show that the
convolutions play a leading role in the learned representations. In contrast to
the common belief, local pooling is not responsible for the success of GNNs on
relevant and widely-used benchmarks.
