Understanding Pooling in Graph Neural Networks

TL;DR

An operational framework to unify this vast and diverse literature by describing pooling operators as the combination of three functions: selection, reduction, and connection (SRC).

Abstract

Inspired by the conventional pooling layers in convolutional neural networks, many recent works in the field of graph machine learning have introduced pooling operators to reduce the size of graphs. The great variety in the literature stems from the many possible strategies for coarsening a graph, which may depend on different assumptions on the graph structure or the specific downstream task. In this paper we propose a formal characterization of graph pooling based on three main operations, called selection, reduction, and connection, with the goal of unifying the literature under a common framework. Following this formalization, we introduce a taxonomy of pooling operators and categorize more than thirty pooling methods proposed in recent literature. We propose criteria to evaluate the performance of a pooling operator and use them to investigate and contrast the behavior of different classes of the taxonomy on a variety of tasks.

Figures (11)

• Figure 1: Schematic view of pooling operators: the selection function ${\rm \textsc{Sel}}$ groups the nodes into supernodes $\mathcal{S}_1, \dots, \mathcal{S}_K$; the reduction function ${\rm \textsc{Red}}$ maps each supernode $\mathcal{S}_k$ to the attribute of node $k$ in the pooled graph; finally, the connection function ${\rm \textsc{Con}}$ computes the edges between each pair of new nodes.
• Figure 2: \ref{['fig:pool-a']} Sparse supernodes have a constant cardinality ($|\mathcal{S}_k|=1$) while dense supernodes scale with the size of the graph. Hierarchical methods reduce the graph gradually, while global methods always return one node. \ref{['fig:pool-b']} Fixed methods return the same number of nodes ($K=3$) while adaptive methods return graphs of size proportional to the input.
• Figure 3: Maximum number of nodes that can be processed by dense (in green) and sparse (in blue) methods.
• Figure 4: Node attributes (point locations) reconstructed with different operators in the autoencoder experiment.
• Figure 5: Graphs pooled with different operators in the autoencoder experiment with the modified Red function, and the associated selection matrices ${\mathbf S}$.