Lower and Upper Bounds for Numbers of Linear Regions of Graph Convolutional Networks

TL;DR

This paper provides bounds on the number of linear regions in graph convolutional networks, demonstrating that deeper GCNs are exponentially more expressive than shallow ones based on these bounds.

Contribution

It presents the first optimal upper bound for one-layer GCNs and bounds for multi-layer GCNs, advancing understanding of GCN expressivity.

Findings

Multi-layer GCNs have exponentially more linear regions than one-layer GCNs.

The estimated maximum number of linear regions is close to the lower bound.

Deeper GCNs exhibit greater expressivity than shallow GCNs.

Abstract

The research for characterizing GNN expressiveness attracts much attention as graph neural networks achieve a champion in the last five years. The number of linear regions has been considered a good measure for the expressivity of neural networks with piecewise linear activation. In this paper, we present some estimates for the number of linear regions of the classic graph convolutional networks (GCNs) with one layer and multiple-layer scenarios. In particular, we obtain an optimal upper bound for the maximum number of linear regions for one-layer GCNs, and the upper and lower bounds for multi-layer GCNs. The simulated estimate shows that the true maximum number of linear regions is possibly closer to our estimated lower bound. These results imply that the number of linear regions of multi-layer GCNs is exponentially greater than one-layer GCNs per parameter in general. This suggests that deeper GCNs have more expressivity than shallow GCNs.