Rethinking the Polynomial Filter of GNNs via Graph Information Activation Theory

TL;DR — This paper focuses on analyzing the polynomial filter in GNNs theoretically and then propose a new GNN with a simpler basis.

Abstract

Recently, it has been a hot research topic to design different polynomial filters in graph neural networks (GNNs). Most of the existing GNNs only pay attention to the properties of polynomials when designing the polynomial filter, thus not only bringing additional computational costs but also ignoring embedding the graph structure information into the construction process of the basis. To address these issues, we theoretically prove that any polynomial basis with the same degree has the same expressive ability and the finely designed polynomial basis that only considers the polynomial property can at most bring linear benefit for GNNs. Then, we propose a graph information activation (GIA) theory that provides a new perspective for interpreting polynomial filters and then analyse some popular bases using the GIA theory. Based on the GIA theory and analysis, we design a simple basis by utilizing the graph structure information and further build a simple GNN (i.e., SimpleNet), which can be applied to both homogeneous and non-homogenous graphs. Experiments on real datasets demonstrate that our SimpleNet can achieve better or comparable performance with relatively less running time compared to other state-of-the-art GNNs.