The expressive power of kth-order invariant graph networks

TL;DR

This paper proves that kth-order invariant graph networks have the same expressive power as the k-dimensional Weisfeiler-Leman test, meaning they can distinguish exactly the same graphs.

Contribution

It generalizes previous results to show that kth-order invariant graph networks are not more powerful than k-WL in graph distinction.

Findings

k-IGNs are bounded in expressive power by k-WL

k-IGNs and k-WL are equally powerful in graph distinction

The result extends to arbitrary k, not just k=2

Abstract

The expressive power of graph neural network formalisms is commonly measured by their ability to distinguish graphs. For many formalisms, the k-dimensional Weisfeiler-Leman (k-WL) graph isomorphism test is used as a yardstick. In this paper we consider the expressive power of kth-order invariant (linear) graph networks (k-IGNs). It is known that k-IGNs are expressive enough to simulate k-WL. This means that for any two graphs that can be distinguished by k-WL, one can find a k-IGN which also distinguishes those graphs. The question remains whether k-IGNs can distinguish more graphs than k-WL. This was recently shown to be false for k=2. Here, we generalise this result to arbitrary k. In other words, we show that k-IGNs are bounded in expressive power by k-WL. This implies that k-IGNs and k-WL are equally powerful in distinguishing graphs.