On the Computational Capability of Graph Neural Networks: A Circuit Complexity Bound Perspective

TL;DR

This paper investigates the fundamental computational limitations of Graph Neural Networks by analyzing their circuit complexity, revealing intrinsic expressivity constraints under certain architectural conditions, and providing a new theoretical framework.

Contribution

It introduces a circuit complexity perspective to analyze GNNs, establishing limitations on their ability to solve problems like graph connectivity and isomorphism.

Findings

GNNs with constant-depth layers cannot solve graph connectivity.

GNNs cannot distinguish certain graph isomorphisms unless complexity classes collapse.

The paper proposes a novel circuit complexity framework for GNN analysis.

Abstract

Graph Neural Networks (GNNs) have become the standard approach for learning and reasoning over relational data, leveraging the message-passing mechanism that iteratively propagates node embeddings through graph structures. While GNNs have achieved significant empirical success, their theoretical limitations remain an active area of research. Existing studies primarily focus on characterizing GNN expressiveness through Weisfeiler-Lehman (WL) graph isomorphism tests. In this paper, we take a fundamentally different approach by exploring the computational limitations of GNNs through the lens of circuit complexity. Specifically, we analyze the circuit complexity of common GNN architectures and prove that under constraints of constant-depth layers, linear or sublinear embedding sizes, and polynomial precision, GNNs cannot solve key problems such as graph connectivity and graph isomorphism unless $\mathsf{TC}^0 = \mathsf{NC}^1$. These results reveal the intrinsic expressivity limitations of GNNs behind their empirical success and introduce a novel framework for analyzing GNN expressiveness that can be extended to a broader range of GNN models and graph decision problems.