CYCLE TO CLIQUE (CY2C) GRAPH NEURAL NET-WORK: A SIGHT TO SEE BEYOND NEIGHBORHOOD AGGREGATION

Abstract

Graph neural networks have been successfully adapted for learning vector representations of graphs through various neighborhood aggregation schemes. Previous researches suggest, however, that they possess limitations in incorporating key non-Euclidean topological properties of graphs. This paper mathematically identifies the caliber of graph neural networks in classifying isomorphism classes of graphs with continuous node attributes up to their local topological properties. In light of these observations, we construct the Cycle to Clique graph neural network, a novel yet simple algorithm which topologically enriches the input data of conventional graph neural networks while preserving their architectural components. This method theoretically outperforms conventional graph neural networks in classifying isomorphism classes of graphs while ensuring comparable time complexity in representing random graphs. Empirical results further support that the novel algorithm produces comparable or enhanced results in classifying benchmark graph data sets compared to contemporary variants of graph neural networks.

1. INTRODUCTION

Graph neural networks (GNN) are prominent deep learning methods for learning vector representation of graphs. Research in GNNs explores their empirical capabilities and effectiveness in classifying node labels, classifying graphs, and predicting links by modifying the message passing layers or pooling methods. These experiments support that GNNs can achieve state-of-the-art performances in executing these tasks and ensure equivalent performance to that of the Weisfeiler-Lehman (WL) isomorphism test in representing graphs with discrete node labels Xu et al. (2018) . However, they have limited capabilities in incorporating global topological properties of graphs, thereby exhibiting restricted discriminative power in distinguishing isomorphism classes of graphs Bouritsas et al. To overcome these limitations, this paper presents a mathematical framework that examines which topological properties of graphs with continuous node attribute that conventional GNNs can encapsulate. Inspired by the works of Krebs and Verbitsky Krebs & Verbitsky (2015) and Xu et al Xu et al. (2018) , we use the theory of covering spaces to prove that under some constraints, a pair of graphs with continuous node attributes is distinguishable by GNNs if and only if there exist isomorphisms among the collection of their finite depth unfolding trees that induce equality of induced node attributes. This gives a universal formulation which pinpoints the discriminative power of a wide range of variants of GNNs and the topological enrichments these models endow over the graph data set. Such approaches include enriching node attributes, using persistent homological techniques, gluing high dimensional complexes, and keeping track of recurring subgraph structures. Among these candidates, we focus on the procedure of transforming the cycle bases of graphs to complete subgraphs or cliques. This operation can be easily implemented by adding suitable edges to transform a cyclic subgraph into a clique and masking any other edges not included in the subgraph. The



(2022); Rieck et al. (2019).

