Local, global and scale-dependent node roles

TL;DR

This paper introduces a scale-dependent framework for defining node roles in graphs, showing that shallow, local roles are sufficient for high-accuracy node classification, aligning with recent graph neural network successes.

Contribution

It formalizes node roles with a scale parameter, bridging local and global perspectives, and demonstrates that shallow roles can achieve state-of-the-art classification performance.

Findings

Shallow roles of depth 3 or 4 are sufficient for high-accuracy node classification.

A shallow classifier based on local roles matches recent GNN performance.

Scale-dependent roles provide a unified view of local and global node equivalences.

Abstract

This paper re-examines the concept of node equivalences like structural equivalence or automorphic equivalence, which have originally emerged in social network analysis to characterize the role an actor plays within a social system, but have since then been of independent interest for graph-based learning tasks. Traditionally, such exact node equivalences have been defined either in terms of the one hop neighborhood of a node, or in terms of the global graph structure. Here we formalize exact node roles with a scale-parameter, describing up to what distance the ego network of a node should be considered when assigning node roles - motivated by the idea that there can be local roles of a node that should not be determined by nodes arbitrarily far away in the network. We present numerical experiments that show how already "shallow" roles of depth 3 or 4 carry sufficient information to perform node classification tasks with high accuracy. These findings corroborate the success of recent graph-learning approaches that compute approximate node roles in terms of embeddings, by nonlinearly aggregating node features in an (un)supervised manner over relatively small neighborhood sizes. Indeed, based on our ideas we can construct a shallow classifier achieving on par results with recent graph neural network architectures.