  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.
