  Network topology is a fundamental aspect of network science that allows us to
gather insights into the complicated relational architectures of the world we
inhabit. We provide a first specific study of neighbourhood degree sequences in
complex networks. We consider how to explicitly characterise important physical
concepts such as similarity, heterogeneity and organisation in these sequences,
as well as updating the notion of hierarchical complexity to reflect previously
unnoticed organisational principles. We also point out that neighbourhood
degree sequences are related to a powerful subtree kernel for unlabelled graph
classification. We study these newly defined sequence properties in a
comprehensive array of graph models and over 200 real-world networks. We find
that these indices are neither highly correlated with each other nor with
classical network indices. Importantly, the sequences of a wide variety of real
world networks are found to have greater similarity and organisation than is
expected for networks of their given degree distributions. Notably, while
biological, social and technological networks all showed consistently large
neighbourhood similarity and organisation, hierarchical complexity was not a
consistent feature of real world networks. Neighbourhood degree sequences are
an interesting tool for describing unique and important characteristics of
complex networks.
