# On neighbourhood degree sequences of complex networks

## TL;DR

This paper introduces a detailed study of neighbourhood degree sequences in complex networks, revealing their unique properties and relationships with network organization, across various models and real-world networks.

## Contribution

It is the first comprehensive analysis of neighbourhood degree sequences, linking them to network similarity, heterogeneity, and hierarchical complexity, and demonstrating their relevance in real-world networks.

## Key findings

- Neighbourhood degree sequences are largely uncorrelated with classical network indices.
- Real-world networks exhibit higher similarity and organization in their neighbourhood degree sequences than expected.
- Biological, social, and technological networks show consistent patterns of similarity and organization.

## Abstract

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.