On neighbourhood degree sequences of complex networks

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 organization in these sequences, as well as updating the notion of hierarchical complexity to reflect previously unnoticed organizational principles. We also point out that neighbourhood degree sequences are related to a powerful subtree kernel for unlabeled 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.

Hierarchical complexity of the macro-scale neonatal brain

“…The neighbourhood of a network node is the set of all nodes to which that node shares links. The neighbourhood degree sequence of the node is then the ordered sequence of degrees of the nodes neighbours, which is a particularly useful tool for studying organisational principles of networks [17]. The HC of a network, then, involves computing the variability of neighbourhood degree sequences of nodes of the same degree.…”

“…As a post-hoc analysis, and to characterise better the network topology and understand the results showed by HC, we investigated the effect of cross-hemisphere neighbourhood symmetry within tiers to probe deeper into the complex organisation underlying the neonatal connectomes, following the insight that higher symmetry is associated with higher order and thus lower complexity [17] (Supplementary Materials). We also studied the percentage of common and uncommon connections within tiers, following the hypothesis than adults have more well established network architecture and have more common connections within tiers than neonates (Supplementary Materials).…”

“…Hierarchical complexity (HC) is a measure of the diversity of connectivity patterns found across hierarchically equivalent (same degree) network nodes, and it reflects the hierarchical and functionally diverse capacities of the human brain. HC provides a tractable signature of brain network architecture in the adult connectome [16,17]: four hierarchical tiers broadly comprise different categories of functional processing -cognitive, sensorimotor, integrative, and memory and emotion [16]. To investigate the possibility that HC is established in the perinatal period, and that alterations in HC patterning might characterize atypical brain development in preterm infants, we have developed an approach to assess HC in the newborn period.…”

Hierarchical Complexity of the Macro-Scale Neonatal Brain

“…The neighborhood of a network node is the set of all nodes with which that node shares links, and the number of the neighbors (and links) of a node is the nodal degree. The neighborhood degree sequence of the node is then the ordered sequence of degrees of the node’s neighbors, which is a particularly useful tool for studying organizational principles of networks ( Smith, 2019b ). The hierarchical complexity of a network involves computing the variability of neighborhood degree sequences of nodes of the same degree.…”

“…nodes that have the same degree). Importantly, it distinguishes connectomes from different random null models where other common metrics fail and is observed to reflect the hierarchical and functionally diverse capacities of the human brain ( Smith et al , 2019 ), while not being a general feature of most other real-world networks ( Smith, 2019b ). Studying the degree hierarchy has also provided a tractable signature of brain network architecture in the adult connectome: four hierarchical tiers broadly comprise different categories of functional processing—cognitive, sensorimotor, integrative, and memory and emotion ( Smith et al , 2019 ).…”