Abstract
In Network Science, node neighbourhoods, also called ego-centered networks, have attracted significant attention. In particular the clustering coefficient has been extensively used to measure their local cohesiveness. In this paper, we show how, given two nodes with the same clustering coefficient, the topology of their neighbourhoods can be significantly different, which demonstrates the need to go beyond this simple characterization. We perform a large scale statistical analysis of the topology of node neighbourhoods of real networks by first constructing their clique complexes, and then computing their Betti numbers. We are able to show significant differences between the topology of node neighbourhoods of real networks and the stochastic topology of null models of random simplicial complexes revealing local organisation principles of the node neighbourhoods. Moreover we observe that a large scale statistical analysis of the topological properties of node neighbourhoods is able to clearly discriminate between power-law networks, and planar road networks.
Highlights
There has been significant recent interest in the topology and the geometry of networks [1, 2, 3, 4]
We propose a new network topology framework to investigate the local structure of real networks determined by the neighbourhoods of their nodes, which we often refer to as node neighbourhoods
We have analysed the topology of node neighbourhoods in large network datasets
Summary
There has been significant recent interest in the topology and the geometry of networks [1, 2, 3, 4]. We propose a new network topology framework to investigate the local structure of real networks determined by the neighbourhoods of their nodes, which we often refer to as node neighbourhoods. In particular the clustering coefficient has been used extensively to quantify to what extent a network satisfies the principle of triadic closure This principle was originally formulated in the context of social networks, where it is observed that two friends of a common person are more likely to be friends of each other than in a set of random relations. Higher-order clustering coefficients have been formulated in order to measure the density of cliques larger than triangles in a given node neighbourhood. Any two simplices of the clique complex have an intersection that is either the null set or it is a simplex of the clique complex
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