  In Network Science node neighbourhoods, also called ego-centered networks
have attracted large 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.
