  The classic clustering coefficient and the lately proposed closure
coefficient quantify the formation of triangles from two different
perspectives, with the focal node at the centre or at the end in an open triad
respectively. As many networks are naturally rich in triangles, they become
standard metrics to describe and analyse networks. However, the advantages of
applying them can be limited in networks, where there are relatively few
triangles but which are rich in quadrangles, such as the protein-protein
interaction networks, the neural networks and the food webs. This yields for
other approaches that would leverage quadrangles in our journey to better
understand local structures and their meaning in different types of networks.
Here we propose two quadrangle coefficients, i.e., the i-quad coefficient and
the o-quad coefficient, to quantify quadrangle formation in networks, and we
further extend them to weighted networks. Through experiments on 16 networks
from six different domains, we first reveal the density distribution of the two
quadrangle coefficients, and then analyse their correlations with node degree.
Finally, we demonstrate that at network-level, adding the average i-quad
coefficient and the average o-quad coefficient leads to significant improvement
in network classification, while at node-level, the i-quad and o-quad
coefficients are useful features to improve link prediction.
