Measuring Quadrangle Formation in Complex Networks

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.