The relationship between the degrees in a connected node pair is very interesting. Since real-world networks are nonrandom, we can expect this relationship to be significant. However, degree distribution only involves the degree of each node. Thus, we need an alternative measure for characterizing such a relationship of degrees.
To characterize the relationship between node degrees, Newman proposed the assortative coefficient , defined as
where ki and kj are the degrees of two nodes at the ends of an edge, and denotes the average over all edges. This is simply the Pearson correlation coefficient of degrees between a connected node pair, and it lies in the range −1 ≤ r ≤ 1.
The relationship between the assortative coefficient and network structures is described as follows: