Statistical and Machine Learning Approaches for Network Analysis

2.9 Assortativity

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.

2.9.1 Assortative Coefficient

To characterize the relationship between node degrees, Newman proposed the assortative coefficient [33], defined as

(2.28)

where k_i and k_j 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:

i. In the case of r > 0, the network shows “assortativity,” in which high-degree nodes tend to connect to high-degree nodes (see Fig. 2.12a).

ii. In the case of r = 0, there is no correlation between the degrees in a connected node pair (see Fig. 2.12b). That ...