Statistical and Machine Learning Approaches for Network Analysis

2.7 Hierarchical Modularity

A high clustering coefficient indicates highly interconnected subnetworks (modules) in real-world networks. However, the next question that arises is regarding the way in which these modules are organized in the network.

Ravasz et al. [25] and Ravasz and Barabási [26] provided an answer for this question by using the degree-dependent clustering coefficient.

This statistical property is defined as

(2.24)

where C_i is the clustering coefficient of node i (see Eq. 2.19).

Ravasz et al. found that the degree-dependent clustering coefficient follows a power–law function in several real-world networks:

(2.25)

where α is a constant and is empirically equal to about 1.

In a random network, the clustering coefficients of all nodes are p, which is independent of the node degree k, because an edge is drawn between a given node pair with the probability p: C^rand(k) = p. Thus, the random network cannot explain this statistical property.

This power–law degree-dependent clustering coefficient indicates that the edge density among neighbors of nodes with a small degree is high and that the edge density among the neighbors of nodes with a large degree is low. In order to explain the relation between this property and the network structure, Ravasz et al. used a simple example, ...