Statistical and Machine Learning Approaches for Network Analysis

2.4 Scale-Free Network

Real-world networks possess remarkable statistical properties that are not explained by random networks and lattice networks. A representative example is the scale-free properties that relate to the distribution of node degrees.

2.4.1 Degree Distribution

The node degree, the simplest measure of a network, is defined as the number of edges (neighbors) that a node has. In the case of a simple network consisting of N nodes, in which A_ij = A_ji = 1, if an edge is drawn between nodes i and j, the degree of node i, k_i, is expressed as

(2.4)

A simple question might arise, regarding the way in which degree is distributed in real-world networks. Barabási and Albert [6] answered this question by defining degree distribution. This distribution is defined as

(2.5)

where δ(x) is the Kronecker's delta function. This function returns 1 when x = 0 and returns 0 otherwise. Hence, the term corresponds to the number of nodes with degree k.

2.4.2 Degree Distribution of Random Network

The degree distribution of Erdös–Rényi random networks corresponds to the probability that a node has k edges. Since an edge is independently drawn between two given nodes with probability p, we can ...