Statistical and Machine Learning Approaches for Network Analysis

2.11 Weighted Networks

The weight (or strength) of edges is also an important factor in complex networks. Here, we introduce statistical measures and statistical relationships in weighted networks without the direction of edges.

For a convenient explanation, we divide the adjacency matrix defined in Equation 2.1 into two matrices, b_ij and . The matrix b_ij corresponds to an adjacency matrix in which b_ij = 1 if there is an edge between nodes i and j, and b_ij = 0 otherwise. The weight of the edge drawn between nodes i and j is stored in . That is, the relationship between the original adjacency matrix A_ij and these matrices is .

2.11.1 Strength

We first focus on two simple measures: the degree of node i, and the “strength” of node i[43,44], defined as

(2.42)

In real-world weighted networks, we observe the power–law relationship between the degree k and the average strength over nodes with degree k:

(2.43)

Assuming no correlation between the weight of edges and the node degree, the weight ...