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, bij and . The matrix bij corresponds to an adjacency matrix in which bij = 1 if there is an edge between nodes i and j, and bij = 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 Aij 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 ...
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