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 is considered as the average weight , where E denotes the total number of edges. In this case, therefore, there is a linear relationship (i.e., ), indicating β = 1.

However, the exponent β is larger than 1 in real-world weighted networks. This result is nontrivial as it indicates that the weight of edges leading to high-degree nodes (hubs) is high.

Furthermore, it was found that the average weight can be expressed as a function of the end-point degrees:

(2.44)

Since , we obtain β = 1 + θ.

2.11.2 Weighted Clustering Coefficient

The concepts of weight and strength can be applied to the classical clustering coefficient (see also Section 2.6.1). The weighted clustering coefficient [43,44] is defined as

(2.45)

In accordance with the classical clustering coefficient, the average weighted clustering coefficient and the degree-dependent weighted clustering coefficient are defined.

In the case where there is no correlation between weights and topology (i.e., randomized networks), and . However, we may observe two opposite cases. If , the edges with larger weights tend to form highly interconnected subnetworks such as modules. On the other hand, if , such modules are likely to be formed by the edges with lower weights. Similarly, the above explanation is applicable to for evaluating the average weighted clustering coefficient over nodes with degree k.

In real-world weighted networks [43] (e.g., international airport networks, in which nodes and weighted edges correspond to airports and flights with traffics, respectively), was observed. Furthermore, a higher weighted clustering coefficient is significant for hub nodes, that is, for large k. This result indicates that edges with high weights (e.g., flights with higher traffics) are densely drawn among hub nodes (e.g., airports). The scientific collaboration networks (in which the nodes and weighted edges are author and coauthor relationships, respectively, considering the number of papers) represent the same tendency.

2.11.3 Weighted Degree Correlation

Similarly, we can also consider the weighted version of degree correlation. This weighted degree correlation is obtained by modifying the original average nearest-neighbor degree Γ_i (see also Section 2.9.2). The weighted average nearest-neighbor degree [43,44] is defined as

(2.46)

This definition implies that if the edges with larger weights are connected to the neighbors with larger degrees and that in the opposite case.

Substituting the above equation in Equation 2.29, in which is referred to as Γ_i, we obtain the weighted degree correlation . The interpretation for magnitude relations between and is similar to that between and Γ_i.

In real-world weighted networks [43] (airport networks and scientific collaboration networks), was observed for large k, suggesting that the edges with larger weights lead to high-degree nodes (hub nodes).