Statistical and Machine Learning Approaches for Network Analysis

6.2 Weighted Spectral Distribution

We now derive our metric, the weighted spectral distribution, relating it to another common structural metric, the clustering coefficient, before showing how it characterizes networks with different mixing properties.

Denote an undirected graph as G = (V, E) where V is the set of vertices (nodes) and E is the set of edges (links). The adjacency matrix of G, A(G), has an entry of one if two nodes, u and , are connected and zero otherwise

(6.1)

Let be the degree of node and D = diag(sum(A)) be the diagonal matrix having the sum of degrees for each node (column of matrix) along its main diagonal. Denoting by I the identity matrix (I)_i,j = 1 if i = j, 0 otherwise, the normalized Laplacian L associated with graph G is constructed from A by normalizing the entries of A by the node degrees of A as

(6.2)

or equivalently

(6.3)

As L is a real symmetric matrix ...