Machine Learning in Complex Networks by Thiago Christiano Silva & Liang Zhao

Author:Thiago Christiano Silva & Liang Zhao
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(6.4)

in which the indicator function makes sure that only those edges crossing different communities are considered in the computation of the cut size R.

Consider the index vector s, whose component s i is + 1 if vertex i is in one group and − 1 if it is in the other group:

(6.5)

Then, R can be rewritten as:

(6.6)

As the degree of vertex i is , then we have .

Then, R can be rewritten as:

(6.7)

In matrix form, we have:

(6.8)

in which s T is the transpose of s and is the Laplacian matrix.

Let us write s as a linear combination of the orthonormal eigenvectors v i of the Laplacian:

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