NMI (normalized mutual information)

NMI (normalized mutual information):

在information theory的理论框架下比较两个可重叠划分(overlapping clusters)的方法。

有两个不同划分C'={X1, X2, ...X|C'|}, C''={Y1, Y2, ..., Y|C''|}

H(X) 表示X的熵(entrophy)，H(X|Y)表示条件熵 (conditilnal entrophy)

步骤

To sum up, all the procedure reduces to

1. for a given k, compute H(Xk|Yl) for each l using the probabilities given by equations (B.4)–(B.7)

2. compute H(Xk|Y) from equation (B.9) taking into account the constraint given in equation (B.14); note that if this condition is never fulfilled, we decided to set H(Xk|Y)= H(Xk);

3 for each k, repeat the previous step to compute H(X|Y)norm according to equation (B.11); 

4 repeat all this for Y and put everything together in equation (B.12).

参考文献：

Lancichinetti, A., Fortunato, S., & Kertész, J. (2009). Detecting the overlapping and hierarchical community structure in complex networks. New Journal of Physics, 11(3), 033015. doi:10.1088/1367-2630/11/3/033015