Understanding the z-score concept

Hi Group

In paper utilizing Louvain-like community detection technique, often z-rand was referred as a measure of stability/ partition quality.

Often, the z-rand partition similarity averaged across all iterations for a given gamma value were used a measure of stability. Higher z-rand score for that gamma value indicated higher partition stability

However, I’m not that sure whether I understand correctly about the implementation,

From the reading, I believe the implementation should be something as below.

Say, for example, we perform community detection for 4 iterations for a given gamma value , which will produce, say, these four partition:

Par_a = [1, 2, 1, 1, 2, 3, 3]

Par_b = [1, 2, 4, 1, 2, 3, 3]

Par_c = [1, 3, 1, 3, 2, 3, 3]

Par_d = [1, 4, 4, 1, 4, 3, 3]

The partition comparison is essentially the combination without repetition of the partition at a time:

Par_a: Par_b

Par_a: Par_c

Par_a: Par_d

Par_b: Par_c

Par_b: Par_d

Par_c: Par_d

Or, mathematically represented as nC r .

Really appreciate if someone can confirm whether my understanding is correct.