What is the intuition behind the variation of information (VI) metric for cluster validation?

Question

For non-statisticians like me, it is very difficult to capture the idea of VI metric (variation of information) even after reading the relevant paper by Marina Melia "Comparing clusterings - An information based distance" (Journal of Multivariate Analysis, 2007). In fact, I am not familiar with many of the clusterings' terms out there.

Below is a MWE and I would like to know what does the output mean in the different metrics used. I have these two clusters in R and in the same order of id:

> dput(a)
structure(c(4L, 3L, 4L, 4L, 4L, 4L, 4L, 3L, 4L, 4L, 3L, 3L, 3L, 
4L, 4L, 4L, 4L, 4L, 3L, 4L, 4L, 4L, 3L, 4L, 4L, 3L, 4L, 4L, 4L, 
1L, 1L, 4L, 4L, 4L, 4L, 4L, 4L, 3L, 4L, 4L, 3L, 4L, 4L, 2L, 2L, 
4L, 3L, 3L, 2L, 2L, 2L, 4L, 3L, 4L, 4L, 3L, 1L, 4L, 3L, 4L, 4L, 
4L, 3L, 4L, 4L, 4L, 4L, 2L, 2L, 2L, 4L, 3L, 4L, 4L, 4L, 4L, 4L, 
4L, 4L, 4L, 4L, 3L, 4L, 4L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 3L, 4L, 
4L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 3L, 4L, 4L, 4L, 4L, 2L, 2L, 4L
), .Label = c("1", "2", "3", "4"), class = "factor")
> dput(b)
structure(c(4L, 3L, 4L, 4L, 4L, 4L, 4L, 3L, 4L, 4L, 3L, 3L, 3L, 
4L, 4L, 3L, 4L, 4L, 3L, 4L, 4L, 4L, 3L, 4L, 4L, 3L, 4L, 4L, 4L, 
1L, 1L, 4L, 4L, 4L, 4L, 4L, 4L, 3L, 4L, 4L, 3L, 4L, 4L, 2L, 2L, 
4L, 3L, 3L, 2L, 2L, 2L, 4L, 3L, 4L, 4L, 3L, 1L, 4L, 3L, 4L, 4L, 
3L, 3L, 4L, 4L, 4L, 4L, 2L, 2L, 2L, 4L, 3L, 3L, 4L, 4L, 4L, 4L, 
4L, 4L, 4L, 4L, 3L, 4L, 4L, 3L, 4L, 4L, 4L, 4L, 4L, 4L, 3L, 4L, 
4L, 3L, 4L, 4L, 4L, 4L, 4L, 3L, 3L, 4L, 4L, 4L, 4L, 2L, 2L, 4L
), .Label = c("1", "2", "3", "4"), class = "factor")

Now doing comparisons based on the VI as well as other metrics / indices and in chronological order of their appearance in literature.

library(igraph)
  # Normalized Mutual Information (NMI) measure 2005:
compare(a, b, method = c("nmi")) 
[1] 0.8673525
  # Variation of Information (VI) metric 2003:
compare(a, b, method = c("vi")) 
[1] 0.2451685
  # Jaccard Index 2002:
clusteval::cluster_similarity(a, b, similarity = c("jaccard"), method = "independence") 
[1] 0.8800522
  # van Dongen S metric 2000:
compare(a, b, method = c("split.join")) 
[1] 8
  # Adjusted Rand Index 1985:
compare(a, b, method = c("adjusted.rand")) 
[1] 0.8750403
  # Rand Index 1971:
compare(a, b, method = c("rand")) 
[1] 0.9374788

As you can see, the VI value was different from all the others.

• What does this value tell (and how is it related to the figure below)?
• What are the guidelines for considering this value low or high?
• Are there any guidelines defined?

Maybe experts in the field can provide some sensible descriptions for laymen like me when trying to report such results. I would really appreciate if someone would provide also guidelines for other metrics as well (when to consider the value is large or small, i.e., in relation to a similarity between two clusters).

I have read related CV threads here and here, but still couldn't grasp the intuition behind VI. Can someone explain this in plain English?

The below figure is figure 2 from the above mentioned paper about VI.

Answer 1

You need to realize that measures may have different interpretation.

Judging from your plot, a low VI is good.

1 - 0.2451685 = 0.7548315

which is much more in line with the other measures.

However, note that most of these measures measure something different.

There is no reason to assume that just because one measure is 0.8, another should also be 0.8

Answer 2

As you know VOI denotes Variation Of Information and when you reach "0.2451685" that means variation between two selected clusters is low and they have a good similarity.