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I'm developing a model that organises items of different classes into a dendrogram, like the one here:

A heterogeneous dendrogram showing items of 6 different classes

Consider the next dendrogram, it is clearly more homogeneous, i.e. verteces of the same colour are more likely to be next to each other:

A more homogeneous dendrogram

I'm wondering how I can assess the homogeneity (or heterogeneity) of a dendrogram. To illustrate this, consider the following two dendrograms:

Dendrograms with high and low "homogeneity"

The first one (right) shows high “homogeneity”, since items of the some colour (class) are nicely grouped together. The second one (left), shows low homogeneity, since the items with the same colour are not close at all.

To make matters more complex, the distance between the groups matters, too. Consider the following two dendrograms:

Dendrograms with medium and low "homogeneity"

While both dendrograms would have a low-ish homogeneity, the one on the right displays the lower homogeneity, since $x_1$ is farer away from $x_2$ than in the first one.

So, what is a good measure to assess the "homogeneity" of my dendrograms, which takes into account the caveats mentioned above?

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    $\begingroup$ crossposted: math.stackexchange.com/q/4906502/87355 $\endgroup$
    – Carlo Beenakker
    Commented Apr 29 at 20:13
  • $\begingroup$ You should wait at least a week before cross-posting to another site. Duplication of effort across multiple platforms = waste of people's time. $\endgroup$
    – Todd Trimble
    Commented Apr 29 at 22:09
  • $\begingroup$ What determines the vertex colour? $\endgroup$
    – David Roberts ♦
    Commented Apr 29 at 23:54
  • $\begingroup$ Colours identify the vertices' class. To give you some background, both dendgrograms (first and second) show the same classes, but the dendrograms are constructed differently. I want to optimise my method of constructing the dendrograms, so I need a way of assessing the quality - in this cases, better means more "homogeneous" if that's the right word). I can do that by cutting the dendrogram at a certain height, determine the majority colour and calculating the balanced accuracy, but this is rather crude and won't assess the more subtle quality of sub-trees. $\endgroup$
    – Rolf Bänziger
    Commented Apr 30 at 7:16

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