Statistical argument for this cluster measure This quick clustering score discussion presents the following single cluster scoring functions:
$$
c_i = 1-\sqrt{\frac{\sum_{j}^{N}\left(1-\phi(i,j) \right)^2}{N-1}}
$$
and
$$
C = 1-\sqrt{\frac{\sum_{i}^{N}\left(1-c_i \right)^2 (N-1)}{2N^2}}
$$
where $c_i$ is the score for element $i$ in the cluster, $C$ is the total cluster score, $N$ is the size of the cluster, and $\phi(i,j) \in [0,1]$ is a pair-wise similarity between elements $i$ and $j$.
The linked article mentions that variance is used somehow to obtain the formula for $C$.
My questions are:

*

*Is this a common measure to score single clusters? If so, are there references.

*How is variance used to derive the second formula from the first?


At first sight I don't think the second formula is "derived" from the first. The formulae look like just constructed to have the desirable property that "high similarities $\Rightarrow$ large $c_i$, and "large $c_i$ overall $\Rightarrow$ large $C$", and then calibrating numbers to be in the desired $[0,1]$ range. The use of squares will just give low values of $c_i$ (or $\phi(i,j)$ in the first equation) more weight - whether this makes sense depends on the exact definition of $\phi(i,j)$ and the intended interpretation of the outcomes.
It isn't so obvious how variance comes in (actually it makes sense, see last paragraph), but I'd think that this is inspired by the variance formula Var$(X)=\frac{1}{n^2}\sum_i\sum_{j>i}(x_i-x_j)^2$ in the sense that $\sum_i (1-c_i)^2(N-1)$ involves $N(N-1)$ terms, with $(1-\phi(i,j))^2$ occurring twice for every pair $(i, j)$ in $\sum_i (1-c_i)^2$, whereas $\sum_{j>i}(x_i-x_j)^2$ has every distance between $x_i$ and $x_j$ once. That thing is divided by $n^2$ in the variance formula, so here it is divided by $2N^2$ to account for the fact that it's twice the terms, and the squares may make one think of a variance-like thing (for sure it's a measure of within-cluster variation, as is the variance).
Thinking about it more, in fact this seems to be directly based on the variance formula if  $(1-\phi(i,j))^2=(x_i-x_j)^2$ in a Euclidean setting. Then we have
$$
C=1-\sqrt{\frac{(N-1)\sum_i\sum_j(x_i-x_j)^2}{2N^2(N-1)}}=1-\sqrt{V},
$$
with $V$ being the within-cluster variance, so the formula is a generalisation (to more general choices of $\phi(i,j)$) of a "natural" variance-based measure of compactness in case that $(x_i-x_j)^2\in[0,1]$.
I hadn't seen this before and it's certainly not a standard in cluster analysis. The only thing I can imagine is that in applications in a specific field this is used, maybe together with a specific choice of $\phi(i,j)$ (on which it will depend whether this way of generalising from the Euclidean case makes sense).
