I've noticed that if I'm doing k-means clustering (in MATLAB) on basically any set of data (not randomness), the mean and variance in centroid linkage distance appears to always be approximately proportional to k.

Does k always correlate with the centroid linkage distance mean and variance? If so, why exactly is this true? For example, is it to do with the fact that k-means separates data into veronoi cells? Is it the convexity of those cells that necessitates this scaling?

If this scaling doesn't always happen, in what cases does it fail?

I've noticed that if I'm doing k-means clustering (in MATLAB) on basically any set of data (not randomness), the mean and variance in centroid linkage distance appears to always be approximately proportional to k.

The centroid linkage distance is the distance between two cluster centroids. So there are $\frac{k(k-1)}{2}$ distances in total.

Does k always correlate with the centroid linkage distance mean and variance? If so, why exactly is this true? For example, is it to do with the fact that k-means separates data into veronoi cells? Is it the convexity of those cells that necessitates this scaling?

If this scaling doesn't always happen, in what cases does it fail?

I've noticed that if I'm doing k-means clustering (in MATLAB) on basically any set of data, the mean and variance in centroid linkage distance appears to always be approximately proportional to k.

Is k always proportional to the centroid linkage distance mean and variance? If so, why exactly is this true? For example, is it to do with the fact that k-means separates data into veronoi cells? Is it the convexity of those cells that necessitates this scaling?

If this scaling doesn't always happen, in what cases does it fail?

I've noticed that if I'm doing k-means clustering (in MATLAB) on basically any set of data (not randomness), the mean and variance in centroid linkage distance appears to always be approximately proportional to k.

Does k always correlate with the centroid linkage distance mean and variance? If so, why exactly is this true? For example, is it to do with the fact that k-means separates data into veronoi cells? Is it the convexity of those cells that necessitates this scaling?

If this scaling doesn't always happen, in what cases does it fail?