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?