While I was studying Hierarchical Agglomerative Clustering in the book Elements of Statistical Learning in the chapter Unsupervised Learning, I came through the following :

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The statement

The clusters produced by single linkage can violate the “compactness” property that all observations within each cluster tend to be similar to one another, based on the supplied observation dissimilarities {dii'}.

how is violating the "compactness"? How does single linkage produce clusters with very large diameters?

It will tend to produce compact clusters with small diameters

And how does "complete linkage cluster" produce compact clusters with small diameters?

These things are not making any sense to me.

  • 1
    $\begingroup$ Linkage methods in HAC are overviewed here. Single linkage is characterized as prone to form "chain"-type clusters while complete linkage - "circle (a community by interest)"-type clusters. Don't particularly like the terms used in your text "compactness" and "diameter" because complete method does not guarantee clusters "compact" in the intuitive sense of the word, and "diameter" is hardly understandable notion in, say, a curvilinear or a snowflake shape chain. $\endgroup$
    – ttnphns
    Commented Jun 19, 2018 at 7:45
  • $\begingroup$ "diameter " here means the maximum distance between the two items of single cluster. I am now in more confusion after going from those answers. Would you please explain these statements ? $\endgroup$
    – ironman
    Commented Jun 19, 2018 at 7:56
  • $\begingroup$ What is the issue the author is pointing out related to single linkage? $\endgroup$
    – ironman
    Commented Jun 19, 2018 at 8:02
  • $\begingroup$ Answers to your questions follow from the descriptions in the link I suggested to read. $\endgroup$
    – ttnphns
    Commented Jun 19, 2018 at 8:20

1 Answer 1


Thry defined compactness by the "diameter", the maximum distance.

Now single-link is prone to the single-link chaining effect.

If a is close to b, b is close to c, c is close to done, ... these points can all be on a line. And the diameter is in the worst case very large.

Complete link, which tries to minimize the diameter, surprise, yields a smaller diameter...

  • $\begingroup$ Sir, would you explain your answer a little bit more! The last two sentences is not so clear to me. Please! $\endgroup$
    – ironman
    Commented Jun 20, 2018 at 13:52

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