If we use agglomerative (hierarchical) clustering on data (you can assume data with an X and Y variable in a normal scatterplot, and can assume something like 10 observations) using complete linkage, and then we do it again 100 times, and the data stays the same every time, and every time we use complete linkage, will the clusters that exist when there are only 3 clusters left always be the same (meaning the clusters contain the same observations)?

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Question restatement: You have data with X and Y coordinates (about 10 observations) and you perform agglomerative (hierarchical) clustering using complete linkage. You then repeat the same clustering process 100 times on the exact same data, and each time you use complete linkage. When there are 3 clusters left, will the clusters consist of the same observations across all runs?

Option 1: The clusters could be different.
- In principle, hierarchical clustering with complete linkage is designed to be deterministic given a fixed dataset and a fixed implementation. If every run starts with the same distance matrix and uses the same rule for selecting the next pair to merge, the resulting dendrogram should be the same.
- However, many software libraries implement tie-breaking rules that resolve equal distances in a nondeterministic or user-configurable way. If the dataset happens to produce ties in the linkage distances at certain steps, different runs could choose different clusters to merge first, potentially leading to different cluster memberships when you look at the state with 3 clusters.
- In practice, for a small, low-noise dataset (like 10 points in a scatterplot with clean distances), ties are less likely, and many standard implementations are deterministic. That reduces the chance of variation across runs, but it is still conceptually possible if a tie occurs or if the library uses a randomization seed for tie-breaking.
- Misconception to watch out for here: variance across runs would imply some inherent randomness in the algorithm or implementation, not just natural data structure. Pure deterministic algorithms with fixed inputs do not spontaneously vary unless ties or stochastic elements are introduced.

Option 2: The clusters will be the same.
- Under a deterministic interpretation, given the exact same data and a fixed complete-linkage rule, the sequence of merges is fully determined by the pairwise distances, so the final partition at the stage of 3 clusters should be identical across runs.
- Complete linkage merges generally rely on the maximum distance between clusters when deciding which clusters to join next. Since these distances are computed from the same data each time, the merge order should repeat identically, assuming no randomness and no unstable tie-breaking.
- If the software enforces a consistent tie-breaking scheme (for example, always choosing the first pair in a predefined ordering when distances are equal), you would expect identical clusters across the 100 repetitions.
- Therefore, in typical scenarios with fixed data and a deterministic implementation, the 3-cluster partition would be reproducible across runs.

Balancing the two perspectives:
- It is important to recognize that the core claim hinges on determinism and tie-handling. If a given implementation guarantees deterministic behavior with a fixed seed for any nondeterminism (or guarantees no ties occur at the critical steps), the clusters will be the same.
- If there is any nondeterminism at any merge step due to equal distances and a nondeterministic tie-breaking rule (or if the library exposes randomized parameters or an unspecified random seed), then different runs could yield different 3-cluster partitions.

In short, the belief that the clusters will be the same rests on assuming a deterministic, tie-consistent implementation and identical data. The alternate view—that they could be different—appeals to the possibility of ties or nondeterministic tie-breaking in some tools. Both possibilities arise from how the specific software handles equal distances during hierarchical clustering, not from the inherent mathematics of complete linkage.

Additional notes:
- If you want to ensure reproducibility, you can (a) use a library option that fixes tie-breaking deterministically, (b) set a fixed random seed if the library supports it for clustering, and (c) verify the merges by examining the linkage matrix to confirm the same sequence of merges across runs.

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