Tensions in Practice: Local similarity in tension with consistent grouping¶
Three invented numeric records
Call two records near when their values differ by at most 1. Values 1 and 2 are near; so are 2 and 3. But 1 and 3 are not. Keeping pairwise comparisons preserves that distinction. Treating every linked chain as one group produces a consistent partition, but also groups the endpoints. That is a change in the meaning of “same,” not a new measurement.
Respect the local similarity test
Keep the direct near/not-near answers for each pair.
Form consistent disjoint groups
Let chains of links define a class with transitive membership.
Why these aims pull against each other
Pairwise nearness need not support one-class-per-record grouping. Closing chains supplies transitivity by introducing additional within-class identifications.
Choose an arrangement to see what changes and what remains difficult.
Arrows express the declared relations, not measured effect sizes. Examples and quantities are illustrative.
What this choice protects
What it costs
When it fits
Compare the arrangements
Keep the pairwise relation
Use gap at most 1 as a direct comparison, without claiming it defines equivalence classes.
- What it protects
- A and C remain non-near.
- What it costs
- One cannot safely treat every connected pair as interchangeable under the original test.
- When it fits
- The task needs local comparisons rather than a partition into sameness classes.
Illustration note: Self-nearness and reverse directions are understood; the diagram draws only the two distinct near pairs.
Group each connected chain
Put all three connected records into one class.
- What it protects
- Membership is consistent across the chain and defines a partition.
- What it costs
- A and C become class-equivalent although they failed the direct similarity test.
- When it fits
- Connectedness is the intended grouping meaning, and endpoint differences are acceptable for the task.
Illustration note: Closure does not certify that these are the same real-world entity or that every operation can use one representative.
What this illustration does—and does not—establish
The source supplies the stated tension; the selected arrangements are bounded editorial illustrations. Costs and conditions remain part of the comparison.
- This is a deliberately exact three-value toy with threshold 1.
- Connectedness, pairwise proximity and real-world identity are different claims.
- A class is useful only for operations that respect its chosen equivalence; the diagram supplies no deduplication policy.
Source entries
Equivalence Relation
This source passage supplies the contextual tension. The concrete arrangements and schematic examples are editorial illustrations, not measured findings.
Strict transitivity versus similarity-based real-world grouping
T1 — Strict transitivity versus similarity-based real-world grouping. Most real-world sameness-judgements start from similarity scores or from domain-expert rules that are reflexive and symmetric but typically not transitive. The reflexive-symmetric-transitive closure construction converts a thresholded similarity into an equivalence relation, but the closure can chain marginally-similar pairs into operationally-unhelpful equivalence classes whose members are not in fact the same entity. Conversely, refusing to enforce transitivity (working with the underlying similarity relation directly, without closure) abandons the partition-and-quotient structure that licences the "study by representative" reduction and produces a relation whose downstream uses are limited to pairwise comparison.
The source operation
Every equivalence relation on $S$ partitions $S$ into pairwise-disjoint, non-empty *equivalence classes* whose union is all of $S$, and the partition view and the relation view are equivalent in a strict mathematical sense — every equivalence relation determines a unique partition and every partition determines a unique equivalence relation.
Similarity is not transitive
The transitivity failure is the defining structural difference between similarity-thresholding and equivalence relation, and it is the principal source of bugs in real-world sameness-claims: an analyst who treats a thresholded similarity as if it were an equivalence relation is implicitly invoking transitivity that the relation does not actually satisfy, and the resulting partition (computed, e.g., as the connected components of the thresholded similarity graph) may chain dissimilar items together through a sequence of bridging records.