What is the "mosaic effect" (linkage attack), and what is the key insight it forces about anonymous data sharing?

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Accepted Answer

The mosaic effect is re-identifying someone by combining several non-identifying attributes, often across datasets — so processing the quasi-identifiers is required for safe sharing.
No single tile of a mosaic shows the picture; assembled, they do. Likewise, gender alone, age alone, or a ZIP alone reveals little — but gender + age + ZIP uniquely identifies most people. A linkage attack joins your "anonymized" dataset to an external one sharing those quasi-identifiers (voter rolls, IMDb, social media) and re-attaches names.
Key insight: because the danger lives in the combination of quasi-identifiers, you cannot anonymize by only deleting direct identifiers. You must process (generalize/suppress) the quasi-identifiers themselves — that's the price of admission for anonymous data sharing.
Tip: Always ask "what could an attacker join this to?" — anonymity is a property of your data plus the rest of the world's data, not of your data alone.
Go deeper:

Robust De-anonymization (Narayanan & Shmatikov, 2008) — the canonical linkage-attack demonstration.
 Data re-identification (Wikipedia) — defines re-identification via auxiliary-data joins.