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Despite proportionality being one of the tenets of data protection laws, wecurrently lack a robust analytical framework to evaluate the reach of moderndata collections and the network effects at play. We here propose agraph-theoretic model and notions of node- and edge-observability to quantifythe reach of networked data collections. We first prove closed-form expressionsfor our metrics and quantify the impact of the graph's structure onobservability. Second, using our model, we quantify how (1) from 270,000compromised accounts, Cambridge Analytica collected 68.0M Facebook profiles;(2) from surveilling 0.01\% the nodes in a mobile phone network, alaw-enforcement agency could observe 18.6\% of all communications; and (3) anapp installed on 1\% of smartphones could monitor the location of half of theLondon population through close proximity tracing. Better quantifying the reachof data collection mechanisms is essential to evaluate their proportionality.

Detrimental Network Effects in Privacy: A Graph-theoretic Model for Node-based Intrusions