Critical retention thresholds under alternative observation models

Determine, for alternative observation models that generate the prior subgraph G from the ground-truth graph G*, the critical edge-retention levels η(P) at which graph-theoretic tasks P (e.g., s–t connectivity, Steiner tree recovery, triangle detection, or counting) transition from requiring superconstant query complexity to permitting constant expected query complexity. Analyze concrete models including radius-dependent thinning in random geometric k-NN graphs and adversarial edge deletions, and rigorously characterize the corresponding query-complexity phase transitions.

Background

The main analysis uses an i.i.d. edge-retention process to obtain G from G*, yielding explicit phase transitions for retrieval friendliness. The authors emphasize that more realistic observation mechanisms could preserve local edges while suppressing long ones or involve adversarial deletions.

They state that each such mechanism induces a distinct critical retention level η(P), and explicitly note that this poses open problems connecting random graph theory and query complexity. These problems aim to generalize the phase-transition characterization beyond i.i.d. retention.