Joint Conformal Source Localization: Source Set Inference and Order-Adaptive Set Scoring

Source localization models assign scores to candidate origins of an observed spreading process on a network, but these scores alone provide no guarantee of recovering the true sources. Existing conformal methods provide source recall guarantees, yet selecting candidates by individual confidence can require large sets because it does not directly account for how their combined selection meets the recall target. We propose Joint Conformal Source Localization (JCSL), a post hoc framework that uses propagation constraints to infer a distribution over source sets and prioritizes nodes by their expected contribution to normalized source recovery. Order-adaptive set scoring uses information from the original model to determine how far to extend the refined ordering to meet the recall target after calibration. Theoretically, we establish finite-sample marginal recall guarantees under exchangeability and prove that each prefix of the proposed ordering maximizes the expected recall-aware objective among all candidate sets of the same size. Across 15 spreading conditions on three real-world networks, JCSL reduces mean set size by 3.45–95.35% against the best baseline in the main comparison. The mean rate of meeting the recall target is 0.8999 at nominal confidence 0.90. Ablations show that propagation constraints and order-adaptive scoring each reduce mean prediction set size.