A Graph-Based Framework for Extending Metric Differential Privacy Mechanisms

Metric differential privacy (mDP) is well suited to structured secret domains, but directly constructing utility-aware mechanisms over large or fine-grained domains is often computationally prohibitive. We study extension-based mDP design, where a mechanism is first specified on a finite set of seed records and then extended to a larger target domain. To our knowledge, this is the first work to systematically formulate extension as a general design paradigm for mDP rather than a method-specific construction. We present a graph-based extension framework, identify three requirements for correctness, local mDP constraints, overlap consistency, and successor-level mDP preservation, and show that, under these conditions, the induced global mechanism is well defined and satisfies $\epsilon$-mDP on the target domain. We further instantiate the framework with a tree-based extension algorithm for multi-resolution grids, where multi-dimensional extension is realized through one-dimensional interpolation and dimension-wise composition. Experiments on road-map datasets demonstrate that our approach achieves a strong utility-scalability trade-off while preserving exact mDP guarantees.