  The proliferation of real-time applications has motivated extensive research
on analyzing and optimizing data freshness in the context of \textit{age of
information}. However, classical frameworks of privacy (e.g., differential
privacy (DP)) have overlooked the impact of data freshness on privacy
guarantees, which may lead to unnecessary accuracy loss when trying to achieve
meaningful privacy guarantees in time-varying databases. In this work, we
introduce \textit{age-dependent DP}, taking into account the underlying
stochastic nature of a time-varying database. In this new framework, we
establish a connection between classical DP and age-dependent DP, based on
which we characterize the impact of data staleness and temporal correlation on
privacy guarantees. Our characterization demonstrates that \textit{aging},
i.e., using stale data inputs and/or postponing the release of outputs, can be
a new strategy to protect data privacy in addition to noise injection in the
traditional DP framework. Furthermore, to generalize our results to a
multi-query scenario, we present a sequential composition result for
age-dependent DP under any publishing and aging policies. We then characterize
the optimal tradeoffs between privacy risk and utility and show how this can be
achieved. Finally, case studies show that to achieve a target of an arbitrarily
small privacy risk in a single-query case, combing aging and noise injection
only leads to a bounded accuracy loss, whereas using noise injection only (as
in the benchmark case of DP) will lead to an unbounded accuracy loss.
