Freshness-Regulated Adaptive Noise (FRAN)

• Freshness-Regulated Adaptive Noise (FRAN) is defined as a mechanism that regulates injected noise in multivariate data streams by accounting for temporal and cross-sequence correlations.
• It employs a Coupling Markov Chain model coupled with sensitivity analysis to quantify leakage and ensure robust privacy with controlled noise levels.
• Experimental evaluations demonstrate that FRAN significantly improves the privacy-utility trade-off, achieving up to 40% MSE reduction compared to standard differential privacy methods.

Freshness-Regulated Adaptive Noise (FRAN) is a privacy-preserving mechanism devised within the Correlated-Sequence Differential Privacy (CSDP) framework to address the privacy leakage that arises from both temporal and cross-sequence correlations in multivariate data streams. Conventional Differential Privacy (DP) mechanisms assume record independence and become suboptimal or even fail for highly correlated streams, often requiring excessive noise that degrades utility. FRAN provides a mathematically rigorous approach by leveraging explicit correlation models and dynamically regulating both the “freshness” of released data and the scale of injected noise according to data and correlation characteristics.

1. The Correlated-Sequence Problem: Motivating FRAN

Multivariate streaming datasets such as multi-sensor feeds or user activity logs typically exhibit two types of dependencies: temporal (the current value in each sequence depends on previous values) and cross-sequence (different sources are statistically coupled). Standard $\epsilon$-DP mechanisms fail to provide privacy when these dependencies allow adversaries to infer more than the claimed privacy budget or force practitioners to add prohibitively large amounts of noise, thus destroying data utility (Luo et al., 22 Nov 2025). FRAN's design directly targets these failures, offering effective privacy guarantees tailored for sequences with rich correlation structure.

2. Coupling Markov Chain Model and Sensitivity Analysis

FRAN relies on the Coupling Markov Chain (CMC) formalism, modeling $s$ categorical sequences $$\boldsymbol x^{(i)} = (x^{(i)}_1, \dots, x^{(i)}_T)$$, with joint state vector $\boldsymbol x_t = (x^{(1)}_t, \dots, x^{(s)}_t)$ and corresponding distribution $\boldsymbol{\pi}_t$. The transition is governed by a block matrix $Q$ with coupling weights $\lambda_{jk}$ and transition blocks $P^{(jk)} \in \mathbb{R}^{m \times m}$:

$\boldsymbol{\pi}_{t+1} = Q \boldsymbol{\pi}_t,$

where $Q$’s spectral properties (dominant eigenvalue $1$, spectral gap $1-\rho$ with $\rho = \max_{i\geq2}|\lambda_i(Q)|$) enable explicit quantification of mixing rates and the decoupling effect of aging. The mechanism computes $k$-sensitivity $d(k)$, formally:

$$d(k) = \max_{X, X': \mathrm{dist}(X, X') \leq k} \|f(X) - f(X')\|_1,$$

where $f$ is the query function and the parameter $k$ reflects the local correlation degree.

3. Leakage Bounds and Theoretical Guarantees

The privacy leakage of FRAN is contingent on both the sensitivity under correlation and the strength/structure of those correlations as aged by the mechanism. For a Laplace mechanism with scale $\epsilon_C$, the loose leakage bound is expressed,

$$\epsilon_S(\boldsymbol A_t, \epsilon_C) = d(k)\,\Delta_k(\Lambda, \boldsymbol A_t)\,\epsilon_C,$$

where $\Delta_k(\Lambda, \boldsymbol A_t)$ denotes the aged correlation (total variation distance after time-shifting by Age-of-Information $\boldsymbol A_t$). A tighter bound can be obtained via an optimized aged-correlation metric $\bar\Delta(\Lambda, \boldsymbol A_t)$, yielding

$$\epsilon_S^{\mathrm{tight}} = \bar\Delta(\Lambda, \boldsymbol A_t)\,\epsilon_C.$$

Spectral analysis enables bounding $\Delta_k$ as

$$\Delta_k(\Lambda, \boldsymbol A_t) \leq C\,\rho^{\min_i A^{(i)}_t},$$

indicating that appropriately chosen coupling strengths $\lambda_{jk}$ (thus spectral gap) and aging can reduce leakage.

4. FRAN Mechanism: Phases and Computation

FRAN operates in two distinct phases:

Phase 1: Data-Aging

Each sequence $i$ is time-shifted by its Age-of-Information $A^{(i)}_t$ to produce $\tilde x^{(i)}_t = x^{(i)}_{t-A^{(i)}_t}$, which mitigates correlations through temporal decay at the expense of freshness.

Phase 2: Correlation-Aware Noise Injection

The mechanism computes the correlation-adjusted sensitivity,

$$\Delta_f^{\mathrm{corr}} = d(k)\,\Delta_k(\Lambda, \boldsymbol A_t),$$

and injects independent Laplace noise, $$\eta \sim \mathrm{Lap}(\Delta_f^{\mathrm{corr}}/\epsilon_C)$$, per query dimension.

Algorithm Complexity: Linear in the number of sequences and query dimensions, $O(s+d)$ per time step; thus scalable to thousands of streams and real-time rates (Luo et al., 22 Nov 2025).

Phase	Purpose	Key Operations
Data-Aging	Reduce temporal/cross-sequence correlation	Time-shift each sequence by AoI
Noise Injection	Hide residual sensitivity/leakage	Compute $\Delta_f^{\mathrm{corr}}$, add Laplace noise

5. Parameterization and Practical Deployment

Several parameters and trade-offs define FRAN's deployment:

• Privacy Budget Allocation: Choose $\epsilon_C$ such that $\epsilon_S = d(k)\,\Delta_k\,\epsilon_C$ remains within a target threshold.
• AoI Tuning: Larger $A^{(i)}_t$ (i.e., older data) reduces $\Delta_k$ exponentially in the spectral mixing rate, permitting less noise for equivalent privacy, but with stale output.
• Coupling Strength Selection: Adjusting $\lambda_{jk}$ tunes the spectral gap; moderate/strong coupling accelerates mixing and reduces leakage.
• Query Sensitivity Considerations: Queries with high $k$-sensitivity (e.g., sums) require more noise; low-sensitivity queries are easier to protect.
• Adaptive Scheduling and Monitoring: Online estimation of $Q$ and spectral parameters, adaptive AoI scheduling in response to detected correlation bursts, real-time utility tracking (e.g., MSE monitoring), and seamless integration with standard DP engines by substituting Laplace scales.

6. Experimental Results and Empirical Insights

Experiments performed on two-sequence binary CMCs ($s=2, m=2$) with identical block transitions and variable self-coupling $\lambda$ demonstrated:

• Privacy-Utility Trade-off: FRAN (CSDP) achieves $\epsilon_S \approx 4.5 \times 10^{-5}$, representing approximately 50% improvement over Age‐DP and two orders of magnitude over standard DP and DDP for the same accuracy constraint.
• Mean Squared Error (MSE): Up to 30–40% reduction in error for an equivalent leakage budget via careful tuning of AoI and $\epsilon_C$.
• Temporal Decay: Leakage decreases by roughly 70% over the first four aging steps; the leakage vs. coupling strength exhibits a U-shaped curve with minimal leakage at $\lambda=0.5$.

Method	Privacy Leakage $\epsilon_S$	MSE Improvements
Standard DP/DDP	$\approx 0.08$	Baseline
Age-DP	$\approx 9 \times 10^{-5}$	Improved
FRAN (CSDP)	$\approx 4.5 \times 10^{-5}$	30–40% lower

7. Integration and Scalability in Streaming Systems

FRAN natively supports integration into existing privacy infrastructures via simple replacement of Laplace noise scaling ($$\Delta_f/\epsilon \rightarrow \Delta_f^{\mathrm{corr}}/\epsilon_C$$), and its linear-time complexity ($O(s+d)$) facilitates application to high-dimensional, high-rate streaming contexts. Offline precomputation of spectral bounds and adaptive real-time AoI scheduling are central for efficiency. Monitoring deployment utility via error metrics and dynamic adjustment of parameters enables Service Level Agreement (SLA) maintenance. This suggests FRAN is suitable for operational deployment in domains with strict privacy and utility demands, such as smart-city sensing, healthcare, and financial analytics (Luo et al., 22 Nov 2025).