Differentially Private Online Community Detection for Censored Block Models: Algorithms and Fundamental Limits

TL;DR

This work tackles private online change detection in dynamic networks modeled by the censored block model, addressing both local and central edge differential privacy. It proposes Privatized Adaptive CUSUM schemes that jointly estimate the post-change community and perform sequential detection, using graph perturbation under LDP and a stability-based mechanism under CDP, with SDP-based recovery guiding the detection statistic. Theoretical results establish sufficient and necessary conditions for exact community recovery under privacy, along with ARL and delay guarantees and fundamental lower bounds on detection delay under privacy constraints. The methods are validated through simulations and real-data case studies (agriculture and U.S. aviation), illustrating the privacy–utility tradeoffs and showing practical effectiveness even under stringent privacy budgets. Overall, the paper advances privacy-preserving online network analysis by linking precise recovery thresholds, adaptive detection performance, and computationally efficient private mechanisms.

Abstract

We study the private online change detection problem for dynamic communities, using a censored block model (CBM). We consider edge differential privacy (DP) in both local and central settings, and propose joint change detection and community estimation procedures for both scenarios. We seek to understand the fundamental tradeoffs between the privacy budget, detection delay, and exact community recovery of community labels. Further, we provide theoretical guarantees for the effectiveness of our proposed method by showing necessary and sufficient conditions for change detection and exact recovery under edge DP. Simulation and real data examples are provided to validate the proposed methods.

Figures (10)

• Figure 1: Overview of the proposed change‑detection scheme. We develop online community‑detection frameworks that detect changes in community memberships via CUSUM-based tests. The framework also incorporates an SDP-based community recovery component to estimate the unknown post-change communities. Both the estimation and detection components satisfy edge differential privacy to protect individual node interactions.
• Figure 2: Illustration of the graph perturbation-based mechanism: The original edges are first randomized via a randomized response scheme, and the resulting noisy adjacency matrix is then supplied to the community recovery algorithm.
• Figure 3: Illustration of the subsampling stability-based mechanism: From the original graph we generate $m$ correlated subgraphs by including each edge independently with probability $q_s$. A non-private community estimator is then applied to every subgraph, producing label vectors $\{\hat{\bm{\sigma}}({\mathbf{A}}_k)\}_{k=1}^m$. These labelings are tallied in a histogram, and the modal labeling— provably stable under appropriately chosen $(q_s,m,\epsilon,\delta)$—is returned as the private output.
• Figure 4: Exact Recovery Regime that depends on $(a,\zeta)$, where $n = 100$ and $\epsilon = \log(n)$.
• Figure 5: Sample statistics trajectories and detection delay of the method $T_{\rm L}$ in \ref{['eq:stop_time']} for both cases.

Theorems & Definitions (31)

• Remark 2.1: Estimator of $\bm{\sigma}^{*}$
• Definition 1: $\epsilon$-edge LDP
• Definition 2: $(\epsilon, \delta)$-edge CDP
• Remark 3.1: Unknown Parameters
• Definition 3: Stability of $\hat{\bm{\sigma}}$
• Remark 3.2: Efficient Computation
• Definition 4: Exact Recovery
• Theorem 4.1
• Theorem 4.2
• Theorem 4.3