PrIsing: Privacy-Preserving Peer Effect Estimation via Ising Model

TL;DR

This work tackles privacy in Ising-model-based network analysis by targeting the node-outcome privacy of a single network realization. It introduces PrIsing, an objective-perturbation, $(\\varepsilon,\\delta)$-DP algorithm that perturbs the maximum pseudo-likelihood estimation to protect individual spins $\\sigma_i$ while estimating the Ising parameter $\\beta$. The authors prove regret bounds showing the private estimator achieves $|\\hat{\\beta}^{priv}-\\beta_0| = O_p\left(\frac{1}{\sqrt{a_n}} + \frac{\\lambda_n}{a_n\\varepsilon}\right)$, with $\\lambda_n = 1 \vee \|\\mathbf{J}_n\|_{1\to\infty}^2$, and provide concrete corollaries for degree-regular and Erdős–Rényi graphs that reveal a phase transition at $\\beta_0=1$ and the privacy-utility trade-offs shaped by graph density. Through simulations on ER graphs and real networks (HIV status and political blogs), the method demonstrates practical privacy guarantees with predictable performance degradation tied to the privacy budget. This work advances privacy-preserving network inference by delivering node-outcome DP for Ising models, quantifying the privacy cost, and validating applicability to real-world sensitive data.

Abstract

The Ising model, originally developed as a spin-glass model for ferromagnetic elements, has gained popularity as a network-based model for capturing dependencies in agents' outputs. Its increasing adoption in healthcare and the social sciences has raised privacy concerns regarding the confidentiality of agents' responses. In this paper, we present a novel $(\varepsilon,δ)$-differentially private algorithm specifically designed to protect the privacy of individual agents' outcomes. Our algorithm allows for precise estimation of the natural parameter using a single network through an objective perturbation technique. Furthermore, we establish regret bounds for this algorithm and assess its performance on synthetic datasets and two real-world networks: one involving HIV status in a social network and the other concerning the political leaning of online blogs.

Figures (13)

• Figure 1: Private and non-private MPLE in an Ising model on and Erdős-Rényi random graph.
• Figure 2: Effect of $n$ on MSE of MPLE in an Ising model on and Erdős-Rényi random graph with $\beta=0.5$
• Figure 3: Effect of $n$ on MSE of MPLE in an Ising model on and Erdős-Rényi random graph with $\beta=1.5$
• Figure 4: Effect of $p_n$ on MSE of MPLE in an Ising model on and Erdős-Rényi random graph with $\beta=0.5$
• Figure 5: Effect of $p_n$ on MSE of MPLE in an Ising model on and Erdős-Rényi random graph with $\beta=1.5$

Theorems & Definitions (12)

• Definition 2.1
• Theorem 3.1
• Theorem 3.2: Simpler Version of Theorem \ref{['thm:upperbdddetailed']}
• Corollary 3.1
• Corollary 3.2
• Lemma A.1
• Lemma A.2
• Lemma A.3
• Theorem B.1
• proof