[Paper Review] Locally Differentially Private Analysis of Graph Statistics

Jacob Imola, Takao Murakami+1 more|arXiv (Cornell University)|Oct 17, 2020

Privacy-Preserving Technologies in Data54 references9 citations

TL;DR

This paper presents locally differentially private (LDP) algorithms for estimating subgraph counts—specifically triangles and k-stars—in graphs, enabling private analysis without a trusted curator. It introduces a two-round protocol that significantly reduces estimation error compared to non-interactive methods, achieving order-optimal error for k-star counts and demonstrating low relative error (e.g., 0.30 for triangles at ε=1) on real-world datasets like IMDB and Orkut.

ABSTRACT

Differentially private analysis of graphs is widely used for releasing statistics from sensitive graphs while still preserving user privacy. Most existing algorithms however are in a centralized privacy model, where a trusted data curator holds the entire graph. As this model raises a number of privacy and security issues -- such as, the trustworthiness of the curator and the possibility of data breaches, it is desirable to consider algorithms in a more decentralized local model where no server holds the entire graph. In this work, we consider a local model, and present algorithms for counting subgraphs -- a fundamental task for analyzing the connection patterns in a graph -- with LDP (Local Differential Privacy). For triangle counts, we present algorithms that use one and two rounds of interaction, and show that an additional round can significantly improve the utility. For $k$-star counts, we present an algorithm that achieves an order optimal estimation error in the non-interactive local model. We provide new lower-bounds on the estimation error for general graph statistics including triangle counts and $k$-star counts. Finally, we perform extensive experiments on two real datasets, and show that it is indeed possible to accurately estimate subgraph counts in the local differential privacy model.

Motivation & Objective

• To enable private analysis of graph statistics under the local differential privacy (LDP) model, avoiding reliance on a trusted curator.
• To address the challenge of estimating subgraph counts—such as triangles and k-stars—when edges are sensitive and users must locally perturb their data.
• To develop efficient, practical LDP algorithms that maintain high utility even with limited privacy budgets.
• To establish theoretical lower bounds on estimation error for general graph statistics, including triangles and k-stars.
• To empirically validate the utility of the proposed algorithms on real-world graph datasets under varying privacy budgets.

Proposed method

• Uses a local model where each user independently perturbs their edge information using randomized response or Laplace noise before sending it to a server.
• Proposes a two-round interaction protocol for triangle counting that improves utility by allowing users to adapt their responses based on aggregated feedback.
• Applies the Laplacian mechanism to achieve order-optimal estimation error for k-star counts in the non-interactive LDP model.
• Introduces a private estimation of maximum degree (d_max) using LDP, which is used to calibrate the noise in subgraph counting algorithms.
• Employs a privacy budget allocation strategy, splitting ε between estimating d_max and estimating subgraph counts to balance accuracy and privacy.
• Uses noisy max degree (d̂_max) as a proxy for true d_max in estimation, showing minimal utility loss compared to using the true value.

Experimental results

Research questions

• RQ1 Can subgraph counts such as triangles and k-stars be accurately estimated under the local differential privacy model without a trusted curator?
• RQ2 How does interaction (one vs. two rounds) affect the estimation error in locally private subgraph counting?
• RQ3 What is the fundamental limit (lower bound) on estimation error for general graph statistics under LDP?
• RQ4 Can order-optimal error be achieved for k-star counts in the non-interactive LDP model?
• RQ5 How does privately estimating the maximum degree impact the accuracy of subgraph count estimation?

Key findings

• The two-round protocol for triangle counting reduces estimation error significantly compared to the non-interactive version, demonstrating the benefit of interaction in LDP.
• For k-star counts, the proposed Laplacian-based algorithm achieves order-optimal estimation error in the non-interactive LDP model.
• The relative error for triangle counts is as low as 0.30 at ε=1 on the IMDB dataset, indicating high utility even with moderate privacy budgets.
• The relative error for 2-star counts is as low as 0.0028 at ε=1 on IMDB, showing excellent accuracy for dense graphs.
• Using a privately estimated d_max (d̂_max) results in relative error close to that of using the true d_max, validating its practicality.
• The clustering coefficient can be estimated with a relative error of 0.30 at ε=2, demonstrating feasibility for meaningful graph analysis under LDP.