[Paper Review] Communication-Efficient Triangle Counting under Local Differential Privacy

Jacob Imola, Takao Murakami+1 more|arXiv (Cornell University)|Oct 13, 2021

Privacy-Preserving Technologies in Data4 citations

TL;DR

This paper proposes communication-efficient triangle counting algorithms under local differential privacy (LDP) by introducing edge sampling and double clipping to drastically reduce communication costs while maintaining low estimation error. The proposed ARROneNS△ with double clipping reduces download time from 6 hours to under 8 seconds at 20 Mbps, making LDP-based triangle counting practical for large-scale graphs.

ABSTRACT

Triangle counting in networks under LDP (Local Differential Privacy) is a fundamental task for analyzing connection patterns or calculating a clustering coefficient while strongly protecting sensitive friendships from a central server. In particular, a recent study proposes an algorithm for this task that uses two rounds of interaction between users and the server to significantly reduce estimation error. However, this algorithm suffers from a prohibitively high communication cost due to a large noisy graph each user needs to download. In this work, we propose triangle counting algorithms under LDP with a small estimation error and communication cost. We first propose two-rounds algorithms consisting of edge sampling and carefully selecting edges each user downloads so that the estimation error is small. Then we propose a double clipping technique, which clips the number of edges and then the number of noisy triangles, to significantly reduce the sensitivity of each user's query. Through comprehensive evaluation, we show that our algorithms dramatically reduce the communication cost of the existing algorithm, e.g., from 6 hours to 8 seconds or less at a 20 Mbps download rate, while keeping a small estimation error.

Motivation & Objective

• Address the high communication cost of existing local differential privacy (LDP) triangle counting algorithms, which require users to download large noisy graphs.
• Reduce estimation error in LDP-based triangle counting, especially in dense or large-scale graphs where existing methods suffer from high error due to 4-cycles.
• Develop a practical LDP framework for triangle counting that supports decentralized and privacy-preserving graph analysis without relying on trusted servers.
• Enable accurate clustering coefficient estimation under LDP by leveraging improved triangle counting.

Proposed method

• Propose a two-round LDP protocol where users first download a sampled noisy graph and then compute local noisy triangle counts.
• Introduce edge sampling to reduce the size of the noisy graph each user must download, minimizing communication overhead.
• Apply a double clipping technique that first clips the number of edges per user and then clips the number of noisy triangles to reduce sensitivity and estimation error.
• Use carefully selected sampling probabilities to balance communication cost and estimation accuracy.
• Leverage the fact that each user can observe two edges of her incident triangles to reduce noise impact, improving accuracy over full edge obfuscation.
• Formulate the global sensitivity of triangle counts using the number of 4-cycles (C₄) and show that reducing C₄ via sampling improves estimation error.

Experimental results

Research questions

• RQ1 How can communication cost in LDP-based triangle counting be reduced without sacrificing estimation accuracy?
• RQ2 What is the impact of 4-cycles on estimation error in LDP triangle counting, and how can they be mitigated?
• RQ3 Can double clipping significantly reduce sensitivity and estimation error in LDP triangle counting?
• RQ4 How does the performance of the proposed algorithms scale with graph size and density?
• RQ5 Can the proposed method enable practical deployment of LDP-based triangle counting in real-world networks?

Key findings

• The ARROneNS△ algorithm with double clipping achieves the smallest estimation error across all datasets, especially outperforming others in large or dense graphs like Gplus.
• The double clipping technique reduces estimation error by two to three orders of magnitude compared to baseline methods.
• Communication cost is reduced from 6 hours to less than 8 seconds at a 20 Mbps download rate, making the protocol practical for real-world deployment.
• The number of 4-cycles (C₄) is a key factor in estimation error, and reducing C₄ via edge sampling significantly improves accuracy.
• The method enables accurate clustering coefficient estimation under LDP, extending its applicability to network analysis tasks.
• The algorithm maintains low relative error (e.g., 0.21) even at high privacy budgets, demonstrating strong privacy-utility trade-offs.