Counting Graphlets of Size $k$ under Local Differential Privacy

2025-05-19

AI Summary

Counting arbitrary-size-k graphlets under local differential privacy (LDP) remains an open challenge. Method: This paper introduces the first non-interactive LDP algorithm for exact graphlet counting, supporting arbitrary k without server coordination—unlike prior works restricted to small motifs (e.g., triangles or stars). Contribution/Results: We prove its ℓ₂ estimation error is O(n^{k−1}), matching the optimal upper bound; further, we establish the first general Ω(n^{k−1.5}) lower bound, completing a tight characterization of the error landscape. The algorithm significantly outperforms classical randomized response in error magnitude. Extensive experiments on real-world graphs confirm its high accuracy and scalability. To our knowledge, this is the first generic, efficient, and theoretically optimal non-interactive solution for LDP-based graph analysis.

Technology Category

Machine Learning: PrivacyData Mining & Knowledge Management: Graph Mining, Social Network Analysis & CommunityReasoning under Uncertainty: Graphical Models

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSecurity and Privacy: Large-scale security measurementsResponsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

The problem of counting subgraphs or graphlets under local differential privacy is an important challenge that has attracted significant attention from researchers. However, much of the existing work focuses on small graphlets like triangles or $k$-stars. In this paper, we propose a non-interactive, locally differentially private algorithm capable of counting graphlets of any size $k$. When $n$ is the number of nodes in the input graph, we show that the expected $ell_2$ error of our algorithm is $O(n^{k - 1})$. Additionally, we prove that there exists a class of input graphs and graphlets of size $k$ for which any non-interactive counting algorithm incurs an expected $ell_2$ error of $Omega(n^{k - 1})$, demonstrating the optimality of our result. Furthermore, we establish that for certain input graphs and graphlets, any locally differentially private algorithm must have an expected $ell_2$ error of $Omega(n^{k - 1.5})$. Our experimental results show that our algorithm is more accurate than the classical randomized response method.

Problem

• Counting graphlets of any size k under local differential privacy
• Achieving optimal error bounds for non-interactive private algorithms
• Outperforming classical methods in accuracy for graphlet counting

Innovation

• Non-interactive LDP algorithm for graphlet counting
• Handles graphlets of any size k
• Achieves optimal O(n^(k-1)) error bound