Counting Graphlets of Size $k$ under Local Differential Privacy

TL;DR

This paper introduces a non-interactive local differential privacy algorithm for counting graphlets of any size in graphs, achieving near-optimal error bounds and outperforming classical methods in accuracy.

Contribution

The paper presents the first non-interactive algorithm for counting arbitrary-sized graphlets under local differential privacy with proven optimal error bounds.

Findings

Expected $\, ext{l}_2$ error is $O(n^{k-1})$

Proves lower bounds of $\, ext{Omega}(n^{k-1})$ and $\, ext{Omega}(n^{k-1.5})$ for certain cases

Algorithm outperforms classical randomized response in experiments

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.