Counting Graphlets of Size $k$ under Local Differential Privacy

TL;DR

The paper addresses counting graphlets of size $k$ under edge-local differential privacy by introducing a non-interactive algorithm that achieves an expected graphlet count with $\ell_2$-error $O(n^{k-1})$. It proves matching lower bounds for non-interactive methods, $\Omega(n^{k-1})$, and a stronger bound $\Omega(n^{k-1.5})$ for general algorithms, establishing asymptotic optimality under the model. The approach builds on unbiased randomized response and automorphism-based counting to extend triangle-counting techniques to arbitrary $k$, with experiments showing substantial improvements over randomized response, especially for larger graphs. The work also provides detailed lower-bound constructions and discusses practical limitations such as computational complexity, while outlining potential future improvements through sampling and tighter bounds involving $\epsilon$.

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 $Ω(n^{k - 1})$, 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 $Ω(n^{k - 1.5})$. Our experimental results show that our algorithm is more accurate than the classical randomized response method.

Figures (6)

• Figure 1: The gadget we use to show the lower bound of $\ell_2$-error when counting $K_4$.
• Figure 2: Root mean square error of our Algorithm \ref{['alg1']} compared with the randomized response mechanism (a) in the stochastic block model when the privacy budget $\epsilon = 1$, (b) in the Barabási–Albert model when $\epsilon = 1$, (c) in the stochastic block model when $\epsilon = 5$, and (d) in the Barabási–Albert model when $\epsilon = 5$. We omit the result for the randomized response in (c) when $n = 10$ as the error is zero, making it impossible to display on a log scale plot.
• Figure 3: Relative root mean square error of our Algorithm \ref{['alg1']} compared with the randomized response mechanism in the stochastic block model (a) in the stochastic block model when the privacy budget $\epsilon = 1$, (b) in the Barabási–Albert model when $\epsilon = 1$, (c) in the stochastic block model when $\epsilon = 5$, and (d) in the Barabási–Albert model when $\epsilon = 5$. We omit the result for the randomized response in (c) when $n = 10$ as the error is zero, making it impossible to display on a log scale plot.
• Figure 4: (a) Computation time of our Algorithm 1 as a function of the number of nodes (b) Standard deviation of our results for graphs derived from the stochastic block model when the privacy budget is set to one.
• Figure 5: Gadget used in eden2023triangle for showing the lower bound in $\ell_2$-error of estimating the number of triangles.

Theorems & Definitions (25)

• Definition 1: Local differentially private query
• Definition 2: qin2017generating
• Definition 3: Non-interactive algorithm
• Definition 4: warner_randomized_1965wang2016using
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof