Differentially Private Densest-$k$-Subgraph

TL;DR

This work addresses the problem of computing the densest-$k$-subgraph under edge differential privacy by privatizing the principal component of the graph adjacency matrix. It introduces two main approaches: a Propose-Test-Release (PTR) framework that leverages a data-dependent local-sensitivity bound to permit near-private PC outputs with fast closed-form steps, and a Private Power Method (PPM) that iteratively adds noise during power iterations for a DP principal component. The authors derive tight local-sensitivity bounds via spectral-gap analysis, show that smooth sensitivity bounds can be close to global sensitivity on real graphs, and provide a practical PTR implementation with theoretical DP guarantees and favorable runtimes, alongside a slower but sometimes advantageous PPM alternative. Experiments on real networks, including graphs with millions of nodes, demonstrate that PTR can achieve privacy-utility parity with the non-private baseline while offering substantial speedups (about 180x faster than PPM), making DP-D_kS feasible for large-scale graph data with sensible privacy budgets.

Abstract

Many graph datasets involve sensitive network data, motivating the need for privacy-preserving graph mining. The Densest-$k$-subgraph (D$k$S) problem is a key primitive in graph mining that aims to extract a subset of $k$ vertices with the maximum internal connectivity. Although non-private algorithms are known for D$k$S, this paper is the first to design algorithms that offer formal differential privacy (DP) guarantees for the problem. We base our general approach on using the principal component (PC) of the graph adjacency matrix to output a subset of $k$ vertices under edge DP. For this task, we first consider output perturbation, which traditionally offer good scalability, but at the expense of utility. Our tight on the local sensitivity indicate a big gap with the global sensitivity, motivating the use of instance specific sensitive methods for private PC. Next, we derive a tight bound on the smooth sensitivity and show that it can be close to the global sensitivity. This leads us to consider the Propose-Test-Release (PTR) framework for private PC. Although computationally expensive in general, we design a novel approach for implementing PTR in the same time as computation of a non-private PC, while offering good utility for \DkS{}. Additionally, we also consider the iterative private power method (PPM) for private PC, albeit it is significantly slower than PTR on large networks. We run our methods on diverse real-world networks, with the largest having 3 million vertices, and show good privacy-utility trade-offs. Although PTR requires a slightly larger privacy budget, on average, it achieves a 180-fold improvement in runtime over PPM.

Figures (4)

• Figure 1: Comparison of the edge-density (y-axis) versus subgraph size (x-axis) trade-off on the Orkut social network with $3M$ vertices and $120M$ edges. PPM (blue) and PTR (red) attain performance comparable with their non-private counterpart (yellow). However, PTR is $\approx 700$ times faster than PPM. Note that adding noise calibrated to global sensitivity (purple) produces unsatisfactory results, even with large privacy parameters.
• Figure 2: Edge density (y-axis) vs. subgraph size $k$ (x-axis): Experimental performance of PTR (red) and PPM (blue) in DP-D$k$S problem, compared to the non-private solution (yellow) and the upper bound on the maximum attainable edge density (black). Note that this bound is not always attainable in general. ($m$: the number of edges.)
• Figure 3: Edge density vs. size ($k$). Performance of Algorithm \ref{['alg:ptr']} for different values of epsilon on real-world datasets. $\delta = 1/m$, where $m$ is the number of edges in each network.
• Figure 4: Edge density vs. size ($k$). Performance of Algorithm \ref{['alg:algo1']} for different values of epsilon on real-world datasets. $\delta = 10^{-12}$.

Theorems & Definitions (18)

• Definition 1
• Proposition 3.1: Adapted from papailiopoulos2014finding
• Lemma 4.1
• Theorem 4.2
• proof
• Theorem 5.1
• proof
• Theorem 5.2
• proof
• Theorem 5.3