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Differentially Private Densest Subgraph Detection

Dung Nguyen, Anil Vullikanti

TL;DR

This work studies the densest subgraph problem in the edge privacy model, in which the edges of the graph are private, and presents the first sequential and parallel differentially private algorithms for this problem.

Abstract

Densest subgraph detection is a fundamental graph mining problem, with a large number of applications. There has been a lot of work on efficient algorithms for finding the densest subgraph in massive networks. However, in many domains, the network is private, and returning a densest subgraph can reveal information about the network. Differential privacy is a powerful framework to handle such settings. We study the densest subgraph problem in the edge privacy model, in which the edges of the graph are private. We present the first sequential and parallel differentially private algorithms for this problem. We show that our algorithms have an additive approximation guarantee. We evaluate our algorithms on a large number of real-world networks, and observe a good privacy-accuracy tradeoff when the network has high density.

Differentially Private Densest Subgraph Detection

TL;DR

This work studies the densest subgraph problem in the edge privacy model, in which the edges of the graph are private, and presents the first sequential and parallel differentially private algorithms for this problem.

Abstract

Densest subgraph detection is a fundamental graph mining problem, with a large number of applications. There has been a lot of work on efficient algorithms for finding the densest subgraph in massive networks. However, in many domains, the network is private, and returning a densest subgraph can reveal information about the network. Differential privacy is a powerful framework to handle such settings. We study the densest subgraph problem in the edge privacy model, in which the edges of the graph are private. We present the first sequential and parallel differentially private algorithms for this problem. We show that our algorithms have an additive approximation guarantee. We evaluate our algorithms on a large number of real-world networks, and observe a good privacy-accuracy tradeoff when the network has high density.

Paper Structure

This paper contains 24 sections, 27 theorems, 22 equations, 7 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem statement
  4. Additional background on differential privacy
  5. Related Works
  6. Private Densest Subgraph
  7. Parallel Private Densest Subgraph
  8. Algorithm ParDenseDP
  9. Algorithm PhaseDenseDP: Logarithmic Bounded Runtime Algorithm
  10. Map-Reduce Implementation
  11. Lower bound
  12. Experiments
  13. Baselines and measurements
  14. Experimental Results
  15. Conclusions
  16. ...and 9 more sections

Key Result

Theorem 2.4

Utility of the Exponential Mechanism. (Theorem 3.11 and Corollary 3.12 of dwork:fttcs14) For a given dataset $x$, let $\text{OPT}=max_{r\in\mathcal{R}}u(x, r)$. For the exponential mechanism $M(\cdot)$, we have:

Figures (7)

  • Figure 1: Accuracy in term of relative density of our private algorithms. The number in each graph indicates the density of the graph as whole (which equals the average degree).
  • Figure 2: Jaccard similarity coefficient of private subgraphs and non-private baselines.
  • Figure 3: Recall: Fraction of nodes of the baseline's subgraphs are included in our algorithm's outputs.
  • Figure 4: Number of iterations taken by ParDenseDP, relative to SeqDenseDP.
  • Figure 5: The relationship between accuracy (relative density) and network density. We use $\epsilon\in\{1, 2\}, \delta=10^{-6}$. Each point corresponds to one network. The solid lines show linear models for the relative density and the network density. The plot confirms that our algorithms have better accuracy on higher density networks.
  • ...and 2 more figures

Theorems & Definitions (52)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Lemma 2.5
  • Corollary 2.5.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 42 more