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Differentially Private Distance Query with Asymmetric Noise

Weihong Sheng, Jiajun Chen, Chunqiang Hu, Bin Cai, Meng Han, Jiguo Yu

TL;DR

This paper considers edges as privacy and proposes distance publishing mechanisms based on edge DP, and formally gives the definition of asymmetric neighborhoods and proposes Individual Asymmetric Differential Privacy with higher privacy guarantees in combination with smooth sensitivity.

Abstract

With the growth of online social services, social information graphs are becoming increasingly complex. Privacy issues related to analyzing or publishing on social graphs are also becoming increasingly serious. Since the shortest paths play an important role in graphs, privately publishing the shortest paths or distances has attracted the attention of researchers. Differential privacy (DP) is an excellent standard for preserving privacy. However, existing works to answer the distance query with the guarantee of DP were almost based on the weight private graph assumption, not on the paths themselves. In this paper, we consider edges as privacy and propose distance publishing mechanisms based on edge DP. To address the issue of utility damage caused by large global sensitivities, we revisit studies related to asymmetric neighborhoods in DP with the observation that the distance query is monotonic in asymmetric neighborhoods. We formally give the definition of asymmetric neighborhoods and propose Individual Asymmetric Differential Privacy with higher privacy guarantees in combination with smooth sensitivity. Then, we introduce two methods to efficiently compute the smooth sensitivity of distance queries in asymmetric neighborhoods. Finally, we validate our scheme using both real-world and synthetic datasets, which can reduce the error to $0.0862$.

Differentially Private Distance Query with Asymmetric Noise

TL;DR

This paper considers edges as privacy and proposes distance publishing mechanisms based on edge DP, and formally gives the definition of asymmetric neighborhoods and proposes Individual Asymmetric Differential Privacy with higher privacy guarantees in combination with smooth sensitivity.

Abstract

With the growth of online social services, social information graphs are becoming increasingly complex. Privacy issues related to analyzing or publishing on social graphs are also becoming increasingly serious. Since the shortest paths play an important role in graphs, privately publishing the shortest paths or distances has attracted the attention of researchers. Differential privacy (DP) is an excellent standard for preserving privacy. However, existing works to answer the distance query with the guarantee of DP were almost based on the weight private graph assumption, not on the paths themselves. In this paper, we consider edges as privacy and propose distance publishing mechanisms based on edge DP. To address the issue of utility damage caused by large global sensitivities, we revisit studies related to asymmetric neighborhoods in DP with the observation that the distance query is monotonic in asymmetric neighborhoods. We formally give the definition of asymmetric neighborhoods and propose Individual Asymmetric Differential Privacy with higher privacy guarantees in combination with smooth sensitivity. Then, we introduce two methods to efficiently compute the smooth sensitivity of distance queries in asymmetric neighborhoods. Finally, we validate our scheme using both real-world and synthetic datasets, which can reduce the error to .
Paper Structure (27 sections, 21 theorems, 38 equations, 5 figures, 1 table, 2 algorithms)

This paper contains 27 sections, 21 theorems, 38 equations, 5 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Differential Privacy in Graph
  5. Sensitivity
  6. Shortest Path
  7. Asymmetric Properties
  8. Asymmetric Neighborhood
  9. Asymmetric Noise
  10. Asymmetric Neighborhood Differential Privacy
  11. Global Asymmetry
  12. Individual Asymmetry
  13. Privately Distance Query
  14. Problem
  15. Adding an Edge
  16. ...and 12 more sections

Key Result

Theorem 3.1

(Diameter bound mukwembi2012note) Let $G$ be a connected graph with $n$ vertices, for diameter $d$, with $d \neq 3,4$, we have: where $t$ is the number of distinct entries of the degree sequence and $\Bar{\delta}$ is the minimum degree of $G$ (for conflict avoidance, we override the normal degree notation $\delta$ to $\Bar{\delta}$ ).

Figures (5)

  • Figure 1: Add an edge.
  • Figure 2: All-pair distance mean relative error for SDP, ADP and IADP under three real-world datasets EIES, BOTC and TDE with the $\varepsilon$ increases from $1$ to $8$.
  • Figure 3: All-pair distance mean relative error for SDP, ADP and IADP under three synthesized datasets SHG, MHG and LHG with the $\varepsilon$ increases from $1$ to $18$.
  • Figure 4: All-pair distance mean relative error for IADP under three real-world datasets EIES, BOTC and TDE with the $\varepsilon$ increases from $1$ to $8$.
  • Figure 5: All-pair distance mean relative error for IADP under three real-world datasets SHG, MHG and LHG with the $\varepsilon$ increases from $1$ to $18$.

Theorems & Definitions (42)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 4.1
  • Definition 4.1
  • ...and 32 more