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Private Edge Density Estimation for Random Graphs: Optimal, Efficient and Robust

Hongjie Chen, Jingqiu Ding, Yiding Hua, David Steurer

TL;DR

This work tackles private edge-density estimation for Erdős-Rényi and inhomogeneous random graphs under node differential privacy, achieving polynomial-time, private, and robust estimation with near-optimal accuracy. The authors develop a two-stage approach: a coarse SOS-based robust estimator followed by a fine estimator, both integrated into a privacy-preserving framework via a sum-of-squares exponential mechanism. A core contribution is the reduction from privacy to robustness, enabling private output to inherit adversarial resilience; this is realized through a level-8 SOS relaxation that certifies key regularity properties and concentration phenomena. The resulting error bounds are information-theoretically optimal up to logarithmic factors, with matching lower bounds established for both ER and inhomogeneous graph models. The methods extend beyond ER to broad random-graph models, offering a principled private-robust inference tool powered by SOS proofs and privacy-preserving exponential mechanisms, with practical implications for privately releasing graph-structure statistics at scale.

Abstract

We give the first polynomial-time, differentially node-private, and robust algorithm for estimating the edge density of Erdős-Rényi random graphs and their generalization, inhomogeneous random graphs. We further prove information-theoretical lower bounds, showing that the error rate of our algorithm is optimal up to logarithmic factors. Previous algorithms incur either exponential running time or suboptimal error rates. Two key ingredients of our algorithm are (1) a new sum-of-squares algorithm for robust edge density estimation, and (2) the reduction from privacy to robustness based on sum-of-squares exponential mechanisms due to Hopkins et al. (STOC 2023).

Private Edge Density Estimation for Random Graphs: Optimal, Efficient and Robust

TL;DR

This work tackles private edge-density estimation for Erdős-Rényi and inhomogeneous random graphs under node differential privacy, achieving polynomial-time, private, and robust estimation with near-optimal accuracy. The authors develop a two-stage approach: a coarse SOS-based robust estimator followed by a fine estimator, both integrated into a privacy-preserving framework via a sum-of-squares exponential mechanism. A core contribution is the reduction from privacy to robustness, enabling private output to inherit adversarial resilience; this is realized through a level-8 SOS relaxation that certifies key regularity properties and concentration phenomena. The resulting error bounds are information-theoretically optimal up to logarithmic factors, with matching lower bounds established for both ER and inhomogeneous graph models. The methods extend beyond ER to broad random-graph models, offering a principled private-robust inference tool powered by SOS proofs and privacy-preserving exponential mechanisms, with practical implications for privately releasing graph-structure statistics at scale.

Abstract

We give the first polynomial-time, differentially node-private, and robust algorithm for estimating the edge density of Erdős-Rényi random graphs and their generalization, inhomogeneous random graphs. We further prove information-theoretical lower bounds, showing that the error rate of our algorithm is optimal up to logarithmic factors. Previous algorithms incur either exponential running time or suboptimal error rates. Two key ingredients of our algorithm are (1) a new sum-of-squares algorithm for robust edge density estimation, and (2) the reduction from privacy to robustness based on sum-of-squares exponential mechanisms due to Hopkins et al. (STOC 2023).
Paper Structure (69 sections, 54 theorems, 185 equations, 8 algorithms)

This paper contains 69 sections, 54 theorems, 185 equations, 8 algorithms.

Table of Contents

  1. Introduction
  2. Results
  3. Erdős-Rényi random graphs
  4. Inhomogeneous random graphs
  5. Techniques
  6. Reduction from privacy to robustness
  7. Robust algorithm via sum-of-squares
  8. Sum-of-squares exponential mechanism
  9. Notation
  10. Organization
  11. Private and robust algorithm for Erdős-Rényi random graphs
  12. General algorithm framework
  13. Coarse estimation
  14. Identifiability proof for robust estimation
  15. Robust algorithm via sum-of-squares
  16. ...and 54 more sections

Key Result

Theorem 1.4

There are constants $C_1, C_2, C_3$ such that the following holds. For any $\eta\leqslant C_1$, $\varepsilon \geqslant C_2\log(n)/n$, and $p^\circ\geqslant C_3/n$, there exists a polynomial-time $\varepsilon$-differentially node-private algorithm which, given an $\eta$-corrupted Erdős-Rényi random g with probability $1-n^{-\Omega(1)}}$.

Theorems & Definitions (103)

  • Definition 1.1: Node distance, neighboring graphs
  • Definition 1.2: Node differential privacy
  • Definition 1.3: Node corruption model
  • Theorem 1.4: Erdős-Rényi random graphs, combination of \ref{['thm:coarse_estimation_inhomo']} and \ref{['thm:fine_estimation_er']}
  • Theorem 1.5: Privacy lower bound for Erdős-Rényi random graphs
  • Theorem 1.6: Inhomogeneous random graphs, combination of \ref{['thm:coarse_estimation_inhomo']} and \ref{['thm:fine_estimation_inhomo']}
  • Theorem 1.7: Robustness lower bound for inhomogeneous random graphs
  • Theorem 1.8: Privacy lower bound for inhomogeneous random graphs
  • Lemma 2.1: Privacy
  • proof
  • ...and 93 more