# On Differentially Private Graph Sparsification and Applications

@inproceedings{Arora2019OnDP, title={On Differentially Private Graph Sparsification and Applications}, author={R. Arora and Jalaj Upadhyay}, booktitle={Neural Information Processing Systems}, year={2019} }

In this paper, we study private sparsification of graphs. In particular, we give an algorithm that given an input graph, returns a sparse graph which approximates the spectrum of the input graph while ensuring differential privacy. This allows one to solve many graph problems privately yet efficiently and accurately. This is exemplified with application of the proposed meta-algorithm to graph algorithms for privately answering cut-queries, as well as practical algorithms for computing {\scshape…

## 27 Citations

### Differential Privacy from Locally Adjustable Graph Algorithms: k-Core Decomposition, Low Out-Degree Ordering, and Densest Subgraphs

- Computer Science2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)
- 2022

This paper defines locally adjustable graph algorithms and shows that algorithms of this type can be transformed into differentially private algorithms, and presents an $\varepsilon$-locally edge differentiallyPrivate (LEDP) algorithm for k-core decompositions.

### Differentially Private Analysis on Graph Streams

- Computer ScienceAISTATS
- 2021

Since di↵erential privacy is preserved under post-processing, the results can be used as a subroutine in many tasks, most notably solving cut functions and spectral clustering.

### Differentially Private Algorithms for Graphs Under Continual Observation

- Computer ScienceESA
- 2021

This work presents event-level and user-level private algorithms for partially dynamic counting-based problems such as triangle count that improve the additive error by a polynomial factor on the state of the art, resulting in the first algorithms with additive error polylogarithmic in T .

### SoK: Differential Privacy on Graph-Structured Data

- Computer ScienceArXiv
- 2022

This work systematise different formulations of DP on graphs, discuss challenges and promising applications, including the GNN domain, and compares and separate works into graph analysis tasks and graph learning tasks with GNNs.

### Breaking the Linear Error Barrier in Differentially Private Graph Distance Release

- Computer Science, MathematicsArXiv
- 2022

This paper proposes an algorithm that releases a constructed synthetic graph privately and shows that the new graph is diﬀerentially private and can be published to answer all pairwise shortest path distances with ˜ O ( n 1 / 2 ) approximation error using standard APSP computation.

### Differentially-Private Hierarchical Clustering with Provable Approximation Guarantees

- Computer ScienceArXiv
- 2023

This work focuses on the stochastic block model, a popular model of graphs, and proposes a private $1+o(1)$ approximation algorithm which also recovers the blocks exactly, and shows strong lower bounds for the problem.

### Distances Release with Differential Privacy in Tree and Grid Graph

- Computer Science2022 IEEE International Symposium on Information Theory (ISIT)
- 2022

This paper revisits the problem of privately releasing approximate distances between all pairs of vertices in [1] and proposes improved solutions to that problem for several cases and proposes a method to release all-pairs distances with additive error.

### Differentially private graph publishing with degree distribution preservation

- Computer ScienceComput. Secur.
- 2021

### Principle of Relevant Information for Graph Sparsification

- Computer ScienceUAI
- 2022

This paper proposes the first general and effective information-theoretic formulation of graph sparsiﬁcation, by taking inspiration from the Principle of Relevant Information (PRI), and extends the PRI from a standard scalar random variable setting to structured data (i.e., graphs).

### A Framework for Private Matrix Analysis

- Computer Science, MathematicsArXiv
- 2020

A lower bound on space required to compute low-rank approximation even if the algorithm gives multiplicative approximation and incurs additive error is shown, which follows via reduction to a certain communication complexity problem.

## References

SHOWING 1-10 OF 54 REFERENCES

### Random Projections, Graph Sparsification, and Differential Privacy

- Computer ScienceASIACRYPT
- 2013

It is shown that the technique of Blocki et al. BBDS can be adapted to preserve DP for answering cut-queries on sparse graphs, with an asymptotically efficient sanitizer thani¾?BBDS, and achieves a better utility guarantee than Gupta, Roth, and Ullman.

### Analyzing Graphs with Node Differential Privacy

- Computer ScienceTCC
- 2013

A generic, efficient reduction is derived that allows us to apply any differentially private algorithm for bounded-degree graphs to an arbitrary graph, based on analyzing the smooth sensitivity of the 'naive' truncation that simply discards nodes of high degree.

### Private Analysis of Graph Structure

- Computer Science, MathematicsTODS
- 2014

This work extends the approach of Nissim et al. to a new class of statistics, namely k-star queries, and gives hardness results indicating that the approach used for triangles cannot easily be extended to k-triangles.

### Spectral Sparsification of Graphs

- Computer Science, MathematicsSIAM J. Comput.
- 2011

It is proved that every graph has a spectral sparsifier of nearly linear size, and an algorithm is presented that produces spectralSparsifiers in time $O(m\log^{c}m)$, where $m$ is the number of edges in the original graph and $c$ is some absolute constant.

### A differentially private estimator for the stochastic Kronecker graph model

- Computer Science, MathematicsEDBT-ICDT '12
- 2012

A parametric graph model is used, the stochastic Kronecker graph model, to model the observed graph and an estimator of the "true parameter" is constructed in a way that both satisfies the rigorous requirements of differential privacy and demonstrates experimental utility on several important graph statistics.

### Graph sparsification by effective resistances

- Mathematics, Computer ScienceSIAM J. Comput.
- 2008

A key ingredient in the algorithm is a subroutine of independent interest: a nearly-linear time algorithm that builds a data structure from which the authors can query the approximate effective resistance between any two vertices in a graph in O(log n) time.

### Twice-ramanujan sparsifiers

- Mathematics, Computer ScienceSTOC '09
- 2009

It is proved that every graph has a spectral sparsifier with a number of edges linear in its number of vertices, and an elementary deterministic polynomial time algorithm is given for constructing H, which approximates G spectrally at least as well as a Ramanujan expander with dn/2 edges approximates the complete graph.

### Iterative Constructions and Private Data Release

- Computer ScienceTCC
- 2012

New algorithms (and new analyses of existing algorithms) in both the interactive and non-interactive settings are given, and a reduction based on the IDC framework shows that an efficient, private algorithm for computing sufficiently accurate rank-1 matrix approximations would lead to an improved efficient algorithm for releasing private synthetic data for graph cuts.

### An SDP-based algorithm for linear-sized spectral sparsification

- Mathematics, Computer ScienceSTOC
- 2017

An algorithm is presented that outputs a (1+ε)-spectral sparsifier of G with O(n/ε2) edges in Ο(m/εO(1)) time, based on a new potential function which is much easier to compute yet has similar guarantees as the potential functions used in previous references.

### Approximate Undirected Maximum Flows in O(mpolylog(n)) Time

- Computer ScienceSODA
- 2016

We give the first O(mpolylog(n)) time algorithms for approximating maximum flows in undirected graphs and constructing polylog(n)-quality cut-approximating hierarchical tree decompositions. Our…