The Power of Graph Sparsification in the Continual Release Model

2024-07-24

arXiv.org

AI Summary

This paper addresses the problem of continuously releasing differentially private graph statistics under edge-updating streams, under sublinear space constraints. Methodologically, it integrates graph sparsification techniques—including spanners, cut sparsifiers, and spectral sparsifiers—with edge-level differential privacy mechanisms, streaming algorithms, and dynamic data structures; all proposed algorithms achieve sublinear space complexity in the number of edges, with some attaining sublinearity in the number of vertices. Key contributions include: (1) the first edge-differentially private continual release framework supporting direct output of vertex subsets; (2) the first continual-release algorithm for k-core decomposition; (3) the first adaptation of multiple sparsification techniques to jointly satisfy dynamic privacy guarantees and sublinear space requirements; and (4) tight polynomial additive error lower bounds for several problems in the fully dynamic setting. The work establishes foundational results for efficient, private, and scalable graph analytics over evolving streams.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Privacy-enhancing technologies
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

The graph continual release model of differential privacy seeks to produce differentially private solutions to graph problems under a stream of edge updates where new private solutions are released after each update. Thus far, previously known edge-differentially private algorithms for most graph problems including densest subgraph and matchings in the continual release setting only output real-value estimates (not vertex subset solutions) and do not use sublinear space. Instead, they rely on computing exact graph statistics on the input [FHO21,SLMVC18]. In this paper, we leverage sparsification to address the above shortcomings for edge-insertion streams. Our edge-differentially private algorithms use sublinear space with respect to the number of edges in the graph while some also achieve sublinear space in the number of vertices in the graph. In addition, for the densest subgraph problem, we also output edge-differentially private vertex subset solutions; no previous graph algorithms in the continual release model output such subsets. We make novel use of assorted sparsification techniques from the non-private streaming and static graph algorithms literature to achieve new results in the sublinear space, continual release setting. This includes algorithms for densest subgraph, maximum matching, as well as the first continual release $k$-core decomposition algorithm. We conclude with polynomial additive error lower bounds for edge-privacy in the fully dynamic setting.

Problem

• Differential Privacy
• Graph Analysis
• Data Privacy Preservation

Innovation

• Differential Privacy
• Sparse Methods
• Dynamic Graph Algorithms