Sublinear Space Graph Algorithms in the Continual Release Model

TL;DR

This paper introduces sublinear space differentially private algorithms for graph problems in the continual release model, enabling private, real-time updates of solutions including vertex subsets, using sparsification techniques.

Contribution

It presents the first sublinear space algorithms for multiple graph problems in the continual release model, including vertex subset outputs for densest subgraph, and introduces a new sparse data structure for k-core decomposition.

Findings

Algorithms achieve sublinear space in edges and vertices.

First continual release algorithms output vertex subsets for densest subgraph.

Polynomial lower bounds for dynamic edge-privacy are established.

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 also develop a novel sparse level data structure for $k$-core decomposition that may be of independent interest. To complement our insertion-only algorithms, we conclude with polynomial additive error lower bounds for edge-privacy in the fully dynamic setting, where only logarithmic lower bounds were previously known.