Sublinear Space Graph Algorithms in the Continual Release Model
Alessandro Epasto, Quanquan C. Liu, Tamalika Mukherjee, Felix Zhou
TL;DR
This work studies differential privacy for graphs under continual edge updates, focusing on insertion-only streams in the continual-release model and aiming for sublinear space while producing useful vertex-level outputs. The authors develop sparsification-based techniques, including stable edge sampling and adaptive multidimensional sparse vector methods, to design private, single-pass algorithms for k-core decomposition, densest subgraph (including vertex subsets), and maximum matching, with space near-linear or sublinear in the number of vertices and polylog additive errors. A key contribution is the first continual-release $\varepsilon$-DP algorithm for k-core decomposition and the first DP continual-release densest-subgraph method that returns private vertex subsets, both achieving approximation guarantees close to non-private streaming and static DP results. They also provide lower bounds showing polynomial additive error is necessary in the fully dynamic setting, underscoring fundamental privacy and space trade-offs, and discuss concurrent work and future directions, including removing public arboricity bounds and extending to fully dynamic environments. Overall, the paper advances sublinear-space private graph streaming, bridging a gap between private continual-release and non-private streaming algorithms, with implications for scalable private analytics on large, evolving networks.
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.