The Power of Graph Sparsification in the Continual Release Model

The graph continual release model of differential privacy seeks to produce differentially private solutions to graph problems under a stream of updates where new private solutions are released after each update. Streaming graph algorithms in the non-private literature also produce (approximately) accurate solutions when provided updates in a stream, but they additionally try to achieve two other goals: 1) output vertex or edge subsets as approximate solutions to the problem (not just real-valued estimates) and 2) use space that is sublinear in the number of edges or the number of vertices. Thus far, all previously known edge-differentially private algorithms for graph problems in the continual release setting do not meet the above benchmarks. Instead, they require computing exact graph statistics on the input [SLMVC18, FHO21, JSW24]. In this paper, we leverage sparsification to address the above shortcomings. 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 most of our problems, we also output differentially private vertex subsets. We make novel use of assorted sparsification techniques from the non-private streaming and static graph algorithms literature and achieve new results in the sublinear space, continual release setting for a variety of problems including densest subgraph, $k$-core decomposition, maximum matching, and vertex cover. In addition to our edge-differential privacy results, we use graph sparsification based on arboricity to obtain a set of results in the node-differential privacy setting, illustrating a new connection between sparsification and privacy beyond minimizing space. We conclude with polynomial additive error lower bounds for edge-privacy in the fully dynamic setting.