  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.
