  We present DegreeSketch, a semi-streaming distributed sketch data structure
and demonstrate its utility for estimating local neighborhood sizes and local
triangle count heavy hitters on massive graphs. DegreeSketch consists of
vertex-centric cardinality sketches distributed across a set of processors that
are accumulated in a single pass, and then behaves as a persistent query engine
capable of approximately answering graph queries pertaining to the sizes of
adjacency set unions and intersections. The $t$th local neighborhood of a
vertex is the number of vertices reachable in $G$ from $v$ by traversing at
most $t$ edges, whereas the local triangle count is the number of 3-cycles in
which it is included. Both metrics are useful in graph analysis applications,
but exact computations scale poorly as graph sizes grow. We present efficient
algorithms for estimating both local neighborhood sizes and local triangle
count heavy hitters using DegreeSketch. In our experiments we implement
DegreeSketch using the celebrated hyperloglog cardinality sketch and utilize
the distributed communication tool YGM to achieve state-of-the-art performance
in distributed memory.
