A graph processing system for concurrent property graph queries of a property graph implemented in a distributed network computes on respective nodes a subgraph shard represented as edge-sets containing vertices within a certain range. Each node stores data for a subgraph shard that contains a range of local vertices that are a subset of all vertices of the property graph. Each subgraph shard also has boundary vertices having edges that connect the subgraph shard to boundary vertices of another subgraph shard. Upon receipt of concurrent queries of the property graph, a query of the subgraph shards is scheduled in accordance with an initial vertex for each concurrent user query. The property graph is traversed by traversing edge-sets within a subgraph shard on each node and during traversal messaging is used to send values of boundary vertices to at least one other node having another subgraph shard sharing the boundary vertices.