The paper provides a unified co-design of 1) a programming and execution model that allows spawning tasks from within the vertex data at runtime, 2) language constructs for \textit{actions} that send work to where the data resides, combining parallel expressiveness of local control objects (LCOs) to implement asynchronous graph processing primitives, 3) and an innovative vertex-centric data-structure, using the concept of Rhizomes, that parallelizes both the out and in-degree load of vertex objects across many cores and yet provides a single programming abstraction to the vertex objects. The data structure hierarchically parallelizes the out-degree load of vertices and the in-degree load laterally. The rhizomes internally communicate and remain consistent, using event-driven synchronization mechanisms, to provide a unified and correct view of the vertex. Simulated experimental results show performance gains for BFS, SSSP, and Page Rank on large chip sizes for the tested input graph datasets containing highly skewed degree distribution. The improvements come from the ability to express and create fine-grain dynamic computing task in the form of \textit{actions}, language constructs that aid the compiler to generate code that the runtime system uses to optimally schedule tasks, and the data structure that shares both in and out-degree compute workload among memory-processing elements.