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This work provides the first study to explore the interaction of update propagation with and without fine-grained synchronization (push vs. pull), emerging coherence protocols (GPU vs. DeNovo coherence), and software-centric consistency models (DRF0, DRF1, and DRFrlx) for graph workloads on emerging integrated GPU-CPU systems with native unified shared memory. We study 6 graph applications with 6 graph inputs for a total of 36 workloads running on 12 system (hardware+software) configurations reflecting the above design space of update propagation, coherence, and memory consistency. We make three key contributions. First, we show that there is no single best system configuration for all workloads, motivating systems with flexible coherence and consistency support. Second, we develop a model to accurately predict the best system configuration -- this model can be used by software designers to decide on push vs. pull and the consistency model and by flexible hardware to invoke the appropriate coherence and consistency configuration for the given workload. Third, we show that the design dimensions explored here are inter-dependent, reinforcing the need for software-hardware co-design in the above design dimensions. For example, software designers deciding on push vs. pull must consider the consistency model supported by hardware -- in some cases, push maybe better if hardware supports DRFrlx while pull may be better if hardware does not support DRFrlx.