Vertex-centric graph processing systems such as Pregel, PowerGraph, or GraphX recently gained popularity due to their superior performance of data analytics on graph-structured data. These systems exploit the graph structure to improve data access locality during computation, making use of specialized graph partitioning algorithms. Recent partitioning techniques assume a uniform and constant amount of data exchanged between graph vertices (i.e., uniform vertex traffic) and homogeneous underlying network costs. However, in real-world scenarios vertex traffic and network costs are heterogeneous. This leads to suboptimal partitioning decisions and inefficient graph processing. To this end, we designed GrapH, the first graph processing system using vertex-cut graph partitioning that considers both, diverse vertex traffic and heterogeneous network, to minimize overall communication costs. The main idea is to avoid frequent communication over expensive network links using an adaptive edge migration strategy. Our evaluations show an improvement of 60% in communication costs compared to state-of-the-art partitioning approaches.