Does such model exists?
The model studied in the following work should be a fairly close match with the model that you described (see in particular graph problems "without edge duplication"):


Woodruff &amp; Zhang: "When Distributed Computation is Communication Expensive" http://arxiv.org/abs/1304.4636


See also this work for closely related models:


Klauck et al.: "Distributed Computation of Large-scale Graph Problems" http://arxiv.org/abs/1311.6209


You may also be interested in the "congested clique" model, which is the special case such that $k = |V|$ and computer number $i$ is aware of the edges incident to node number $i$.