Iterative arrays with restricted internal inter-cell communication are investigated. A quantitative measure for the communication is defined by counting the number of uses of the links between cells and it is differentiated between the sum of all communications of an accepting computation and the maximum number of communications per cell occurring in accepting computations. The computational complexity of both classes of devices is investigated and put into relation. In addition, a strict hierarchy depending on the maximum number of communications per cell is established. Finally, it is shown that almost all commonly studied decidability questions are not semidecidable for iterative arrays with restricted communication and, moreover, it is not semidecidable as well whether a given iterative array belongs to a given class with restricted communication.