What is the big version of NC? $\mathsf{NC}$ captures the idea of efficiently parallelizable, and one interpretation of it is problems that are solvable in time $O(\log^c n)$ using $O(n^k)$ parallel processors for some constants $c$, $k$. My question is if there is an analogous complexity class where time is $n^c$ and number of processors is $2^{n^k}$. As a fill-in-the-blank question:


  $\mathsf{NC}$ is to $\mathsf{P}$ as __ is to $\mathsf{EXP}$


In particular, I am interested in a model where we have an exponential number of computers arranged in a network with polynomially bounded degree (lets say the network is independent of the input/problem or atleast somehow easy to construct, or any other reasonable uniformity assumption). At each time step:


Every computer reads the polynomial number of polynomial sized messages it received in the previous time step.
Every computer runs some polytime computation that can depend on these messages.
Every computer passes a message (of polylength) to each of its neighbours.


What is the name of a complexity class corresponding to these sort of models? What is a good place to read about such complexity classes? Are there any complete-problems for such a class?
I believe the class you are looking for is $PSPACE$. Suppose you have $exp(n^k) = 2^{O(n^k)}$ processors fitting the requirements:


  
  Every computer reads the polynomial number of polynomial sized messages it received in  the previous time step.
  Every computer runs some polytime computation that can depend on these messages.
  Every computer passes a message (of polylength) to each of its neighbours.
  


This can be modeled by having a circuit with $poly(n)$ layers, where each layer has $exp(n^k)$ "gates", and each "gate" does a polynomial time computation (satisfying part 2) with polynomial fan-in (satisfying part 1), and has polynomial fan-out (satisfying part 3).  

Since each gate computes a polynomial time function, they each can be replaced by a polynomial size circuit (with AND/OR/NOT) in the usual way. Note the polynomial fan-ins and fan-outs can be made to be 2, by only increasing the depth by a $O(\log n)$ factor. What remains is a $poly(n)$ depth uniform circuit with $exp(n^k)$ AND/OR/NOT gates. This is precisely alternating polynomial time, which is precisely $PSPACE$.