Assuming, I have a network game (where the players are e.g. users in a network), how is it possible to model user-specific a-priori information (which results in different utility functions for each user)?
E.g. user $i$'s utility is given by some term $\tilde u_0$ which is common for all plus the user specific term $\tilde u_i$ which is assumed to be unknown to the others:
$u_i = \tilde u_0 + \tilde u_i, \quad \forall i$.
As each user's utility function is not known to the other users, it is a game with incomplete information, right? Which solution concepts are applicable in this case, assuming each user's utility functions is supermodular?
Your problem is not so much incomplete information as it a problem of asymmetric information: Every player knows at least the possible distribution of $\tilde{u}_i$, but not the actual realization. Thus, there is asymmetric information about $\tilde{u}_i$: $i$ knows it, all others ($-i$) only know its distribution.
The typical solution concepts for games of asymmetric information, besides Nash and subgame perfect Nash equilibrium, are Bayesian Nash equilibrium and Perfect Bayesian Nash equilibrium (sequential move games).
There are many examples in economics where the utility/payoff function of players depends on private information of the players. For example, firms' production cost may be private information, or the value of an item to bidders in an auction.