  We discuss a generalization of the dephasing-type spin-boson model in which N
qubits are connected to K bosonic modes in an arbitrary way. The model can be
solved exactly for any initial state and coupling type. We found that this
leads to two important effects, which are specific of the multipartite nature
of the problem. First, the bosons can mediate an effective interaction among
the spins, which corresponds to a type of time-dependent Lamb-shift. By tuning
the types of couplings, this interaction can also be varied at will. Second,
the dephasing rate of the qubits is found to depend on the order of the
coherence, with superpositions that have drastically different magnetizations
being exponentially more affected. The relative strength of the Lamb-shift and
dephasing can also be tuned by changing the spin-boson couplings.
