  We propose a generalization of the linked-cluster expansions to study
driven-dissipative quantum lattice models, directly accessing the thermodynamic
limit of the system. Our method leads to the evaluation of the desired
extensive property onto small connected clusters of a given size and topology.
We first test this approach on the isotropic spin-1/2 Hamiltonian in two
dimensions, where each spin is coupled to an independent environment that
induces incoherent spin flips. Then we apply it to the study of an anisotropic
model displaying a dissipative phase transition from a magnetically ordered to
a disordered phase. By means of a Pad\'e analysis on the series expansions for
the average magnetization, we provide a viable route to locate the phase
transition and to extrapolate the critical exponent for the magnetic
susceptibility.
