Before revision:  Here  we introduce a general formalism that allows to easily write down mean-field equations for any diffusive dynamics on weighted networks , and to recognize the concept of an annealed weighted network , in which such equations become exact.
After revision:  Here we introduce a general formalism that allows to easily write down mean-field equations for any diffusive dynamics on weighted networks . We also propose the concept of an annealed weighted network , in which such equations become exact.