  We consider networks of massive particles connected by non-linear springs.
Some particles interact with heat baths at different temperatures, which are
modeled as stochastic driving forces. The structure of the network is
arbitrary, but the motion of each particle is 1D. For polynomial interactions,
we give sufficient conditions for H\"ormander's "bracket condition" to hold,
which implies the uniqueness of the steady state (if it exists), as well as the
controllability of the associated system in control theory. These conditions
are constructive; they are formulated in terms of inequivalence of the forces
(modulo translations) and/or conditions on the topology of the connections. We
illustrate our results with examples, including "conducting chains" of variable
cross-section. This then extends the results for a simple chain obtained in
Eckmann, Pillet, Rey-Bellet (1999).
