  We consider stochastic mean field games for which the state space is a
network. In the ergodic case, they are described by a system coupling a
Hamilton-Jacobi-Bellman equation and a Fokker-Planck equation, whose unknowns
are the invariant measure m, a value function u, and the ergodic constant
$\rho$. The function u is continuous and satisfies general Kirchhoff conditions
at the vertices. The invariant measure m satisfies dual transmission
conditions: in particular, m is discontinuous across the vertices in general,
and the values of m on each side of the vertices satisfy special compatibility
conditions. Existence and uniqueness are proven, under suitable assumptions.
