  We consider stochastic reaction-diffusion equations on a finite network
represented by a finite graph. On each edge in the graph a multiplicative
cylindrical Gaussian noise driven reaction-diffusion equation is given
supplemented by a dynamic Kirchhoff-type law perturbed by multiplicative scalar
Gaussian noise in the vertices. The reaction term on each edge is assumed to be
an odd degree polynomial, not necessarily of the same degree on each edge, with
possibly stochastic coefficients and negative leading term. We utilize the
semigroup approach for stochastic evolution equations in Banach spaces to
obtain existence and uniqueness of solutions with sample paths in the space of
continuous functions on the graph. In order to do so we generalize existing
results on abstract stochastic reaction-diffusion equations in Banach spaces.
