  We investigate the parabolic Cauchy problem associated with quantum graphs
including Lipschitz or polynomial type nonlinearities and additive Gaussian
noise perturbed vertex conditions. The vertex conditions are the standard
continuity and Kirchhoff assumptions in each vertex. In the case when only
Kirchhoff conditions are perturbed, we can prove existence and uniqueness of a
mild solution with continuous paths in the standard state space $\mathcal{H}$
of square integrable functions on the edges. We also show that the solution is
Markov and Feller. Furthermore, assuming that the vertex values of the
normalized eigenfunctions of the self-adjoint operator governing the problem
are uniformly bounded, we show that the mild solution has continuous paths in
the fractional domain space associated with the Hamiltonian operator,
$\mathcal{H}_{\alpha}$ for $\alpha<\frac{1}{4}$. This is the case when the
Hamiltonian operator is the standard Laplacian perturbed by a potential. We
also show that if noise is present in both type of vertex conditions, then the
problem admits a mild solution with continuous paths in the fractional domain
space $\mathcal{H}_{\alpha}$ with $\alpha<-\frac{1}{4}$ only. These regularity
results are the quantum graph analogues obtained by da Prato and Zabczyk [9] in
case of a single interval and classical boundary Dirichlet or Neumann noise.
