  In this paper we prove a Feynman-Kac-It\^{o} formula for magnetic
Schr\"odinger operators on arbitrary weighted graphs. To do so, we have to
provide a natural and general framework both on the operator theoretic and the
probabilistic side of the equation. On the operator side we identify a very
general class of potentials that allows the definition of magnetic
Schr\"odinger operators. On the probabilistic side, we introduce an appropriate
notion of stochastic line integrals with respect to magnetic potentials. Apart
from linking the world of discrete magnetic operators with the probabilistic
world through the Feynman-Kac-It\^{o} formula, the insights from this paper
gained on both sides should be of an independent interest. As applications of
the Feynman-Kac-It\^{o} formula, we prove a Kato inequality, a Golden-Thompson
inequality and an explicit representation of the quadratic form domains
corresponding to a large class of potentials.
