  We analyze the large-$N$ expansion of general non-equilibrium systems with
fluctuating matrix degrees of freedom and $SU(N)$ symmetry, using the
Schwinger-Keldysh formalism and its closed real-time contour with a forward and
backward component. In equilibrium, the large-$N$ expansion of such systems
leads to a sum over topologies of two-dimensional surfaces of increasing
topological complexity, predicting the possibility of a dual description in
terms of string theory. We extend this argument away from equilibrium, and
study the universal features of the topological expansion in the dual string
theory. We conclude that in non-equilibrium string perturbation theory, the sum
over worldsheet topologies is further refined: Each worldsheet surface $\Sigma$
undergoes a triple decomposition into the part $\Sigma^+$ corresponding to the
forward branch of the time contour, the part $\Sigma^-$ on the backward branch,
and the part $\Sigma^\wedge$ that corresponds to the instant in the far future
where the two branches of the time contour meet. The sum over topologies
becomes a sum over the triple decompositions. We generalize our findings to the
Kadanoff-Baym time contour relevant for systems at finite temperature, and to
the case of closed and open, oriented or unoriented strings. Our results are
universal, and follow solely from the features of the large-$N$ expansion
without any assumptions about the worldsheet dynamics.
