  We present the path-sum formulation for
$\mathsf{OE}[\mathsf{H}](t',t)=\mathcal{T}\,\text{exp}\big(\int_{t}^{t'}\!\mathsf{H}(\tau)\,d\tau\big)$,
the time-ordered exponential of a time-dependent matrix $\mathsf{H}(t)$. The
path-sum formulation gives $\mathsf{OE}[\mathsf{H}]$ as a branched continued
fraction of finite depth and breadth. The terms of the path-sum have an
elementary interpretation as self-avoiding walks and self-avoiding polygons on
a graph. Our result is based on a representation of the time-ordered
exponential as the inverse of an operator, the mapping of this inverse to sums
of walks on graphs and the algebraic structure of sets of walks. We give
examples demonstrating our approach. We establish a super-exponential decay
bound for the magnitude of the entries of the time-ordered exponential of
sparse matrices. We give explicit results for matrices with commonly
encountered sparse structures.
