  Linear evolution equations are considered usually for the time variable being
defined on an interval where typically initial conditions or time-periodicity
of solutions are required to single out certain solutions. Here we would like
to make a point of allowing time to be defined on a metric graph or network
where on the branching points coupling conditions are imposed such that time
can have ramifications and even loops. This not only generalizes the classical
setting and allows for more freedom in the modeling of coupled and interacting
systems of evolution equations, but it also provides a unified framework for
initial value and time-periodic problems. For these time-graph Cauchy problems
questions of well-posedness and regularity of solutions for parabolic problems
are studied along with the question of which time-graph Cauchy problems cannot
be reduced to an iteratively solvable sequence of Cauchy problems on intervals.
Based on two different approaches - an application of the Kalton-Weis theorem
on the sum of closed operators and an explicit computation of a Green's
function - we present the main well-posedness and regularity results. We
further study some qualitative properties of solutions. While we mainly focus
on parabolic problems we also explain how other Cauchy problems can be studied
along the same lines. This is exemplified by discussing coupled systems with
constraints that are non-local in time akin to periodicity.
