  Synchronous dynamic systems are well-established models that have been used
to capture a range of phenomena in networks, including opinion diffusion,
spread of disease and product adoption. We study the three most notable
problems in synchronous dynamic systems: whether the system will transition to
a target configuration from a starting configuration, whether the system will
reach convergence from a starting configuration, and whether the system is
guaranteed to converge from every possible starting configuration. While all
three problems were known to be intractable in the classical sense, we initiate
the study of their exact boundaries of tractability from the perspective of
structural parameters of the network by making use of the more fine-grained
parameterized complexity paradigm.
  As our first result, we consider treewidth - as the most prominent and
ubiquitous structural parameter - and show that all three problems remain
intractable even on instances of constant treewidth. We complement this
negative finding with fixed-parameter algorithms for the former two problems
parameterized by treedepth, a well-studied restriction of treewidth. While it
is possible to rule out a similar algorithm for convergence guarantee under
treedepth, we conclude with a fixed-parameter algorithm for this last problem
when parameterized by treedepth and the maximum in-degree.
