I've been skipping around, reading the network theory posts by John Baez and Brendan Fong, such as this one: Part 11: Noether's Theorem for stochastic mechancs. Somewhere you pointed out that a Markov chain is a matrix multiplication taking us from inputs to outputs, which is to say, it is a Sequence (multiplying $U$, $U^2$, $U^3$, etc.) of a Network (whose edges $u_{i,j}$ are given by the entries of the matrix $U$). So that, in a way, conforms and clarifies my comment above about Petri nets, which are a special case of Markov chains.