Several models have been proposed for the modeling of and reasoning about concurrent networks, the most widely known being the several variants of Petri nets. One interesting problem is the dichotomy between the geometric aspects of the network (this can also be seen as the global view) and a compositional algebraic view describing the network as being composed from basic units by algebraic operations.

The approach described in this paper is a category theoretical one, modeling the networks in certain categories of graphs. As the title suggests, networks containing state machines are also modeled (globally and compositionally) and follow quite naturally from the category theoretic machinery.

The paper starts with a short introduction outlining the main contributions of the paper, referencing earlier papers that investigated the basic category theoretic structures used in the paper, namely the algebras Span(Graph) and Cospan(Graph). Accordingly, the second section presents a review of the definition of these algebras, together with a basic example of how to model automata with them. The following section explains the modeling of open and closed networks in a category of monoidal graphs, respectively, in the category of cospans over a quotient of this category. Adding states to these networks is the topic of the fourth section. Again, this follows from certain category theoretic constructions. This section also draws the connection to Petri nets and shows that the network with state needs fewer elements as an equivalent Petri net due to the fact that some aspects can be incorporated into the state machine. The final sections shortly reference a tool for transforming an expression from the underlying algebra into the network described by this expression and presenting a connection of the machinery for describing the networks to mathematical topology and knot theory.

The gist of the approach is quite easy to understand from the paper and looks very interesting. Several aspects are more completely described in the previous papers referenced. Furthermore, a good deal of category theoretic intuition is needed to better understand the paper. I was unable to completely follow the diagrammatic proofs in the last section because the semantics of these string diagrams is not explained; maybe this would become clearer after reading the paper these results are a summary of. In conclusion, I have the impression that this is an interesting avenue, but further work is needed to make the results more accessible.