I'm one of the authors of the paper. Ask me anything!
How would you recommend I go about making friends in the CT community? I'm a full-time parent without a degree and see CT as something worth trying to teach my child now (though without all the jargon).
Also, Uni chooses what they do and when they do it. This includes diaper changes and baths. We seek to, at most, influence through what we say/do and configure the environment. We also moderate their food intake when it comes to things like sugar.
So with that in mind, does that change your answer at all?
What kind of object is the self in this case? I've been contemplating this stuff and I'm not clear on how to model things that evolve over time in this way.
The kind of graphical gadget that generates FDHilb (in the sense that the graphical calculus is sound and complete wrt FDHilb) is called ZX calculus (or one of its equivalent variants, such as ZW). It took roughly 10 years to prove that ZX is complete wrt FDHilb! In any case, a string diagram in ZX calculus looks like a hypegraph with extra properties and equations. But you lose the dynamic interpretation of tokens moving in the net, there are no tokens in ZX!
I made it more than halfway through the article, but without this simple intuition it was really hard to grasp the ideas.
Graphs give you finite state machines in the obvious way: You mark the vertex you are in and walk the arrows.
Hypergraphs give you Petri nets: You mark each vertex as many times as you want and walk the arrows to move marks around. This tells us two things: 1. Petri nets are a calculus of resources. The marking is not telling anymore "what state you are in". A state is an allocation of resources to each vertex in the net. 2. Petri nets are concurrent: you don't have to move stuff around by walking one edge at a time: Two different hyperedges in two different places of the hypergraph can "act at the same time", since the "what state you are in" thing makes no sense anymore.
Anyway, this paper is pretty complicated and for sure there are waaaaay easier places to start. Such as this one: https://arxiv.org/abs/1906.07629
Are any forms where the edges are weighted, or does each edge necessarily have the same weight?
Related to the previous question, if you have a finite number of tokens at a vertex with multiple outgoing edges, how do you choose which edges they follow? I suppose that for any given allocation there may be multiple succeeding allocations.
Finally, the structure seems very similar to neural nets. Are they actually similar, or very different?
About the choosing which edges they follow: You don't. In standard Petri nets firing is concurrent: If tokens can be used by more than one transition at the same time, they will non-deterministically go one way or another. You can actually refine this situation by extending your formalism, e.g. to timed nets.
I am not an expert of neural nets, but I'd guess they are more similar to signal flow graphs. These are related to Petri nets tho, but in a very deep and complicated way that I have no chance of explaining here right now. Check out the work of Sobocinski, Piedeleu and Zanasi about additive relations if you are interested in this!
Applications are not the central focus of this paper. There are about a ton of applied papers out there employing Petri nets in computing, chemistry, epidemiology etc. This paper is not one of them. We just mentioned, in passing, how different flavors of nets have been applied in the last decades. We deem this to be an interesting paper for people working in Petri nets theory, because it systematizes decades of research. I'm pretty sure it will be of little interest for anyone not directly involved in Petri net research. :)