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Benjamin Merlin Bumpus (he/him) (Apr 26 2022 at 08:42):As a graph (structure) theorist, to determine qualitative information (equilibria, etc) about a dynamical system, my instinct would be to study the graph-theoretical structure of a reaction network. I saw a talk of John's in which he mentioned that one wants to be able to pass between two different views: 'networks as processes' and 'networks as objects'. It turns out that, since we're in a bicategory, we can switch between these views by working within a homcategory. This idea got me thinking that perhaps we could use this to our advantage when studying reaction networks: what do the dynamical systems that correspond to tree-like (i.e. low treewidth) reaction networks look like? And what happens when we compose many reaction networks of low tree-width?
(One doesn't need to focus on low tree-width, it's just an example.)
Conveniently, Zoltan Kocsis, Jade Master and I (in preparation) have been studying a certain kind of bicategory (think slice of Span(K)' for some category K) which we use to generalize graph-theoretic invariants which summarise the global shape of graph' to arbitrary categories and I wonder how this relates to John and Blake's gray- and black-box functors. Indeed I'd conjecture that our structural stuff (which measures the global `shape' of a network) should be liftable by the gray- and black-box functors to dynamical systems and equilibria...
These are still very vague thoughts, but I think that there's definitely something worth studying here :)
John Baez (Apr 28 2022 at 19:13):One thing you might like, @Benjamin Merlin Bumpus, is the Deficiency Zero Theorem. This is a theorem that lets you take a reaction network, easily compute an invariant of it called the 'deficiency'... and if it's zero you instantly know a lot about the equilibria for this system of chemical reaction networks, even without knowing their rate constants!
John Baez (Apr 28 2022 at 19:14):I explain this theorem in the book I just sent you (and everyone) an email about!
John Baez (Apr 28 2022 at 19:15):One thing nobody has studied is the "compositional" behavior of deficiency, i.e. how the deficiency of a large reaction network can be computed from those of its parts.
John Baez (Apr 28 2022 at 19:15):One problem is that while reaction networks and Petri nets convey the same information, they are different - and while Petri nets seem better for creating a category of "open" Petri nets, reaction networks are better for computing the deficiency!
John Baez (Apr 28 2022 at 19:16):This is why I haven't already tried defining deficiency for open Petri nets, and studying how it gets along with composing open Petri nets.
John Baez (Apr 28 2022 at 19:17):What do you mean by "the global shape of a graph"?
Benjamin Merlin Bumpus (he/him) (Apr 29 2022 at 10:03):@John Baez thanks for this, I'll definitely have a look at the book and at the deficiency zero thm!
Is there a simple intuition as to why open reaction networks are `better' for deficiency theorems compared to Petri Nets, or is this something best seen from the proof of the 0-def-thm?
By `global shape' I mean something like in the drawing below:
Screenshot-2022-04-29-at-11.56.09.png
The idea is the following. If you remove any edge of a tree, you split the tree into more than one part (same for any internal node); tree decompositions generalise this: if you remove from a graph G any bag (blob in the picture) or any edge (intersection of adjacent blobs) given in a tree decompositon of G, you split G into more than one part. So, insofar as connectivity is concerned, tree decompositions capture the `global tree-shape' of a graph. (I given a imo very gentle intro to this in Sec 1.2 of my thesis.)
This idea can be generalised to other shapes (paths, cycles, etc..) and to other categories (this is part of what Jade, Zoltan and I are doing; this new stuff also generalises and improves upon earlier work which is in the `Spined Categories' chapter of my thesis) .
John Baez (Apr 29 2022 at 18:45):Benjamin Merlin Bumpus said:
John Baez thanks for this, I'll definitely have a look at the book and at the deficiency zero thm!
Is there a simple intuition as to why open reaction networks are `better' for deficiency theorems compared to Petri Nets, or is this something best seen from the proof of the 0-def-thm?
Simply this: the definition of 'deficiency' is phrased in terms of reaction networks, not Petri nets.
A reaction network is a graph where the vertices are "complexes", linear combinations of species like . To compute the deficiency of a reaction network, you need to count complexes.
A Petri net is a bipartite graph where the vertices are of two kinds: "species" like and and "transitions" like this:
I'm not saying this is an impossible obstacle to overcome. You can turn Petri nets into reaction networks and vice versa, and the complexes of the reaction network are hiding in the transitions of the Petri net!
See page 170 of my book for an example of turning a reaction network into a Petri (or vice versa).
John Baez (Apr 29 2022 at 18:47):I'm just saying that I didn't instantly see how to solve this problem, so I decided to think about other things.
John Baez (Apr 29 2022 at 18:49):By the way, @Benjamin Merlin Bumpus , your use of quotation marks `like this' brand you as a LaTeX user. :upside_down:
John Baez (Apr 29 2022 at 18:52):I will have to think about your stuff about "global shape".
Sam Tenka (May 04 2022 at 15:01):@Benjamin Merlin Bumpus I just got to your earlier .png about global shape, and I see that this is a (generalization of?) tree decomposition as used in computational bayesian inference (which is part of my world of machine learning etc)!
Benjamin Merlin Bumpus (he/him) (May 09 2022 at 10:06):Sorry for the delay, I was at a conference :)
@John Baez thanks for the info; I figured that it would be some issue of this kind, but I thought I'd check.
@Sam Tenka I know nothing about "computational bayesian inference", in what kinds of problems do tree decompositions turn up here?
Sam Tenka (May 30 2022 at 15:26):@Benjamin Merlin Bumpus --- in sampling from / doing inference on "probabilistic graphical models"!