Network Theory

John Baez

Scientists and engineers use diagrams of networks in many different ways. Together with many collaborators I am studying networks with the tools of modern mathematics, such as category theory. You can read blog articles, papers and a book about our research. I am collaborating with the Topos Institute to use the resulting math for scientific computation, such as quickly adaptable models of infectious disease.

Papers

Overview:

• Brendan Fong, The Algebra of Open and Interconnected Systems, Ph.D. thesis, University of Oxford, 2016. (Blog article here.)
• Jason Erbele, Categories in Control: Applied PROPs, Ph.D. thesis, U. C. Riverside, 2016.
• Blake Pollard, Open Markov Processes and Reaction Networks, Ph.D. thesis, U. C. Riverside, 2017.
• Brandon Coya, Circuits, Bond Graphs, and Signal-Flow Diagrams: A Categorical Perspective, Ph.D. thesis, U. C. Riverside, 2018. (Blog article here.)
• Kenny Courser, Open Systems: a Double Categorical Perspective, Ph.D. thesis, U. C. Riverside, 2020. (Blog articles here.)
• Jade Master, Composing Behaviors of Networks, Ph.D. thesis, U.C. Riverside, 2021.
• John Baez and Jacob Biamonte, Quantum Techniques for Stochastic Mechanics, World Scientific, Singapore, 2018. (Blog articles here.)

Classical mechanics:

• John C. Baez, David Weisbart and Adam Yassine, Open systems in classical mechanics. (Blog article here.)
• David Weisbart and Adam Yassine, Constructing span categories from categories without pullbacks.

• John Baez and Brendan Fong, A compositional framework for passive linear networks, Theory and Applications of Categories 33 (2018), 1158–1222. (Blog article here.)
• John Baez, Brandon Coya and Franciscus Rebro, Props in network theory, Theory and Applications of Categories 33 (2018), 727–783. (Blog article here.)
• Brandon Coya, A compositional framework for bond graphs.

Control theory:

• John Baez and Jason Erbele, Categories in control, Theory and Applications of Categories 30 (2015), 836–881. (Blog article here.)
• Jason Erbele, Categories in Control: Applied PROPs, Ph.D. thesis, U. C. Riverside, 2016.

Markov processes:

• John Baez and Brendan Fong, A Noether theorem for Markov processes, Journal of Mathematical Physics 54 (2013), 013301. (Blog article here.)
• John Baez, Brendan Fong and Blake S. Pollard, A compositional framework for Markov processes, Journal of Mathematical Physics 57 (2016), 033301. (Blog article here.)
• Blake S. Pollard, A Second Law for open Markov processes, Open Systems and Information Dynamics 23 (2016), 1650006. (Blog article here.)
• Blake S. Pollard, Open Markov processes: A compositional perspective on non-equilibrium steady states in biology, Entropy 18 (2016), 140. (Blog article here.)
• John Baez and Kenny Courser, Coarse-graining open Markov processes, Theory and Applications of Categories 33 (2018), 1223–1268. (Blog article here.)

Reaction networks:

• John Baez, Quantum techniques for reaction networks, Advances in Mathematical Physics 2018 (2018), 7676309. (Blog article here.)
• John Baez and Brendan Fong, Quantum techniques for studying equilibrium in reaction networks, Journal of Complex Networks 3 (2014), 22–34. (Blog article here.)
• John Baez and Blake S. Pollard, A compositional framework for reaction networks, Reviews in Mathematical Physics 29 (2017), 1750028. (Blog article here.)
• John Baez, Jonathan Lorand, Blake S. Pollard and Maru Sarazola, Biochemical coupling through emergent conservation laws. (Blog articles here.)

Petri nets:

• John Baez and Jade Master, Open Petri nets. (Blog articles here.)
• Jade Master, Petri nets based on Lawvere theories, Mathematical Structures in Computer Science 30 (2020), 833–864
• John Baez, John Foley and Joe Moeller, Network models from Petri nets with catalysts, Compositionality 1, 4 (2019). (Blog article here.)
• Categories of nets, with Fabrizio Genovese, Jade Master and Michael Shulman. (Blog article here.)

Stock-flow diagrams:

• John Baez, Xiaoyan Li, Sophie Libkind, Nathaniel Osgood and Evan Patterson, Compositional modeling with stock and flow diagrams.

Graphs and quantale-valued matrices:

• Jade Master, The open algebraic path problem.
• Jade Master, Composing Behaviors of Networks, Ph.D. thesis, U.C. Riverside, 2021.

The mathematics of networks - decorated cospans:

• Brendan Fong, Decorated cospans, Theory and Applications of Categories 30 (2015), 1096–1120. (Blog articles here and here.)
• Brendan Fong, Decorated corelations, Theory and Applications of Categories 33 (2018), 608–643.
• Kenny Courser, A bicategory of decorated cospans, Theory and Applications of Categories 32 (2017), 985–1027. (Blog article here.)
• Daniel Cicala, Spans of cospans, Theory and Applications of Categories 33 (2018), 131–147.
• Daniel Cicala and Kenny Courser, Spans of cospans in a topos, Theory and Applications of Categories 33 (2018), 1–22.

The mathematics of networks - structured cospans:

• John Baez and Kenny Courser, Structured cospans, Theory and Applications of Categories, 35 (2020), 1771–1822. (Blog article here.)
• Daniel Cicala, Rewriting Structured Cospans: a Syntax for Open Systems, Ph.D. thesis, U.C. Riverside, 2019.
• Kenny Courser, Open Systems: a Double Categorical Perspective, Ph.D. thesis, U. C. Riverside, 2020. (Blog articles here.)
• John Baez, Kenny Courser and Christina Vasilakopoulou, Structured versus decorated cospans, Compositionality 4, 3 (2022).

The mathematics of networks - props:

• Simon Wadsley and Nick Woods, PROPs for linear systems. (Blog article here.)
• Brandon Coya and Brendan Fong, Corelations are the prop for extraspecial commutative Frobenius monoids, Theory and Applications of Categories 32 (2017), 380–395. (Blog article here.)
• John Baez, Brandon Coya and Franciscus Rebro, Props in network theory, Theory and Applications of Categories 33 (2018), 727–783. (Blog article here.)

Blog articles

• Parts 1-26: Chemical reaction networks, Petri nets and Markov processes.
• Parts 27-31: Electrical circuits.

You can navigate forwards and back through these using the blue arrows. By clicking the links that say "on Azimuth", you can see blog entries containing these articles. Those let you read comments about my articles—and also make comments or ask questions of your own!

Chemical reaction networks, Petri nets and Markov processes

• Part 1 - toward a general theory of networks. Also available on Azimuth.
• Part 2 - stochastic Petri nets; the master equation versus the rate equation. Also available on Azimuth.
• Part 3 - the rate equation of a stochastic Petri net, and applications to chemistry and infectious disease. Also available on Azimuth.
• Part 4 - the master equation of a stochastic Petri net, and analogies to quantum field theory. Also available on Azimuth.
• Part 5 - the stochastic Petri net for a Poisson process; analogies between quantum theory and probability theory. Also available on Azimuth.
• Part 6 - the master equation in terms of annihilation and creation operators. Also available on Azimuth.
• Part 7 - a stochastic Petri net from population biology whose rate equation is the logistic equation; an equilibrium solution of the corresponding master equation. Guest post by Jacob Biamonte. Also available on Azimuth.
• Part 8 - the rate equation and master equation of a stochastic Petri net; the role of Feynman diagrams. Also available on Azimuth.
• Part 9 - the Anderson–Craciun–Kurtz theorem, which gives equilibrium solutions of the master equation from complex balanced equilibrium solutions of the rate equation; coherent states. Joint post with Brendan Fong. Also available on Azimuth.
• Part 10 - an example of the Anderson-Craciun-Kurtz theorem. Also available on Azimuth.
• Part 11 - a stochastic version of Noether's theorem. Joint post with Brendan Fong. Also available on Azimuth.
• Part 12 - comparing quantum mechanics and stochastic mechanics. Also available on Azimuth.
• Part 13 - comparing the quantum and stochastic versions of Noether's theorem. Also available on Azimuth.
• Part 14 - an example: chemistry and the Desargues graph. Also available on Azimuth, together with a special post on answers to the puzzle.
• Part 15 - Markov processes and quantum processes coming from graph Laplacians, illustrated using the Desargues graph. Also available on Azimuth.
• Part 16 - Dirichlet operators and electrical circuits made of resistors. Also available on Azimuth.
• Part 17 - reaction networks versus Petri nets; the deficiency zero theorem. Joint post with Jacob Biamonte. Also available on Azimuth.
• Part 18 - an example of the deficiency zero theorem: a diatomic gas. Joint post with Jacob Biamonte. Also available on Azimuth
• Part 19 - an example of Noether's theorem and the Anderson–Craciun–Kurtz theorem: a diatomic gas. Joint post with Jacob Biamonte. Also available on Azimuth.
• Part 20 - Dirichlet operators and the Perron–Frobenius theorem. Guest post by Jacob Biamonte. Also available on Azimuth.
• Part 21 - warmup for the proof of the deficiency zero theorem: the concept of deficiency. Also available on Azimuth.
• Part 22 - warmup for the proof of the deficiency zero theorem: reformulating the rate equation. Also available on Azimuth.
• Part 23 - warmup for the proof of the deficiency zero theorem: finding the equilibria of a Markov process, and describing its Hamiltonian in a slick way. Also available on Azimuth.
• Part 24 - proof of the deficiency zero theorem. Also available on Azimuth.
• Part 25 - Petri nets, logic, and computation: the reachability problem for Petri nets. Also available on Azimuth.
• Part 26 - Using chemical reactions for computation—an introduction by Luca Cardelli. Also available on Azimuth.

• John Baez and Jacob Biamonte, Quantum Techniques for Stochastic Mechanics, World Scientific, Singapore, 2018.

• Autocatalysis in Reaction Networks - The concept of critical siphon—a collection of species such that if none are present, then they will remain absent in future—and an interesting result about this concept. This post was written by Manoj Gopalkrishnan. Also available on Azimuth.
• Lyapunov Functions for Complex-Balanced Systems - Sketch of a proof that free energy decreases for complex balanced reaction networks. This post was written by Manoj Gopalkrishnan. Also available on Azimuth.

• The Large-Number Limit for Reaction Networks (Part 1) - How the large-number limit for reaction networks resembles the classical limit of quantum mechanics. This post is based on joint work with Arjun Jain. Available on Azimuth.
• The Large-Number Limit for Reaction Networks (Part 2) - Coherent states and the large-number limit. This post is based on joint work with Arjun Jain. Available on Azimuth.
• The Large-Number Limit for Reaction Networks (Part 3) - Deriving the rate equation from the master equation by taking the large-number limit. Joint with Arjun Jain. Available on Azimuth.

• John Baez, Jonathan Lorand, Blake S. Pollard and Maru Sarazola, Biochemical coupling through emergent conservation laws.

Here's the series:

• Part 1 - Introduction. Available on Azimuth.
• Part 2 - Review of reaction networks and equilibrium thermodynamics. Available on Azimuth.
• Part 3 - What is coupling? Also available on Azimuth.
• Part 4 - Interactions. Also available on Azimuth.
• Part 5 - Coupling in quasiequilibrium states. Also available on Azimuth.
• Part 6 - Emergent conservation laws. Also available on Azimuth.
• Part 7 - The urea cycle. Also available on Azimuth
• Part 8 - The citric acid cycle. Also available on Azimuth.

• Part 27 - Getting differential equations from circuit diagrams. Available on Azimuth.
• Part 28 - The analogy between electronics and mechanics. Available on Azimuth.
• Part 29 - Analogies between the mechanics of translation, the mechanics of rotation, electronics, hydraulics, thermal physics, chemistry, heat flow and economics. Flow versus effort. Available on Azimuth.
• Part 30 - Toward a category with circuits as morphisms. Available on Azimuth, along with a videotaped lecture on YouTube.
• Part 31 - Circuits as cospans, and how to compose cospans. Available on Azimuth and on YouTube.

• Brendan Fong, The Algebra of Open and Interconnected Systems, Ph.D. thesis, University of Oxford, 2016. (Blog article here.)
• Brandon Coya, Circuits, Bond Graphs, and Signal-Flow Diagrams: A Categorical Perspective, Ph.D. thesis, U. C. Riverside, 2018. (Blog article here.)

Text © 2020 John Baez
baez@math.removethis.ucr.andthis.edu

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