Network Theory

John Baez

March 31, 2015

• Parts 1-26: Chemical reaction networks and Petri nets.
• Parts 27-: Electrical circuits and control theory.

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!

You can watch 4 lectures, an overview of network theory, here:

• Network theory: Oxford talks.

• Categorical foundations of network theory.

Parts 1 to 26 are about chemical reaction networks and Petri nets. These parts have been made into a book:

• John Baez and Jacob Biamonte, A Course on Quantum Techniques for Stochastic Mechanics.

I'm also expanding some parts into self-standing papers:

• John Baez and Brendan Fong, A Noether theorem for stochastic mechanics, J. Math. Phys. 54 (2013), 013301.
• John Baez, Quantum techniques for reaction networks, to appear in Natural Computing.
• John Baez and Brendan Fong, Quantum techniques for studying equilibrium in reaction networks, Journal of Complex Networks (2014).

Here are the blog articles for parts 1-26:

• 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.

• 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 and Jason Erbele, Categories in control, Theory and Applications of Categories 30 (2015), 836-881.
• Simon Wadsley and Nick Woods, PROPs for linear systems.
• John Baez and Brendan Fong, A compositional framework for passive linear networks.
• John Baez, Brendan Fong and Blake Pollard, A compositional framework for Markov processes.
• Brendan Fong, Decorated cospans.

Here are the blog articles:

• 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 beteween 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.

Text © 2015 John Baez
Diagram on top by Nicolas Le Novere, illustrating SBGN: Entity Relationship Language
baez@math.removethis.ucr.andthis.edu

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