Network Theory
John Baez
January 7, 2014
Scientists use diagrams of networks in many different ways. To make
sense of this, I'm writing a series of
articles on network theory. You can navigate forwards and back
through these using the blue arrows. And 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!
Parts 2 to 24 are being made into a book:
I'm also expanding some parts into selfstanding papers:
You can also watch 4 lectures, an overview of network theory, here:
The blog articles

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 AndersonCraciunKurtz 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.
This was the official end of the series of posts on stochastic Petri nets and chemical reaction networks, but later we had some closely connected posts by Manoj Gopalkrishnan, which dug deeper into these subjects:
To understand ecosystems, ultimately will be to understand networks. 
B. C. Patten and M. Witkamp
Text © 2014 John Baez
Diagram on top by Nicolas Le Novere, illustrating
SBGN: Entity Relationship Language
baez@math.removethis.ucr.andthis.edu