John Baez

Transactions in Applied Category Theory 2025, June 2, 2025

Graphs with Polarities

In fields ranging from business to systems biology, directed graphs with edges labeled by signs are used to model systems in a very simple way: the nodes represent entities of some sort, and an edge indicates that one entity directly affects another either positively or negatively. Multiplying the signs along a directed path of edges lets us determine indirect positive or negative effects, and if the path is a loop we call this a positive or negative feedback loop. Here we generalize this to graphs with edges labeled by a commutative monoid, whose elements represent "polarities" possibly more general than simply "positive" or "negative". We define and study the "1st homology monoid" of a directed graph, different from the usual 1st homology group in that it only detects directed loops. This is joint work with Adittya Chaudhuri.

You can see my slides here.

For more, read these:

• Daniel H. Kim and Virginia Anderson, Systems Archetype Basics: From Story to Structure, Waltham, Massachusetts, Pegasus Communications Inc., 1998.
• John Sterman, Business Dynamics, McGraw-Hill, Inc., 2000.
• Oren Shoval and Uri Alon, SnapShot: network motifs, Cell 143 (2) (2010), 326–326.
• Rebekah Aduddell, James Fairbanks, Amit Kumar, Pablo Ocal, Evan Patterson and Brandon Shapiro, A compositional account of motifs, mechanisms, and dynamics in biochemical regulatory networks, Compositionality 6 (2) (2024).
• John Baez and Additya Chaudhuri, Graphs with polarities, forthcoming.

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