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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.23375 |
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Table of Contents:
- In fields ranging from business to systems biology, directed graphs with edges labeled by signs are used to model systems in a 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 monoid, whose elements represent `polarities' possibly more general than simply "positive" or "negative". We study three notions of morphism between graphs with labeled edges, each with its own distinctive application: to refine a simple graph into a complicated one, to transform a complicated graph into a simple one, and to find recurring patterns called "motifs". We construct three corresponding symmetric monoidal double categories of "open" graphs. We also study feedback loops using a generalization of the homology of a graph to homology with coefficients in a commutative monoid. In particular, we describe the emergence of new feedback loops when we compose open graphs using a variant of the Mayer-Vietoris exact sequence for homology with coefficients in a commutative monoid.