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Graphs With Polarities

John C. Baez, Adittya Chaudhuri

TL;DR

Three notions of morphism between graphs with labeled edges are studied: 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".

Abstract

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 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.

Graphs With Polarities

TL;DR

Three notions of morphism between graphs with labeled edges are studied: 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".

Abstract

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 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.

Paper Structure

This paper contains 10 sections, 26 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Set-labeled graphs
  3. Monoid-labeled graphs
  4. Motifs in monoid-labeled graphs
  5. Commutative monoid-labeled graphs
  6. Rig-labeled graphs
  7. Open labeled graphs
  8. Feedback loops and homology
  9. Emergent feedback loops
  10. Conclusion

Key Result

Lemma 2.1

Let $L$ be a set and $(G,\ell)$ an $L$-labeled graph. For any map of graphs $f \colon G' \to G$, there is a unique $L$-labeling of $G'$, called the pullback of $\ell$ along $f$ and denoted $f^\ast \ell$, such that $f$ is a map of $L$-labeled graphs from $(G',f^\ast \ell)$ to $(G,\ell)$. The pullbac

Theorems & Definitions (86)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • Example 3.1
  • Example 3.2
  • ...and 76 more