Motifs and Emergent Feedback in Labeled Graphs

TL;DR

This paper generalizes the concept of feedback in labeled directed graphs by introducing monoid-labeled edges, explores morphisms between such graphs, and uses homology theory to analyze emergent feedback loops.

Contribution

It introduces a monoid-based framework for analyzing feedback and motifs in labeled graphs, including new morphisms and homology methods for open graph composition.

Findings

Generalized feedback loops via monoid labels

Constructed symmetric monoidal double categories of open graphs

Applied homology to study emergent feedback in graph compositions

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 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.