Noncommutative Network Models

TL;DR

This paper introduces a noncommutative network model framework that allows for modeling networks with bounded degree by generalizing Green's graph products to pointed categories, overcoming limitations of traditional commutative models.

Contribution

It constructs the free network model on a monoid, enabling the modeling of bounded degree networks, and generalizes Green's graph products to pointed categories.

Findings

Successfully models networks with bounded degree.

Generalizes Green's graph products to pointed categories.

Provides a new framework for noncommutative network modeling.

Abstract

Network models, which abstractly are given by lax symmetric monoidal functors, are used to construct operads for modeling and designing complex networks. Many common types of networks can be modeled with simple graphs with edges weighted by a monoid. A feature of the ordinary construction of network models is that it imposes commutativity relations between all edge components. Because of this, it cannot be used to model networks with bounded degree. In this paper, we construct the free network model on a given monoid, which can model networks with bounded degree. To do this, we generalize Green's graph products of groups to pointed categories which are finitely complete and cocomplete.