Network Models

TL;DR

This paper introduces 'network models' as a formal framework for encoding various methods of combining networks, establishing their connection to operads and providing tools for network design.

Contribution

It formalizes network models as lax symmetric monoidal functors and links them to operads, offering a new mathematical approach to network composition and design.

Findings

Network models encode different network combination methods.

Each network model induces an operad for network assembly.

Operads derived from network models can aid in network design.

Abstract

Networks can be combined in various ways, such as overlaying one on top of another or setting two side by side. We introduce "network models" to encode these ways of combining networks. Different network models describe different kinds of networks. We show that each network model gives rise to an operad, whose operations are ways of assembling a network of the given kind from smaller parts. Such operads, and their algebras, can serve as tools for designing networks. Technically, a network model is a lax symmetric monoidal functor from the free symmetric monoidal category on some set to $\mathbf{Cat}$, and the construction of the corresponding operad proceeds via a symmetric monoidal version of the Grothendieck construction.