Decomposition of admissible functions in weighted coupled cell networks

TL;DR

This paper introduces two decompositions of admissible functions in weighted coupled cell networks, clarifying their degrees of freedom and enabling the design of coupling components with finite or infinite coupling orders.

Contribution

It provides a low-level representation of oracle components by unwrapping them into their degrees of freedom, introducing the concept of coupling order and a limit approach for infinite orders.

Findings

Defined the concept of coupling order.

Developed two decompositions for admissible functions.

Enabled design of coupling components with infinite coupling orders.

Abstract

This work makes explicit the degrees of freedom involved in modeling the dynamics of a network, or some other first-order property of a network, such as a measurement function. In previous work, an admissible function in a network was constructed through the evaluation of what we called oracle components. These oracle components are defined through some minimal properties that they are expected to obey. This is a high-level description in the sense that it is not clear how one could design such an object. The goal is to obtain a low-level representation of these objects by unwrapping them into their degrees of freedom. To achieve this, we introduce two decompositions. The first one is the more intuitive one and allows us to define the important concept of coupling order. The second decomposition is built on top of the first one and is valid for the class of coupling components that have finite coupling order. Despite this requirement, we show that this is still a very useful tool for designing coupling components with infinite coupling orders, through a limit approach.