Commutative Monoid Formalism for Weighted Coupled Cell Networks and Invariant Synchrony Patterns

TL;DR

This paper develops a monoid-based framework for analyzing weighted coupled cell networks, extending invariant synchrony results from unweighted to weighted cases and improving algorithms for balanced partition refinement.

Contribution

It introduces a formal monoid-based approach to weighted networks, generalizes invariant synchrony results, and enhances the coarsest invariant refinement algorithm.

Findings

Invariant synchrony patterns hold for weighted networks.

The framework simplifies reasoning about multiedge networks.

Refinement algorithm has worst-case complexity of O(|C|^3).

Abstract

This paper presents a framework based on matrices of monoids for the study of coupled cell networks. We formally prove within the proposed framework, that the set of results about invariant synchrony patterns for unweighted networks also holds for the weighted case. Moreover, the approach described allows us to reason about any multiedge and multiedge-type network as if it was single edge and single-edge-type. Several examples illustrate the concepts described. Additionally, an improvement of the coarsest invariant refinement algorithm to find balanced partitions is presented that exhibits a worst-case complexity of $ \mathbf{O}(\vert\mathcal{C}\vert^3) $, where $\mathcal{C}$ denotes the set of cells.