# Arrow Contraction and Expansion in Tropical Diagrams

## TL;DR

This paper introduces arrow contraction and expansion in tropical diagrams, relating these operations to rate regions and the entropic cone, providing new tools for analyzing probability space diagrams.

## Contribution

It defines arrow contraction and expansion in tropical diagrams and explores their relationship to rate regions and the entropic cone, advancing diagram modification techniques.

## Key findings

- Arrow contraction replaces a morphism with an isomorphism in tropical diagrams.
- Arrow expansion is the inverse of contraction, enabling diagram modifications.
- Connections between arrow operations and information-theoretic concepts like rate regions and entropic cones.

## Abstract

Arrow contraction applied to a tropical diagram of probability spaces is a modification of the diagram, replacing one of the morphisms by an isomorphims, while preserving other parts of the diagram. It is related to the rate regions introduced by Ahlswede and K\"orner. In a companion article we use arrow contraction to derive information about the shape of the entropic cone. Arrow expansion is the inverse operation to the arrow contraction.

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Source: https://tomesphere.com/paper/1905.05597