# On Multi-source Networks: Enumeration, Rate Region Computation, and Hierarchy

## TL;DR

This paper develops a structural theory and computational tools to analyze the capacity regions of small hypergraph networks, creating a comprehensive database and hierarchy for network coding problems with up to five sources and edges.

## Contribution

It introduces a formal notion of network minimality, an algorithm for classifying network equivalences, and constructs a large database of rate regions for small networks, advancing understanding of network coding complexity.

## Key findings

- Created a database of rate regions for over 9 million networks with up to five sources and edges.
- Developed an algorithm to classify network coding problems into equivalence classes.
- Established a hierarchy of network coding problems using new combination and embedding theories.

## Abstract

Recent algorithmic developments have enabled computers to automatically determine and prove the capacity regions of small hypergraph networks under network coding. A structural theory relating network coding problems of different sizes is developed to make best use of this newfound computational capability. A formal notion of network minimality is developed which removes components of a network coding problem that are inessential to its core complexity. Equivalence between different network coding problems under relabeling is formalized via group actions, an algorithm which can directly list single representatives from each equivalence class of minimal networks up to a prescribed network size is presented. This algorithm, together with rate region software, is leveraged to create a database containing the rate regions for all minimal network coding problems with five or fewer sources and edges, a collection of 744119 equivalence classes representing more than 9 million networks. In order to best learn from this database, and to leverage it to infer rate regions and their characteristics of networks at scale, a hierarchy between different network coding problems is created with a new theory of combinations and embedding operators.